Asymptotic properties of small data solutions of the Vlasov-Maxwell system in high dimensions
Léo Bigorgne
math.AP

    Extrapolating Expected Accuracies for Large Multi-Class Problems
     Charles Zheng
 charles.y.zheng@gmail.com 

       Rakesh Achanta 
 rakesha@stanford.edu 

       Yuval Benjamini 
 yuval.benjamini@mail.huji.ac.il
    December 30, 2023
=======================================================================================================================================================



We prove almost sharp decay estimates for the small data solutions and their derivatives of the Vlasov-Maxwell system in dimension n ≥ 4. The smallness assumption concerns only certain weighted L^1 or L^2 norms of the initial data. In particular, no compact support assumption is required on the Vlasov or the Maxwell fields. The main ingredients of the proof are vector field methods for both the kinetic and the wave equations, null properties of the Vlasov-Maxwell system to control high velocities and a new decay estimate for the velocity average of the solution of the relativistic massive transport equation.

We also consider the massless Vlasov-Maxwell system under a lower bound on the velocity support of the Vlasov field. As we prove in this paper, the velocity support of the Vlasov field needs to be initially bounded away from 0. We compensate the weaker decay estimate on the velocity average of the massless Vlasov field near the light cone by an extra null decomposition of the velocity vector.


    



§ INTRODUCTION


In this paper, we study the asymptotic properties of the small data solutions of the Vlasov-Maxwell system in dimensions n ≥ 4. For K species, this system is given by[During this article, we will use the Einstein summation convention. For instance, e^kJ(f_k)_ν=∑_k=1^K e^kJ(f_k)_ν. Roman indices goes from 1 to n and greek indices from 0 to n. Moreover, we raise and lower indices with respect to the Minkowski metric.]

    √(m_k^2+|v|^2)∂_t f_k+v^i ∂_i f_k + e_k v^μF_μ^ j∂_v^jf_k     =     0,  
    ∇^μ F_μν    =      e^kJ(f_k)_ν, 
    ∇^μ^*  F_μα_1 ... α_n-2    =     0 ,
 with initial data,

    f_k(0,.,.)     =     f_0k, 
     
        F(0,.)     =     F_0.
 

This is a classical model in plasma physics and we refer to <cit.> for an introduction to its analysis. Here,


  * m_k ∈_+ and e_k ∈^* are the mass and the charge of the particles of the species k ∈ 1, K. The function f_k(t,x,v) is their velocity distribution, which is a non-negative function.

  * The Maxwell field is described in geometric form by the 2-form F(t,x) and its Hodge dual ^*  F(t,x).

  * The (n+1)-current J(f_k)_ν in equation (2) is given by

    J(f_k)^ν(t,x)= ∫_v ∈^nv^ν/v^0_k f_k(t,x,v) dv, where v^0_k:=√(m_k^2+|v|^2).


  * The variable t will be taken in ℝ_+, x will be taken in ^n and for the species k, v will be taken either in ^n, if m_k ≠ 0, or in ^n ∖{0 }, if m_k=0.
  
In the 3 dimensional case, we can express the system in terms of the electric and the magnetic vector fields through the relations

    E^i=F_0iand B^i=-^*  F_0i

so that the Vlasov-Maxwell equations take the familiar form


      √(m_k^2+|v|^2) ∂_t f_k+v^i ∂_i f_k + e_k(E+v ×B) ·∇_v f_k = 0,     

     ∇·E = e^k J(f_k)^0,  ∂_t E^j = (∇×B)^j -e^kJ(f_k)^j,     

     ∇·B = 0,  ∂_t B = - ∇×E.    
 



 §.§ Global in time solutions for the Vlasov-Maxwell system


The global existence for classical solutions to the Vlasov-Maxwell system is still an open problem. They are known to be global in certain particular cases such as under a translation symmetry hypothesis on the initial data in one of the space variables. The pioneer works on this two and one half dimensional case originated from Glassey-Schaeffer in <cit.> and required a compact support assumption in v. The result obtained recently by Luk-Strain allows data with non-compact velocity support <cit.>. The solutions to the Vlasov-Maxwell system also appear to be global when they arise from pertubation of spherically symmetric initial data[Recall that for spherically symmetric solutions, the Vlasov-Maxwell system reduces to the relativistic Vlasov-Poisson system.] (see <cit.>). 

For the general case, several continuation criteria are known. The first one, obtained by Glassey and Strauss in <cit.> (see also <cit.> and <cit.> for alternative proofs), expresses that C^1 solutions to the Vlasov-Maxwell system arising from compactly initial data do not develop singularities as long as the velocity supports of the particle densities f_k remain bounded. An improved continuation criteria requires the finiteness of

    √(1+|v|^2)^θ f_k _L^∞( [0, T^*[, L^q_x L^1_v ),

for a certain q and θ, in order to extend the solution beyond T^* >0. Let us mention <cit.> for the cases 6 ≤ q ≤∞ and θ > 4/q, <cit.> for q = + ∞ and θ = 0 as well as <cit.> for q=6 and θ=0. Earlier results of Glassey-Strauss required the boundedness of (<ref>) for q= ∞ and θ =1 and cover data with non-compact support in v (see <cit.>). Recently, Luk-Strain removed in <cit.> all compact support assumptions and extend the continuation criteria (<ref>) for 6 ≤ q ≤ + ∞ and θ > 2/q.



 §.§ Previous work on small data solutions of the Vlasov-Maxwell system


Global existence for small data in dimension 3 was first established by Glassey-Strauss in <cit.> under a compact support assumption (in x and in v). In particular they proved ∫_v f dv ≲1/(1+t)^3, coinciding with the linear decay, but they did not control ∂_μ_1...∂_μ_p∫_v f dv. They also proved decay estimates for the electromagnetic field and its derivatives of first order, but not for the derivatives of higher order. A similar result was proved in <cit.> for the nearly neutral case, i.e. ∑_k e_k m_k^3f_0k(x,m_kv) has to be small and not the individual particle densities. The result established by Schaeffer in <cit.> allowed particles with high velocity but still requires the data to be compactly supported in space[Note also that when the Vlasov field is not compactly supported (in v), the decay estimate obtained in <cit.> for its velocity average contains a loss.]. Finally, let us also mention the earlier result of Bardos-Degond for the more classical Vlasov-Poisson system <cit.>. Under a smallness assumption, they established that the solution of the system is global in time and proved that ∫_v f dv ≲1/(1+t)^3 but they did not obtain informations on the derivatives of f. They also proved decay estimates for the electric field up to second order.






 §.§ Optimal gradient estimates for Vlasov systems



Due to the linearity of the Maxwell equations and the elliptic nature of the Poisson equation or a nonresonant phenomenon[According to <cit.>, the velocity averages of the solutions of a system coupling a linear wave equation with a transport equation are such that the velocity averages are more regular than expected if the speed of propagation of the wave equation is strictly larger than the speed of the particles governed by the transport equation.], the previous results are established without essentially commuting the Vlasov equation and controlling higher derivatives of the solutions. For the Vlasov-Poisson system with small data, the sharp time decay estimates |∫_v ∂_μ_1...∂_μ_p f dv| ≲1/(1+t)^3+p was proved[A similar result is established in <cit.>, using the same techniques, for the Vlasov-Yukawa system in dimension 2.] in <cit.>. A similar result was obtained in <cit.> using a vector field method which led to global bounds for the solution and optimal space and time decay rates for the velocity averages. In the same spirit, optimal decay estimates was proved for the derivatives of the solutions of the Vlasov-Nordström system in <cit.> and <cit.>. The stability of the Minkowski space for the Einstein-Vlasov system was recently, and independently, proved in <cit.> and <cit.>. For both of them, vector field methods was a crucial point in the proof and led in particular to almost optimal decay rates for the derivatives of the solutions.

The goal of this paper is to prove almost optimal decay for the small data solutions and their derivatives of the Vlasov-Maxwell system in dimension n ≥ 4 without any compact support assumption on the initial data.


 §.§ The vector field method for Vlasov fields


In this paper, we will use vector field methods to derive decay estimates for both the electromagnetic field and the Vlasov field. The vector field method was originally developped by Klainerman in <cit.> to study wave equations and was adapted to cover the Maxwell equations (and the spin 2 equations) in 3d in <cit.>. More recently, the method was extended for the free relativistic transport equation in <cit.>.

As in <cit.>, these methods are based on energy estimates, commutation vector fields and weighted Sobolev inequalities. For the transport operator v^μ∂_μ, the set  of commutation vector fields used in <cit.> is composed of the scaling vector field S=x^μ∂_μ and the complete lifts of the generators of the Poincaré group, that is to say the transalations

    ∂_μ,  0 ≤μ≤ n,

the complete lifts of the rotations

    Ω_ij=x^i∂_j-x^j ∂_i+v^i ∂_v^j-v^j ∂_v^i,  1 ≤ i < j ≤ n

and the complete lifts of the Lorentz boosts

    Ω_0k=t∂_k+x^k ∂_t+v^0 ∂_v^k,  1 ≤ k ≤ n.

In <cit.>, vector field methods are applied to derive the behavior of the solutions to the Vlasov-Nordström system in the future the hyperboloid[The use of a hyperboloidal foliation in order to establish decay estimates was introduced in <cit.> in the context of the Klein-Gordon equation.] t^2-r^2=1. However, without any compact support assumption, one cannot reduce the study of a solution in the future of a t=constant slice to its study in the future of a hyperboloidal slice (see for instance <cit.>, Appendix A, for more details). In order to remove all compact support assumption on the data, one of the goal of this paper is to start from a t=0 slice and adapt the vector field method for transport equations to the more common foliation ( {t }×^n )_t ≥ 0. Note that <cit.> (respectively <cit.>) use slight modifications of the commutation vector fields[The modified vector fields are built in order to compensate the worst source terms in the commuted transport equations.] of the operator v^μ∂_μ in order to study the small data solutions of the Vlasov-Nordström (respectively Einstein-Vlasov) system in 3d. They also use a hyperboloidal foliation and restrict the study of the solutions to the future of a hyperboloid.



 §.§ The Lorenz gauge


Recall that a 1-form A is a potential of the electromagnetic field F if F=dA (or F_μν = ∂_μ A_ν-∂_ν A_μ). If moreover

    ∂^μ A_μ =0,

we say that A satisfies the Lorenz gauge condition. As the energy momentum tensor of F is not traceless in dimension n ≥ 4 and the Morawetz vector field K_0:=(t^2+r^2)∂_t+2tr∂_r is merely a conformal Killing vector field, we encounter the same difficulty in using it as for the wave equation in 3d (see <cit.>, Chapter II for more details). To circumvent this difficulty, we will consider in this paper the Vlasov-Maxwell system in the Lorenz gauge. A_μ will then satisfy the equation

    □ A_μ = e^k∫_v v_μ/v^0 f_k dv.

We also make fundamental use of the Lorenz gauge to establish the optimal decay rate on the component α of the null decomposition of the electromagnetic field, as the method used in <cit.> cannot be reproduced in dimension n ≠ 3.



 §.§ Results for the massive Vlasov-Maxwell system


We will consider weighted L^2 norms to control A and F such as[For a tensor G and a multi-index β=(β_1,...,β_p),

    ℒ_Z^βG= ℒ_Z^β_1...ℒ_Z^β_pG,

while 
    Zℒ_Z^β(G)_μ=Z[ℒ_Z^β(G)_μ].
]

    ℰ_N[A] = ∑_μ=0^n ∑_|β| ≤ 1∑_|γ| ≤ NZ^βℒ_Z^γ(A)_μ^2_L^2(^n)
 as well as weighted L^1 norms for the Vlasov field, such as

    𝔼^q_N[g](t)=∑_|β| ≤ N∫__x^n∫_^n_v  (v^0)^q |Z^β g| dv dx+∫_C_u(t)∫_^n_v  (v^0)^q-1v^L |Z^β g | dv d C_u(t),
 
where C_u(t):= {(s,y)  ∈ℝ_+ ×ℝ^n  /  s ≤ t,  s-|y|=u }. For the Vlasov field, we also use extra norms with the additional weights v^μ, x^iv^j-x^jv^i or tv^i-x^iv^0. See Definitions <ref>, <ref> and Section <ref> for an introduction to the other norms and the weights.

We are now ready to present our main result for the massive Vlasov-Maxwell system (for a detailled version, see Theorem <ref>).



Let n ≥ 4, K ≥ 2, and N ≥5/2n+1. Let (f_0,F_0) be an initial data set for the massive Vlasov-Maxwell system with K species. Let (f,F) be the unique classical solution to the system and let A be a potential in the Lorenz gauge. There exists ϵ >0 such that[A smallness condition on F, which implies ℰ_N[A](0) ≤ϵ, is given in Proposition <ref>.], if 


    ℰ_N[A](0) + ℰ_N[F](0) +∑_k=1^K 𝔼^2_N+n,1[f_k](0) ≤ϵ,

then (f,F) exists globally in time and verifies the following estimates. 


  * Energy bounds for A, F and f_k such as  𝔼^2_N[f_k] ≲ϵ on _+.

  * Sharp pointwise decay estimates on the null decomposition of ℒ_Z^β(F) and for the velocity average of f_k and its derivatives. For instance, 
    ∀ |β| ≤ N-3n+2/2,  (t,x) ∈ℝ_+ ×ℝ^n, ∫_v ∈ℝ^n   | Z^β f_k | dv ≲ϵ/(1+t+|x|)^n.





Since we suppose that the initial energy on F is finite, we are necessarily in the neutral case[In other words, the total charge verifies

    e^k ∫_x ∈^n∫_v ∈^n f_0k dv dx =0.
] when the dimension is n = 4.
On the other hand, when the total charge is non zero, Gauss's law implies that the energy ℰ_N[F] is infinite. We refer to <cit.> and <cit.> for a study of the Maxwell-Klein-Gordon system with a non-zero total charge.




Thanks to the vector field method and in view of the definition of our norms, we do not need any compact support assumption on the initial data. We also automatically obtain improved decay rates for the derivatives of both the electromagnetic field and the velocity averages of the particle densities. For instance, for all |β| ≤ N- 3n+2/2 and (t,x) ∈ℝ_+ ×ℝ^n,

    | ∂^β_t,x∫_v ∈ℝ^n    f_k  dv | ≲ϵ/(1+t+|x|)^n(1+|t-|x||)^|β|
 
and (see Section <ref>), assuming more decay on the initial data,

    | ∂^β_t,x∫_v ∈ℝ^n    f_k  dv | ≲ϵ/(1+t+|x|)^n+|β|.





Notice that in dimension n=4, it is sufficient for Z^βf_k to initially decay faster than (1+|v|)^-6-δ(1+|x|)^-5-δ, with δ >0, for our theorem to apply. In <cit.>, which concerns the 3d case, the main result requires the initial particle densities to be compactly supported in x and to decay faster than (1+|v|)^-q, with q > 60 + 12√(17).





 §.§ Results for the massless Vlasov-Maxwell system


We now present an elusive version of our main result for massless particles (we refer to Theorem <ref> for more details).


Let n ≥ 4, K ≥ 2, N ≥ 6n+3 and R>0. Let (f_0,F_0) be an initial data set for the massless Vlasov-Maxwell system with K species, (f,F) be the unique classical solution to the system and A be a potential in the Lorenz gauge. There exists ϵ >0 and R>0 such that, if 

    ℰ_N[A](0) + ℰ_N[F](0) + ∑_k=1^K 𝔼^0_N+n,1[f_k](0) ≤ϵ,


    ∀ 1 ≤ k ≤ K, supp(f_0k) ⊂{(x,v) ∈ℝ^n_x ×ℝ^n_v ∖{ 0 } /  |v| ≥ R } ,

then (f,F) exists globally in time and verifies the following properties.


  * f_k(.,.,v) vanishes for all |v| ≤R/2.

  * Energy bounds are propagated for F and f_k. For instance, if n=4, ℰ_N-8[F](t) ≲ϵ (1+t) for all t ∈ [0,T].

  * Pointwise decay estimates on the null decomposition of ℒ_Z^β(F) and for the velocity average of f and its derivatives. For instance,
 
    ∀ |β| ≤ N-2n,  (t,x) ∈ℝ_+ ×ℝ^n, ∫_v ∈ℝ^n ∖{0 } | Z^β f_k | dv ≲ϵ/τ_+^n-1τ_-.




The hypotheses on the velocity supports of the particle densities appear to be necessary (see Section <ref>).


We are not able to obtain optimal decay estimates for the electromagnetic field in dimension n=4 with our reasoning since the velocity average of the Vlasov field has a weaker decay rate near the light cone when the mass is zero (this is related to the estimate (<ref>) mentionned below, which only applies to massive particles).




 §.§ The main difficulties and ingredients of our proof



  §.§.§ High velocities and null properties of the system

As we use vector field methods, we are brought to commute the equations and prove global bounds on the solutions through energy estimates. After commuting the Vlasov equation once, we are led to estimate terms that could be written schematically as 
    ∫_0^t ∫_x ∫_v |vℒ_Z(F)∂_v f| dv dx ds.
 Unfortunately, ∂_v^i, for 1 ≤ i ≤ n, is not part of the commutation vector fields for the transport equation. We rewrite them in terms of the elements of  as

    ∂_v^i f_k=1/v^0(Ω_0i f_k-t∂_i f_k-x^i ∂_t f_k),

so that ∂_v f essentially behaves like t∂_μf, which is consistent with the behavior of solutions to the free transport equation. This leads us to estimate

    ∫_0^t ∫_x ∫_v (s+|x|)|ℒ_Z(F)∂ f| dv dx ds.
 
As a solution to a wave equation, ℒ_Z(F) only decays near the light cone as 1/(1+t+|x|)^n-1/2 and we cannot prove by a naive estimate that, in dimensions n ≤ 5, (<ref>) is uniformly bounded. However, if f is initially compactly supported, one can expect (for, say, sufficiently small data) the characteristics of the transport equation to have velocities bounded away from 1, and thus the Vlasov field support (in x) to be ultimately remote from the light cone. Now, assuming enough initial decay on the Maxwell field, one can prove that 
    |ℒ_Z(F)| ≲ (1+s+|x|)^-n-1/2(1+|s-|x||)^-3/2

which, combined with the support properties of f, leads to

    ∫_x ∫_v (s+|x|)|ℒ_Z(F) ∂ f| dv dx ≲  (1+s)^-n/2
 and (<ref>) is then uniformly bounded in dimensions[Note that our proof would lead to a √(t)-loss in dimension 3 which is not sufficient to prove the uniform boundedness of (<ref>).] n ≥ 4.

In our work, we do not make any compact support assumption. Instead, we make crucial use of null properties of the Vlasov-Mawxell system[The null structure of the Vlasov-Nordström system is also a main ingredient of the proof of <cit.> for the dimension n=4. ] to deal with the high velocities. More precisely, certain null components of the velocity vector v, the derivatives of the electromagnetic field (as ℒ_Z (F)) or the vector (0, ∂_v^1 f,..., ∂_v^n f) behave better than others and the structure of the system is such that there is no product involving only terms with a bad behavior. Taking advantage of the null structure allows us either


  * to transform a t-r decay in a t+r one. For instance,

    |ρ(ℒ_Z(F))| = | x^i/rℒ_Z(F)_i0| ≲ (1+s+|x|)^-n+1/2(1+|s-|x||)^-1/2.


  * To transform a t+r loss in a t-r loss using x^i/r∂_v^i f ∼ (t-r) ∂ f.

  * Or to exploit the t-r decay. For instance, we will control

    ∫_C_u(t)∫_v ∈^nv^L/v^0 |∂ f| dv dC_u(t) ≤𝔼[f](t),

so that, by the change of variables (u,u)=(s+r,s-r),

    ∫_0^t ∫_x 1/(1+|s-|x||)^2∫_v v^L/v^0 |∂ f| dv dx ds ≤∫_u=-∞^t 𝔼[f](t)/(1+|u|)^2 du.





  §.§.§ Improved decay estimates


∙ For massive particles. In <cit.>, two Klainerman-Sobolev inequalities for velocity averages of Vlasov fields were obtained. They imply in particular that, for g a solution to v^μ∂_μ (g)=0,

    ∀ (t,x) ∈_+ ×^n,       
    ∫_^n_v |g|(t,x,v) dv    ≲   ∑_Z^β∈^k
     |β| ≤ 3 ∫_^n_y∫_^n_v  |Z^β g |(0,y,v) dv dy/(1+t+|x|)^n-1(1+|t-|x||),   
    ∀ (t,x) ∈_+ ×^n,        t^2-|x|^2 ≥ 1, 
    ∫_^n_v |g|(t,x,v) dv    ≲   1/(1+t)^n∑_Z^β∈^k
     |β| ≤ 3 ∫_^n_y∫_^n_v  |Z^β g |(√(1+|y|^2),y,v) dvdy/√(1+|y|^2).

The first one has the advantage to be based on the foliation ({t }×^n)_t ≥ 0 but provides a weak estimate near the light cone. The second one gives a stronger decay rate near the light cone but is based on a hyperboloidal foliation. In this paper, in order to remove all compact support assumption on the data and start from a t=0 slice, we will prove and use a refined version of the Klainerman-Sobolev inequalities of <cit.>. Our estimate will imply that, for g a solution to v^μ∂_μ (g)=0 and for all (t,x) ∈_+ ×^n,


    ∫_^n_v |g|(t,x,v) dv ≲∑_Z^β∈^k
     |β| ≤ 3 ∫_^n_y∫_^n_v |v^0|^2 (1+|y|)   |Z^β g |(0,y,v) dv dy/(1+t+|x|)^n.

Compared to the Klainerman-Sobolev inequalities proved in <cit.>, (<ref>) cumulates the advantages of giving a strong decay in the whole spacetime and being adapted to the foliation ({t }×^n ) _t ≥ 0. On the other hand, our estimate is not a pure Sobolev inequality (we used the transport equation satisfied by g to establish it).


In the exterior of the light cone (where t ≤ |x|), one can in fact obtain arbitrary decay provided we consider additional decay on the initial data (see Section <ref>).

∙ For massless particles. Unfortunately, (<ref>) does not apply to massless particles. Instead, we use weights z ∈𝐤_0 preserved by the relativistic transport operator |v|∂_t+v^i ∂_i in order to gain additional decay. More precisely, in the same spirit as the derivative ∂_t+∂_r (respectively ∂_t-∂_r) provides an extra decay in t+r (respectively t-r) for, say, a solution to □ u =0, one has

    v^0-x^i/rv_i ≤|v|/1+t+r∑_z ∈𝐤_0 |z| and v^0+x^i/rv_i ≤|v|/1+|t-r|∑_z ∈𝐤_0 |z|.




  §.§.§ The problem of the small velocities


For the massless Vlasov-Maxwell system, another problem arises from the small velocities since v^0=|v| is not bounded by below. The velocity part V of the characteristics of

    ∂_t f+v^i/|v|∂_i f+v^μ/|v|F_μ^j ∂_v^j f=0

solves the ordinary differential equation

    V̇^̇j̇=V^μ/|V|F_μ^j,

which can lead to V=0 in finite time[Note that this difficulty does not appear in the Einstein-Vlasov system. Indeed, as in <cit.>, the Vlasov equation can be written, for a metric g, as

    v_μg^μν∂_νf-1/2v_μv_ν∂_i g^μν∂_v_if=0,

so these situations should be compared to the two ordinary differential equations

    ẏ=1 andẏ=y.
]. More precisely, we prove in Section <ref> that there exists smooth initial data such that the particle densities f_k do not vanish for small velocities and for which the massless Vlasov-Maxwell system does not even admit a local classical solution.

An important step of the proof of Theorem <ref> then consists in proving that the velocity supports of f_k remain bounded by below. For this, we make crucial use of the smallness of assumption on the electromagnetic field as well as its strong decay rate[In dimension n ≥ 4, F_L^∞_x≲ (1+t)^-3/2 is time integrable.]




  §.§.§ The perspective of the three dimensional case


Nevertheless, even in making use of the null properties of the system, our proof does not work in dimension 3. One way to treat the 3d massive case would be to use modified vector fields in the spirit of <cit.> for the Vlasov-Poisson system and <cit.> for the Vlasov-Nordström system. This method led to the proof of the stability of the Minkowski space for the Einstein-Vlasov system (cf <cit.>, <cit.>), providing sharp estimates on both the Vlasov field and the metric.



 §.§ Structure of the paper


In section <ref> we introduce the notations used in the paper, the commutation vector fields and the Vlasov-Maxwell system. In Section <ref> we establish various energy estimates for solutions to the relativistic transport equation or the Maxwell equations. Section <ref> contains an integral estimate, some ways to estimate the v derivatives and the tools to prove pointwise decay estimates for the electromagnetic field. Section <ref> is devoted to our new decay estimate for the solution of a massive relativistic transport equation. In Section <ref> (respectively <ref>), we prove the global existence and the asymptotic properties of the small data solutions of the massive (respectively massless) Vlasov-Maxwell system, which is Theorem <ref> (respectively Theorem <ref>). In Section <ref>, we prove that there exists smooth initial data which do not vanish for small velocities and for which the massless Vlasov-Maxwell system do not admit a local classical solution.




 §.§ Acknowledgements


This work forms part of my Ph.D. thesis and I obviously would like to thank my advisor Jacques Smulevici for his guidance.



§ NOTATIONS AND PRELIMINARIES




 §.§ Basic notations


Throughout this article we work on the n+1 dimensional Minkowski spacetime (ℝ^n+1, η) and we consider two types of coordinates on it. The Cartesian coordinates (t,x), in which η=diag(-1,1,...,1), and null coordinates which are defined by


    u=t-r, u=t+r,

and spherical variables (B,C,D,...) (always denoted by capital Latin letters[The letter A will be reserved for the potential of the electromagnetic.]) which are spherical coordinates on the (n-1)-dimensional spheres t, r=constant. These coordinates are defined globally on ^n+1 apart from the usual degeneration of spherical coordinates and at r=0. The null derivatives L and L are defined as 


    L=∂_t+∂_r, L=∂_t-∂_r,

and we designate by (e_B,e_C,e_D,...) an orthonormal basis on the spheres (t,r)=constant. We will use the weights


    τ_+^2=1+u^2 andτ_-^2=1+u^2.

For a 2-form F_μν, its Hodge dual is denoted by ^*  F, with 
    ^*  F_λ_1 ... λ_n-1 = 1/2F^μνε_μνλ_1 ... λ_n-1,

where ε_λ_1 ... λ_n+1 is the Levi-Civita symbol. As in <cit.>, we consider its null-decomposition given by 
    α_B (F) = F_BL, α_B (F) = F_B L, ρ(F) = 1/2 F_LL, σ_BD (F) = F_BD.

We also associate to a 2-form F its energy-momentum tensor


    T[F]_μν = F_μβF_ν^β- 1/4η_μν F_ρσ F^ρσ.

We use Greek letters to denote spacetime indices and Latin letters for space indices.
The velociy vector (v^β)_0 ≤β≤ n is parametrized by (v^i)_1 ≤ i ≤ n and v^0=√(m^2+|v|^2). When we study massive particles, we often take m=1 for simplicty so that v^0=√(1+|v|^2). On the other hand, for massless particles v^0=|v|.

 We designate the null components of the velocity vector (v^β)_0 ≤β≤ n by (v^L,v^L,v^B,...), i.e.

    v=v^L L+v^LL+v^Be_B.

In particular,


    v^L=v^0+v^r/2and v^L=v^0-v^r/2.

We now introduce several subsets of ℝ_+ ×ℝ^n depending on t ∈ℝ_+ or u ∈ℝ. Let Σ_t, C_u(t) and V_u(t) be defined as


    Σ_t := {t}×ℝ^n,  C_u(t):= {(s,y)  ∈ℝ_+ ×ℝ^n /  s ≤ t,  s-|y|=u }


and


    V_u(t) := { (s,y) ∈ℝ_+ ×ℝ^n /  s ≤ t,  s-|y| ≤ u }.

The volum form on C_u(t) is given by dC_u(t)=√(2)r^n-1dud 𝕊^n-1, where d 𝕊^n-1 is the standard volume form on the n-1 dimensional unit sphere.




[-Straight Barb[angle'=60,scale=3.5]] (0,-0.3)–(0,5);
[color=gray!35] (1,0)–(4,3)–(9.8,3)–(9.8,0)–(1,0);
[align=center,font=, yshift=-2em] (title) 
    at (current bounding box.south)
    The sets Σ_t, C_u(t) and V_u(t);
(0,3)–(9.8,3) node[scale=1.5,right]Σ_t;
[-Straight Barb[angle'=60,scale=3.5]] (0,0)–(9.8,0) node[scale=1.5,right]Σ_0;
[densely dashed, blue] (1,0)–(4,3) node[scale=1.5,left, midway] C_u(t);
(5.5,1.5) node[ color=black!100, scale=1.5] V_u(t); 
(0,-0.5) node[scale=1.5]r=0;
(-0.5,4.7) node[scale=1.5]t;
(9.5,-0.5) node[scale=1.5]r;   


We will use the notation Q_1 ≲ Q_2 for an inequality of the form Q_1 ≤ C Q_2, where C>0 is a positive constant independent of the solutions but which could depend on N ∈ℕ, the maximal order of commutation, or fixed parameters. Finally we will raise and lower indices with respect to the Minkowski metric η. For instance, F_μ^j = η^j ν F_μν so that F_μ^j =  F_μ j for all 1 ≤ j ≤ n.



 §.§ The relativistic transport operator


For m>0, we use the notation T_m to refer to the operator defined, for all v ∈ℝ^n, by 
    T_m=v^0 ∂_t+v^i ∂_i,
 with v^0=√(m^2+|v|^2). 

For the massless case (m=0), the relativistic transport operator T_0 is only defined for all v ∈ℝ^n ∖{0 } and we have 
    T_m=v^0 ∂_t+v^i ∂_i,
 with v^0=|v|.

To simplify the notation, we will most of the time take either m=1 or m=0 and we will only use T_1 and T_0.




 §.§ Vector fields




  §.§.§ The conformal isometries and their complete lifts


Let us consider the set 𝕂 composed by the generators of the isometries group of Minkowski spacetime and by the scaling vector field. 𝕂 contains


     the translations  ∂_μ,  0 ≤μ≤n,     

     the rotations  Ω_ij=x^i∂_j-x^j ∂_i,  1 ≤i < j ≤n,     

     the hyperbolic rotations  Ω_0k=t∂_k+x^k ∂_t,  1 ≤k ≤n, 

     the scaling  S=x^μ∂_μ.    

In this paper, we will denote ∂_x^i, for 1 ≤ i ≤ n, by ∂_i and sometimes ∂_t by ∂_0.

Sometimes we will only use the Poincaré group ℙ := 𝕂∖{S } or the set of the generators of the rotation group, 𝕆 (composed of the Ω_ij). These vector fields will be used as commutators whereas ∂_t, S and the vector field K_0, defined by

    K_0=K_0+∂_t=1/2τ_-^2 L+1/2τ_+^2L,

will be used as multipliers as in <cit.>.

These vector fields are well known to commute with the wave operator, i.e. if a smooth function u satisfies □ u =0, then, 
    ∀ Z ∈𝕂, □ Z u =0.
 We will use them to commute the Maxwell equations. However, as in <cit.>, we use another set of vector fields to study the Vlasov equation. For this, we use the complete lift of a vector field, a classical operation in differential geometry (see <cit.>, Appendix C for more details). For us, the following definition in coordinates will be sufficient.



Let Γ be a vector field of the form Γ^β∂_β. Then, the complete lift Γ of Γ is defined by

    Γ=Γ^β∂_β+v^γ∂Γ^i/∂ x^γ∂_v^i.





We then consider ℙ the set of the complete lifts of ℙ given by 

    ℙ= {Z /  Z ∈ℙ}.

The last set of vector fields required is the following 

    ℙ_0=ℙ∪{ S }.


We can list the complete lifts that we will manipulate.


For 0 ≤μ≤ n,

    ∂_μ=∂_μ.

For 1 ≤ i < j ≤ n,

    Ω_ij = x^i ∂_j -x^j ∂_i +v^i ∂_v^j-v^j ∂_v^i.

Finally, for 1 ≤ k ≤ n,

    Ω_0k=t ∂_k + x^k ∂_t+v^0 ∂_v^k.





The following lemma is used in <cit.> to prove a Klainerman-Sobolev inequality.


Let f : [0,T[ ×ℝ^n_x × P →ℝ, with P=ℝ^n_v or P=ℝ^n_v ∖{0 }, be a sufficiently regular function. Almost everywhere, we have

    ∀ Z ∈ℙ, |Z( ∫_v ∈ P  |f| dv ) | ≲∫_v ∈ P  |Zf| dv +∫_v ∈ P  |f| dv,


    | S( ∫_v ∈ P  |f| dv ) | ≲∫_v ∈ P  |Sf| dv.



Similar estimates exist for ∫_v ∈ℝ^n (v^0)^k |f| dv. For instance,

    | S( ∫_v ∈ P  v^0|f| dv ) | ≲∫_v ∈ P  v^0|Sf| dv.







When Z ∈ℙ is not a Lorentz boost, we have

    |Z( ∫_v ∈ P  |f| dv ) | ≲∫_v ∈ P  |Zf| dv .






We consider an ordering on each of the following sets of vector fields : 𝕆, ℙ, 𝕂, ℙ and ℙ_0. For simplicity, we introduce 𝕃 which represents one of those sets. We can suppose that

    𝕃= { L^i /  1 ≤ i ≤ |𝕃| }.

Let β∈{1, ..., |𝕃| }^r, with r ∈ℕ^*. Then we will denote the differential operator Z^β_1...Z^β_r by Z^β. For a vector field Y, we will denote by ℒ_Y the Lie derivative with respect to Y and if Z^γ∈𝕂^q, we will write ℒ_Z^γ for ℒ_Z^γ_1...ℒ_Z^γ_q. We can suppose that the orderings are such that if 
    ℙ={Z^i /  1 ≤ i ≤ |ℙ| },
 then

    ℙ = {Z^i / 1 ≤ i ≤ |ℙ| }and = {Z^i / 1 ≤ i ≤ |ℙ|+1 }, withZ^| ℙ |+1=S.

Note that even if the scaling is not a complete lift, we will for simplicity denote any vector field of  by Z.

We now introduce some pointwise norms.


Let U be a smooth p-covariant tensor field defined in ^n or in ^1+n. For k ∈ℕ, the pointwise norm of U with respect to 𝕆, of order k, is defined by

    | U |_𝕆,k=( ∑_|β| ≤ k |ℒ_Z^βU|^2 )^1/2,

with Z^β∈𝕆^|β| and where

    |ℒ_Z^βU|^2 = ∑_λ_1, ... , λ_p |ℒ_Z^β(U)_λ_1 ... λ_p|^2,

with ℒ_Z^β(U)_λ_1 ... λ_p the Cartesian components of ℒ_Z^β(U).





  §.§.§ Commutation properties


We have the following commutation relations :



Let 𝕃 be either 𝕆, ℙ, 𝕂, ℙ or ℙ_0. Then


    ∀  L, L' ∈𝕃,  ∃ C_LL'Γ∈ℝ,  such that  [L,L']=∑_Γ∈𝕃 C_LL'ΓΓ.



The commutation relations between the vector fields of ℙ_0 and the massive transport operator T_1 (or the massless relativistic transport operator) are similar to those between the vector fields of 𝕂 and the wave operator. 


We have, for m ∈{0,1 },


    ∀Z∈ℙ,  [T_m,Z]=0 and [T_m,S]=T_m.





This follows easily from Lemma <ref> and the definition of the relativistic transport operator.





 §.§ Weights preserved by the flow


We define, as in <cit.>, the two following sets of weights 

      𝐤_1= {v^μ/v^0  /  0 ≤μ≤n } ∪{ x^μv^ν/v^0-x^νv^μ/v^0  /  μ≠ν},     

     𝐤_0= 𝐤_1 ∪{ x^μv_μ/v^0 }.    



These weights are solutions to the free transport equation, i.e.




    ∀ z ∈𝐤_0,  T_0(z)=0,

and

    ∀ z ∈𝐤_1,  T_1(z)=0.


Thus, if f is a regular function and if z ∈𝐤_1, then T_1(zf)=zT_1(f). 

Moreover, these weights have also good interactions with the vector fields of ℙ_0.



If Z∈ℙ_0, m ∈{0,1 } and z ∈𝐤_m, then either 
    Z(v^0z)=0 orZ(v^0z) ∈ v^0𝐤_m.
 
This leads to

    ∀Z∈,  z ∈𝐤_m,  |Z(|z|)| ≤∑_z ∈𝐤_m |z|.







Consider for instance Ω_01 and x^1v^2-x^2v^1 or x^2v^3-x^3v^2. We have


    Ω_01(x^1v^2-x^2v^1)=tv^2-x^2v^0

as well as

    Ω_01(x^2v^3-x^3v^2)=0.

All the other cases are similar.


The next proposition shows how these weights can be used to provide us extra decay (at least in the massless case). 



Denoting x^μv_μ by s and x^νv^μ-x^μ v^ν by z_μν, we have

    2(t-r)v^L= -x^i/rz_0i-s,

and

    2(t+r)v^L= x^i/rz_0i-s.

We also have

    |v^B|/v^0≲1/τ_+∑_z ∈𝐤_1 |z|,  |v^B| ≲√(v^Lv^L)andm^2/4v^0≤ v^L.




This result should be compared with the identities 

    (t-r)L=S-x^i/rΩ_0i,


    (t+r)L=S+x^i/rΩ_0i,

and

    re_B=∑_1 ≤ i < j ≤ n C^i,j_B Ω_ij,

where C^i,j_B are bounded functions on the sphere.





Let us start by the first two equations. On the one hand, 

    (t^2-r^2)v^0=-x^iz_0i-ts.

On the other hand,

    (t^2-r^2)v^r=-t x^i/rz_0i-rs.

It only remains to take the sum and the difference of these two equations. For the third one, use |v^B| ≤ v^0 and that rv^B=C_B^i,j z_ij, which implies

    |v^B|    ≲   v^0/r∑_1 ≤ i < j ≤ n |z_ij| 
        =    v^0/tr∑_1 ≤ i < j ≤ n| x^i( v^j/v^0t-x^j+x^j )  -  x^j( v^i/v^0t-x^i+x^i ) | ≲v^0/t∑_q=1^n |z_0q|.

The fourth inequality ensues from rv^B=C_B^i,j z_ij and

    4r^2v^Lv^L    =     m^2r^2+r^2 |v|^2-|x^i|^2|v_i|^2-2∑_1 ≤ k < l ≤ nx^kx^lv^kv^l 
        =     m^2r^2+∑_1 ≤ k < l ≤ n |z_kl|^2,

since v^0= √(m^2+|v|^2). Finally, using the Cauchy-Schwarz inequality,

    2v^L=v^0-x^i/rv_i ≥m^2/v^0+|v|≥m^2/2v^0.




As for the sets of vector fields, we consider an ordering on 𝐤_0 with x^μv_μ/v^0 being the last weight. It then gives an ordering on 𝐤_1 too. If 𝐤_0= { z^i /  1 ≤ i ≤ |𝐤_0| } and β∈{ 1, ..., |𝐤_0| }^r with r ∈ℕ^*, we denote z^β_1...z^β_r by z^β.



 §.§ Decay estimates




  §.§.§ Norms


With the vector field method, the pointwise decay estimates are obtained through weighted Sobolev inequalities. In view of the above definitions of the vector fields and weights, we are naturally brought to define the following weighted L^1 and L^2 norms.




Let u : [0,T[ ×ℝ^n →ℝ be a smooth function. For k ∈ℕ, we define for all t ∈ [0,T[,


    u _𝕂,k(t) := ∑_μ=0^n ∑_|β| ≤ k ∂_μ Z^β u(t,.) _L^2(ℝ^n) ,

with Z^β∈𝕂^|β|.

Let f : [0,T[ ×ℝ^n_x × P →ℝ be a smooth function, with P=^n_v or P=^n_v ∖{0 }. For k ∈ℕ, we define for all t ∈ [0,T[,


    f _ℙ_0,k(t) := ∑_|β| ≤ k Z^β f(t,.,.) _L^1_x,v,

with Z^β∈ℙ_0^|β|.

We also define, for q ∈ℕ and m ∈{0,1 },


    f _ℙ_0,k,q,m(t):=∑_|β| ≤ k ∑_|γ| ≤ qz^γZ^β f(t,.,.) _L^1_x,v,

with Z^β∈ℙ_0^|β| and z^γ∈𝐤^|γ|_m.


Note that u _𝕂,0 corresponds to the energy ∑_μ=0^n ∂_μ u _L^2(^n).



  §.§.§ Decay estimates for the velocity averages



Recall the Klainerman-Sobolev inequality (see <cit.>, Chapter II). For u a sufficiently regular function such that for all t ∈ [0,T[, u_𝕂,n+2/2(t) < + ∞, we have

    ∀(t,x) ∈ [0,T[ ×ℝ^n,  |∇_t,x u(t,x)| ≲u_𝕂,n+2/2(t)/(1+t+|x|)^n-1/2(1+|t-|x||)^1/2.


In particular, if □ u=0 then u_𝕂,n+2/2 is constant, as □ Z^β u=0 for all Z^β∈𝕂^|β|. It gives us a decay estimate for ∇_t,xu.

However if f is a solution to a relativistic transport equation, we cannot expect decay on f _L^∞_x,v (even for the free transport equation T_1(f)=0 or T_0(f)=0). It is only the velocity averages of f, such as ∫_v f dv, that decay. For instance, we have the following classical estimate.




Let f be the solution of T_1(f)=0 which satisfies f(0,.,.)=f_0, with f_0 a smooth function compactly supported in v. Then, if R is such that f_0(.,v)=0 for all |v| ≥ R,


    ∀ (t,x) ∈ℝ_+ ×ℝ^n, ∫_v ∈ℝ^n |f(t,x,v)| dv ≤√(1+R^2)^n+2/t^nf_0_L^1_xL^∞_v.




 We fix (t,x) ∈ℝ_+ ×ℝ^n. By the method of characteristics, we obtain that


    ∀ v ∈ℝ^n, f(t,x,v) = f_0(x-v/v^0t,v).


We now use the change of variables y=v/v^0. Then, 
    ∫_v ∈ℝ^n |f(t,x,v)|dv =∫_|y| < 1 |f_0(x-ty,y/√(1-|y|^2))|1/√(1-|y|^2)^n+2dy.


Using the hypothesis on the support of f_0, we have 
    ∫_v ∈ℝ^n |f(t,x,v)|dv ≤√(1+R^2)^n+2∫_|y| < R/√(1+R^2)f_0(x-ty,.)_L^∞_vdy.


A last change of variables (z=x-ty) gives the result.






  §.§.§ Klainerman-Sobolev inequalities for velocity averages


As we can expect decay on the velocity average of a solution of a relativistic transport equation (and not on the solution itself), we will then use the following Klainerman-Sobolev inequalities. 




Let T>0 and f be a smooth function defined on [0,T[ ×ℝ^n_x ×ℝ^n_v or [0,T[ ×ℝ^n_x × (ℝ^n_v ∖{0 } ). Then


    ∀ (t,x) ∈ [0,T[ ×ℝ^n, ∫_v ∈ℝ^n |f(t,x,v)| dv ≲ f _ℙ_0,n(t)/τ_+^n-1τ_-.







A proof of this inequality can be found in <cit.> (see Theorem 7). We then deduce the following result.



Let T>0, q ∈ℕ, m ∈{0,1 } and f be a smooth function defined on [0,T[ ×ℝ^n_x ×ℝ^n_v or [0,T[ ×ℝ^n_x × (ℝ^n_v ∖{0 } ). Then


    ∀|γ| ≤ q,  (t,x) ∈ [0,T[ ×ℝ^n, ∫_v ∈ℝ^n |z^γf(t,x,v)| dv ≲ f _ℙ_0,n,q,m(t)/τ_+^n-1τ_-,


with z^γ∈𝐤_m^|γ|.




Let |β| ≤ n, |γ| ≤ q, Z^β∈ℙ_0^|β| and z^γ∈𝐤_m^|γ|. By Lemma <ref>, we have

    |Z^β(z^γf)| ≲∑_|β_0| ≤ |β|∑_|γ_0| ≤ |γ| |w^γ_0Γ^β_0 f |,

with w^γ_0∈𝐤_m^|γ_0| and Γ^β_0∈ℙ_0^|β_0|.
It only remains to apply Theorem <ref>.





All the results of this section are true if we add a v^0-weight (we can for instance study ∫_v ∈ℝ^n (v^0)^k |f| dv, for k ∈ℤ). We just need to modify the norms in the same way. For instance,

    ∀ (t,x) ∈ [0,T[ ×^n, ∫_v ∈^n (v^0)^k |f| dv ≲∑_|β| ≤ n(v^0)^k Z^βf(t,.,.) _L^1_x,v/τ_+^n-1τ_-.






 §.§ The Vlasov-Maxwell system




  §.§.§ Presentation


In order to introduce the Vlasov-Maxwell system, we abusively use the notation

    ∇_v f= [         0; ∂ f/∂ v^1;         ⋮; ∂ f/∂ v^n ] .


For a sufficiently regular function f, we recall that 

    (J(f)^ν)_0 ≤ν≤ n = [       ∫_v f dv; ∫_v fv^1/v^0dv;              ⋮; ∫_v fv^n/v^0dv ] ,

with v^0=√(m^2+|v|^2), where the mass m depends on the species considered.

Let K ∈ℕ^*. The equation (1) of the Vlasov-Maxwell system, for the species k, can be rewritten as


    T_m_k(f_k) + e_kF(v, ∇_v f_k) =0  .



Note that the initial data needs to satisfy 
    ∇^i (F_0)_i0= e^k J(f_0k)_0and∇^i (^*  F_0)_i α_1 ... α_n-3 0=0.



It is well known that in 3d the electric field and the magnetic field are solutions to a wave equation. In dimension n and in the context of the Vlasov-Maxwell system (and more precisely, with equations (<ref>) and (<ref>)), we have

    ∀ 1 ≤ i ≤ n, □ E^i=∑_k=1^Ke_k∫_v ∈^n∂_i f_k+v^i/√(m_k^2+|v|^2)∂_t f_k dv,

with E^i=F_0i, and[In dimension n >3, the magnetic field is a 2-form defined by B_ij=-F_ij but we make the choice to work with F_ij.]

    ∀ 1 ≤ i < j ≤ n, □ F_ij=∑_k=1^Ke_k∫_v ∈^nv^j/√(m_k^2+|v|^2)∂_i f_k-v^i/√(m_k^2+|v|^2)∂_j f_k dv.


We end this subsection by the following proposition, which gives an alternative form of the Maxwell equation.



The Maxwell equations


    {[                ∇^μ G_μν                     M_ν; ∇^μ^*  G_μα_1 ... α_n-2                      0,;                         ].


for a 2-form G and a 1-form M, are equivalent to 


    {[                         ∇_[ λ G_μν ]                                    0;          ∇_ [ λ^*  G_α_1 ... α_n-1 ] (-1)^n+1(n-1)!/2ε_λα_1 ... α_nM^α_n,;                                      ].







That ensues from straightforward calculations. Let us consider the equation ∇^i G_i0 = M_0.
For 1 ≤ i ≤ n, we denote by (m_j^i)_1 ≤ j ≤ n-1 the n-1 integers of 1,n ∖{ i } ranked in ascending order. We have, without any summation,

    ^*  G_m_1^i ... m_n-1^i=G^0iε_0im_1^i ... m_n-1^i=G_i0ε_im_1^i ... m_n-1^i.

Hence,

    ∇^i G_i0=∑_i=1^n ε_im_1^i ... m_n-1^i∇^i ^*  G_m_1^i ... m_n-1^i=2/(n-1)!∇_[1^*  G_2...n].


It only remains to remark that

    M_0=(-1)^n+1ε_1...n0M^0.


For the equation ∇^μ^*  G_μ 3 ... n =0, we note that


    ^*  G_0 3 ... n=G_12, ^*  G_1 3 ... n=G_02, ^*  G_2 ... n =G_10.

So 
    ∇^μ^*  G_μ 3 ... n=∇_0 G_21 + ∇_1 G_02+ ∇_2 G_10.


It then comes that

    ∇_[0 G_12]=0.


The remaining equations can be treated similarly. 




For the remaining of this section, we consider the maximal smooth solution (f:=(f_1,..,f_K), F ) to the Vlasov-Maxwell system, defined on [0, T [, arising from initial data (f_0,F_0), so that f is a vector valued field (f_1,..,f_K). However, to lighten the notations, we will often denote (by a small abuse of notation) by f only one of the f_i and we will suppose, without loss of generality for the results establish below, that the charge of the species associated to f is equal to 1. 




  §.§.§ The electromagnetic potential


In order to establish energy estimates for the electromagnetic field, it is useful to introduce a potential in the Lorenz gauge.



A 1-form A is said to be a potential of the electromagnetic field F if

    F=dA or, in coordinates, F_μν = ∂_μA_ν-∂_νA_μ.

A satisfies the Lorenz gauge condition if moreover

    ∂^μ A_μ=0.




Every electromagnetic field F defined on ^n+1, which is contractible, has a potential since dF=0. Furthermore, if A is a potential then, for χ a regular function, A+dχ is also a potential. In particular, if A is a potential and χ solves

    □χ= -∂^μ A_μ

then A+dχ is a new potential satisfying the Lorenz gauge. The following lemma will be useful to study the derivatives of F in the Lorenz gauge.



If A is a potential satisfying the Lorenz gauge for an electromagnetic field G, i.e.

    dA=G and∂^μ A_μ=0,

then, for all Z ∈𝕂,

    dℒ_Z(A)=ℒ_ZG and∂^μℒ_Z(A)_μ=0.


 

Let us mention the wave equation satisfied by the potential in the Lorenz gauge.



Let (f,F) be a solution to the Vlasov-Maxwell system and A be a potential of the electromagnetic field F which satisfies the Lorenz gauge. Then, for all Z^β∈𝕂^|β| and 0 ≤μ≤ n, there exists constants C_γ^μ such that

    □ℒ_Z^β A_μ = ∑_|γ| ≤ |β| C_γ^μe^k∫_v ∈^nv^μ/v^0Z^γ f_k dv,

with Z^γ∈^|γ|.





As 
    F_μν=∂_μ A_ν-∂_ν A_μand∂^μ A_μ=0,
 we have for 0 ≤ν≤ n

    ∂^μ∂_μ A_ν = ∇^μ F_μν.

It remains to apply this to ℒ_Z^β A (see Lemma <ref>) and to use Proposition <ref> below.




The following proposition shows how we can construct a potential in the Lorenz gauge which is initially controled by the energy (at the time 0) of the electromagnetic field.


We suppose here that n ≥ 4. Let N ∈ℕ and let F be a closed 2-form such that all the norms considered below are finite and F(0,.) ∈ L^2(^n). Then, there exists a potential A in the Lorenz gauge such that, for all |β| ≤ N,

    ℒ_Z^β A _L^2(^n)(0) ≲∑_l|γ| ≤N-1 
  1 ≤i ≤n  (1+|x|)^|γ|+1∂^γF_0i(0,.) _L^2(^n)    

 
    +∑_l|γ| ≤  N 
     1 ≤ i ≤ n (  (1+|x|)^|γ|∂^γ∂^j F_ji(0,.) _L^2_x+(1+|x|)^|γ|+1∂^γ∂^j F_ji(0,.) _L^1_x),

with Z^β∈𝕂^|β|.




We start by a technical lemma.


Let G such that 

    (1+|x|)G_L^1(^n)+G_L^2(^n) < + ∞and∫_^n G dx =0.

Then, denoting by ℱ the Fourier transform (in x),

    ℱ^-1( -1/|ξ|^2ℱ(G) ) _L^2(^n)≲(1+|x|)G_L^1(^n)+G_L^2(^n).




We have

    ℱ^-1( -1/|ξ|^2ℱ(G) ) _L^2(^n)    =    1/|ξ|^2ℱ(G) _L^2(^n)
       ≲   ℱ(G) _L^2(|ξ| ≥ 1)+1/|ξ|^4ℱ(G)^2 ^1/2_L^1(|ξ| ≤ 1).

Note now that ℱ(G) _L^2(|ξ| ≥ 1)≤ G _L^2(^n). Finally, as (1+|x|)G_L^1(^n) is finite, ℱ(G) is of class C^1 and vanishes at 0, so, using the mean value theorem,

    ℱ(G)/|ξ|^4^1/2_L^1(|ξ| ≤ 1)   ≲   ∇_ξℱ(G)_L^∞_ξ1/|ξ|^3^1/2_L^1(|ξ| ≤ 1)
       ≲   |x|G_L^1(^n),

since F(g)_L^∞_ξ≤g_L^1_x for any L^1 function g.


The first step of the construction of the suitable potential is contained in the following lemma.


There exists a potential A of the electromagnetic field F satisfying the Lorenz gauge and such that

    A_0(0,.)=0, ∂_t A_0(0,.)=0,
 and 
    ∀ 1 ≤ k ≤ n, A_k_H^2(^n)(0) ≤∂^j F_jk(0,.) _L^2_x+(1+|x|)  ∂^j F_jk(0,.) _L^1_x.

This implies in particular that

    ∀ 1 ≤ k ≤ n, ∂_t A_k(0,.) = F_0k(0,.) andΔ A_k (0,.) = ∂^i F_ik(0,.).






Suppose that A exists. As ∂_t A_0(0.)=0 and ∂^μ A_μ=0, we have ∂^i A_i(0.)=0. Combined with ∂_μ A_ν-∂_ν A_μ = F_μν and A_0(0,.)=0, it comes that at t=0,


    ∀ 1 ≤ k ≤ n, ∂_t A_k = F_0kandΔ A_k = ∂^i F_ik.

Moreover, recall from the proof of Proposition <ref> that

    ∀ 0 ≤ν≤ n, □ A_ν = ∇^μ F_μν.

We then define A_ν as the solution of the wave equation (<ref>) such that A_0(0,.)=0, ∂_t A_0(0,.)=0 and, for all 1 ≤ k ≤ n, 
    ∂_t A_k(0,.)=F_0k(0,.) and A_k(0,.)=ℱ^-1(-1/|ξ|^2ℱ(∂^j F_jk) )(0,.).
 
Consequently, according to Lemma <ref>, Δ A_k(0,.)=∂^j F_jk and

    A_k_L^2(^n)(0) ≤∂^j F_jk(0,.) _L^2_x+(1+|x|) ∂^j F_jk(0,.) _L^1_x.

From classical elliptic equations theory, we have

    ∇^2 A_k _L^2(^n) = ∂^j F_jk_L^2(^n)

and 

    ∇ A_k ∈ L^2(^n), with∇ A_k_L^2(^n)≲ A_k_L^2(^n)+∇^2 A_k_L^2(^n),

which concludes the proof.





We consider the potential A constructed in Lemma <ref>. In what follows, we omit to specify that all the quantities are considered at t=0. Since, for instance,

    ∀ 1 ≤ i,j ≤ n, Ω_0iΩ_0j A=x^i∂_j A+x^ix^j∂_t ∂_t A,

we have (and it is sufficient) to estimate x^β∂_t,x^γ A _L^2(^n), with |β| ≤ |γ| ≤ N, in order to control ℒ_Z^ξ A _L^2(^n)(0) for all Z^ξ∈𝕂^|ξ|, with |ξ| ≤ N. Note that, as ∂^μ A_μ=0,

    x^β∂_t,x^γ∂_tA_0_L^2(^n)≤∑_k=1^n x^β∂_t,x^γ∂_kA_k_L^2(^n),

so that, since A_0=0, we only have to bound x^β∂_t,x^γ A_k _L^2(^n), for all 1 ≤ k ≤ n.

Let 1 ≤ k ≤ n, |γ| ≤ N-1 and |β| ≤ |γ|+1. Then, since ∂_t A_k=F_0k (see Lemma <ref>),

    x^β∂_t,x^γ∂_t A_k=x^β∂^γ_t,x F_0k  , sox^β∂_t,x^γ∂_t A_k _L^2_x≲(1+|x|)^|γ|+1∂^γF_0k_L^2_x.

The remaining case, where there are only spatial translations, is treated in the following lemma.



For all 1 ≤ k ≤ n, |γ| ≤ N and |β| ≤ |γ|,

    x^β∂^γ A_k _L^2(^n)≲∑_|β_0| ≤ |γ_0| ≤ N x^β_0∂^γ_0∂^j F_jk_L^2_x+(1+|x|) x^β_0∂^γ_0∂^j F_jk_L^1_x,


where γ, β∈ℕ^n,  x^β=x_1^β_1...x_n^β_n and ∂^γ=∂_1^γ_1...∂_n^γ_n, so there are no time derivatives.




We fix 1 ≤ k ≤ n and we proceed by induction on |β|. As Δ A_k= ∂^j F_jk, we have, for all |γ| ≤ N-2,

    ∀ 1 ≤ k ≤ n, Δ∂^γ A_k = ∂^γ∂^j F_jk

So, by classical elliptic equations theory,

    ∀ |γ| ≤ N-2, ∇^2 ∂^γ A_k _L^2(^n) = ∂^γ∂^j F_jk_L^2(^n),

implying the result for |β|=0 (the case of the lower order derivatives is treated in Lemma <ref>).

Let 1 ≤ |β| ≤ N. We suppose that for all |δ| ≤ |γ| ≤ N and |δ| ≤ |β|-1,

    x^δ∂^γ A_k _L^2(^n)≲∑_|β_0| ≤ |γ_0| ≤ N x^β_0∂^γ_0∂^j F_jk_L^2_x+ (1+|x|)x^β_0∂^γ_0∂^j F_jk_L^1_x.

Let γ be a multi-index such that |β| ≤ |γ| ≤ N. We have

    Δ x^β∂^γ A_k = Δ(x^β)∂^γ A_k+2∂_j(x^β)∂^j∂^γ A_k+x^β∂^γ∂^j F_jk.
 
The first two terms of the right hand side are equal to zero or can be rewritten as a linear combination of terms like

    ∂^γ_2(x^δ∂^γ_1 A_k),

with |γ_2|=2, |γ_1| ≤ |γ|-1 and |δ| ≤ |γ_1|. For instance,

    2∂_j(x_1^q)∂^j∂_2^q A_k=2q ∂_1∂_2 ( x_1^q-1∂_2^q-1A_k)-2q(q-1)∂_2^2(x_1^q-2∂_2^q-2A_k).

Let B be the right hand side of (<ref>) and G=x^β∂^γ∂^j F_jk. G satisfies the hypothesis of Lemma <ref> and B-G is a linear combination of terms such as (<ref>), which implies 
    ℱ^-1( -1/|ξ|^2ℱ(B-G) )_L^2(^n)≲∑_l |γ_1| ≤ |γ|-1 
     |δ| ≤ |γ_1| x^δ∂^γ_1 A_k_L^2(^n).
 So we only have to prove that

    x^β∂^γ A_k=ℱ^-1( -1/|ξ|^2ℱ(B) ),

or (it is equivalent) that x^β∂^γ A_k is the L^2 solution of Δφ = B. Recall that the difference of two solutions of this equation is an harmonic polynomial, so that there exists exactly one L^2 solution, given by the right hand side of (<ref>). Consequently, there exists Q_k,β,γ∈ L^2(^n) and P_k,β,γ an harmonic polynomial function such that

    x^β∂^γ A_k=Q_k,β,γ+P_k,β,γ.

By the induction hypothesis, x^δ∂^γ A_k ∈ L^2(^n) for all |δ| = |β|-1, so

    P_k,β,γ/1+|x|=x^β/1+|x|∂^γ A_k-1/1+|x|Q_k,β,γ∈ L^2(^n).

As the dimension is n ≥ 4>1, P_k,β,γ is necessarily zero.




If the dimension n is at least 5, we can do better.


We suppose here that n ≥ 5. Let N ∈ℕ and let F be a 2-form such that all the norms considered below are finite. There exists a potential in the Lorenz gauge such that, for all |β| ≤ N,

    ℒ_Z^β A _L^2(^n)(0) ≲∑_l |γ| ≤N-1 
  1 ≤i ≤n  (1+|x|)^|γ|+1∂^γF_0i(0,.) _L^2(^n)   

 
    +∑_l|γ| ≤  N 
     1 ≤ i ≤ n (  (1+|x|)^|γ|∂^γ∂^j F_ji(0,.) _L^2(^n)+  (1+|x|)^|γ|∂^γ∂^j F_ji(0,.) _L^1(^n)),

with Z^β∈𝕂^|β|.






The proof is similar to the previous one. The difference comes from the fact ξ↦1/|ξ|^4 is integrable around 0 in ^n, with n ≥ 5, which allows us to lower the hypothesis of Lemma <ref>.





  §.§.§ Commutation properties




  §.§.§ Commutation of the transport equation


We fix the mass m ∈_+ and we denote by T_F the operator[Note that if the charge e of the species considered is not equal to 1, one just has to consider T_eF (in other words, one just has to replace F by eF).] 
    T_F : g ↦ T_m(g)+F(v,∇_v g),
 
so that T_F(f)=0.
We are now interested by the nature of the source terms of the equation T_F(Zf)=G.

 If Z∈ℙ, then


    T_F(Z f ) = -ℒ_Z(F)(v,∇_v f) .


For the scaling, we have



    T_F(S f ) = 2F(v,∇_v f)-ℒ_S(F)(v,∇_v f) .









First of all, let us consider the scaling. According to Lemma <ref>,


    T_m(Sf)=-S(F(v,∇_v f))+T_m(f).

But, 
    S(F(v,∇_v f)) = ℒ_S (F)(v, ∇_v f)+F([S,v],∇_v f)+F(v, [S,∇_v f]).

Since 
    [S,v]=-v and [S, ∇_v f]=∇_v S(f)-∇_v f,

we obtain

    T_F(Sf)=2F(v, ∇_v f)-ℒ_S (F)(v, ∇_v f).


Now, let Z∈ℙ and consider Z_v=Z-Z. According to lemma <ref>,


    T_m(Zf)=-Z(F(v,∇_v f))-Z_v(F(v,∇_v f)).


On the one hand, we have

    Z_v(F(v,∇_v f))=F(Z_v(v),∇_v f)+F(v,Z_v(∇_v f)).

On the other hand we have

    Z(F(v,∇_v f))=ℒ_Z(F)(v,∇_v f)+F([Z,v],∇_v f)+F(v,[Z,∇_v f]).


As [Z,v]=-Z_v(v), F(Z_v(v),∇_v f) and F([Z,v],∇_v f) cancel. 

If Z is a translation (we denote it by ∂), then Z_v=0 and [Z, ∇_v f ]=∇_v ∂(f). Thus

    T_F(∂ f)=-ℒ_∂(F)(v,∇_v f).

If Z=Ω_ij, then 

    Z_v(∇_v f)=∇_v Z_v(f)+∂_v^i f ∂_j-∂_v^j f ∂_i
 and 
    [Z,∇_v f]=∇_v Z(f)-∂_v^i f ∂_j+∂_v^j f ∂_i.

Therefore 

    T_F( Ω_ij f ) = - ℒ_Ω_ij(F)(v,∇_v f) .

Finally, if Z= Ω_0i, then

    Z_v(∇_v f)=∇_v Z_v(f)-∂_v^i f v^k/v^0∂_k and [Z,∇_v f]=∇_v Z(f)-∂_v^i f ∂_0.

It comes that

    T_F( Ω_0i f ) = - ℒ_Ω_0i(F)(v,∇_v f) + ∂_v^i f/v^0F(v,v).

It remains to remark that F(v,v)=0 for all v ∈ℝ^n, as F is a 2-form.




Iterating the above, one obtains


If β∈{1,...,|ℙ_0|}^r, with r ≥ 0, there exist integers C_γ,δ^β such that


    T_F(Z^β f)=∑_l |γ|+|δ| ≤ r 
     |δ| ≤ r-1  C_γ,δ^βℒ_Z^γ(F)(v,∇_v Z^δ(f)),


with Z^β∈ℙ_0^r, Z^δ∈ℙ_0^|δ| and Z^γ∈𝕂^|γ|.





If there is a source term G (such that T_F(f)=G), then we need to add a linear combination of terms such as Z^β G, with |β| ≤ r, on the right hand side.






  §.§.§ Commutation of the Maxwell equations


Before studying specifically the Vlasov-Maxwell system, we recall the following general result.



Let M_ν be a smooth 1-form and G_μν a 2-form satisfying


    {[                ∇^μ G_μν                     M_ν; ∇^μ^*  G_μα_1 ... α_n-2                      0.;                         ].


Then, for all Z ∈ℙ,


    {[                ∇^μℒ_Z(G)_μν                    ℒ_Z(M)_ν; ∇^μ^* ℒ_Z(G)_μα_1 ... α_n-2                          0.;                             ].

For the scaling, we have


    {[                ∇^μℒ_S(G)_μν               ℒ_S(M)_ν+2M_ν; ∇^μ^* ℒ_S(G)_μα_1 ... α_n-2                          0.;                             ].




In the Vlasov-Maxwell system, the source term is e^kJ(f_k)_ν (see (<ref>)), with


    (J(f_k)^ν)_0 ≤ν≤ 3 = [       ∫_v f_k dv; ∫_v f_kv^1/v^0dv;                ⋮; ∫_v f_kv^n/v^0dv ] ,


so we need to compute ℒ_Z (J(f)), with Z ∈𝕂 and f a regular function.



For all Z ∈ℙ,


    ℒ_Z (J(f)_ν)=J(Z f)_ν.


For the scaling, we have


    ℒ_S (J(f)_ν)=J(S f)_ν+J(f)_ν.








Let Z ∈𝕂,


    ℒ_Z J(f)_ν = ZJ(f)_ν+J(f)_μ∂ Z^μ/∂ x^ν.


So

    L_∂ J(f) =   J(∂ f),  L_SJ(f) =J(Sf)+J(f).

If Z is a Lorentz boost, say x^1 ∂_t+t ∂_1, then, as


    ∫_v v^0∂_v^1f dv=-∫_v f v^1/v^0 dv=-J(f)_1,
 
    ∫_v v^0v^i/v^0∂_v^1f  dv=-δ_1,i∫_v f dv=δ_1,i J(f)_0

and 

    J(f)_μΩ_01^μ/∂ x^ν =J(f)_1 δ_ν,0+J(f)_0 δ_ν,1,

it comes that

    ℒ_Ω_01 J(f) = J(Ω_01 f ).


The case where Z is a rotation is similar.




Iterating the above, we obtain the following proposition.



Let (f,F) be a smooth solution of the Vlasov-Maxwell system. For all β∈{1,..., |𝕂| }^r, with r ∈ℕ, there exist integers C^β_γ such that


    ∇^μℒ_Z^β(F)_μν    =     e^kJ(Z^βf_k)_ν+∑_|γ| ≤ |β|-1 C_γ^βe^kJ(Z^γ f_k)_ν, 
    ∇^μ^* ℒ_Z^β(F)_μα_1 ... α_n-2    =    0,



with Z^β∈𝕂^r and Z^γ∈ℙ_0^|γ|.






§ ENERGY ESTIMATES FOR THE VLASOV-MAXWELL SYSTEM


For all this section, we consider a sufficiently regular solution (f,F), on [0,T[, to the Vlasov-Maxwell system arising from smooth initial data (f_0,F_0).



 §.§ Energy estimates for the transport equation


We treat here the massless and the massive case together. As the set {v=0 } is of measure zero, we write ∫_v ∈^n h dv, or merely ∫_v h dv, even when the function h is not defined for v=0. We start by introducing the vector field N^μ(g) defined by, for a function g : [0,T[ ×ℝ^n_x ×ℝ_v^n →ℝ, 

    N^μ(g):= ∫_v ∈ℝ^n g v^μdv/v^0.


We have the following energy estimates.



Let g and H be two smooth functions defined on [0,T[ × ℝ^n_x ×ℝ_v^n such that T_F(g)=H and k ∈ℤ.  Then, for all t ∈ [0,T[,

    ∫_Σ_t∫_^n_v |g| dv dx+ √(2)sup_u ≤ t∫_C_u(t)∫_^n_v |g| v^L/v^0 dv dC_u(t)    ≤    2 ∫_Σ_0∫_^n_v |g| dv dx 
          + 2∫_0^t ∫_Σ_s∫_^n_v |H|  dv/v^0 dx ds.



 First, let us compute the (euclidian) divergence of N^μ(|g|). Start by noticing that, in W^1,1,

    T_m(|g|)=v^μ∂_μ |g|=g/|g|H-F(v,∇_v |g|).

By integrations by parts and using F_jj=0 as well as v^i v^j F_ij=0 (recall that F is a 2-form), we have

    ∫_v  F(v,∇_v |g|) dv/v^0 =∫_v v^μ/v^0F_μ^j ∂_v^j |g| dv= ∫_v v^i v^j/(v^0)^3F_ij |g| dv=0.

Consequently,

    ∂_μ N^μ(|g|)=∫_v ∈ℝ^n( g/|g|H-F ( v,∇_v |g|) ) dv/v^0=∫_v ∈ℝ^ng/|g|H dv/v^0.

We now apply the divergence theorem to N^μ(|g|) in several region. Applied to [0,t] ×ℝ^n, it gives

    ∫_Σ_t∫_v  |g| dvdx ≤∫_Σ_0∫_v  |g| dvdx+∫_0^t ∫_Σ_s∫_v  |H|dv/v^0 dx ds .

Applied to V_u(t) and using that 1/√(2)(∂_t-∂_r) is the outward pointing unit normal field to C_u(t), it gives

    √(2)∫_C_u(t)∫_v ∈ℝ^n |g| v^L/v^0 dv d C_u(t)  ≤∫_Σ_0∫_v  |g| dvdx +∫_0^t ∫_Σ_s∫_v  |H| dv/v^0 dx ds .

The estimate then ensues from the combination of the two inequalities. 


This estimate invites us to consider the following energies.



For N ∈ℕ and k ∈ℤ, we define, for g a sufficiently regular function, 


    𝔼^k_N[g](t) = ∑_lZ^β∈ℙ_0^|β|
     |β| ≤ N (v^0)^k Z^β g_L^1_x,v(t)+sup_u ∈ℝ∫_C_u(t)∫_ℝ^n |Z^β g| (v^0)^kv^L/v^0 dv dC_u(t) .


We also need the following norms. For q ∈ℕ and m ∈{0,1 },



     𝔼^k_N,q,m[g](t) = ∑_l Z^β ∈ℙ_0^|β| 
  |β| ≤N  ∑_l z^γ ∈𝐤_m^|γ| 
  |γ| ≤q   (v^0)^kz^γ Z^β g_L^1_x,v(t)     

 
    + ∑_lZ^β∈ℙ_0^|β|
     |β| ≤ N ∑_l z^γ∈𝐤_m^|γ|
     |γ| ≤ q sup_u ∈ℝ∫_C_u(t)∫_v ∈ℝ^n |z^γZ^β g| v^L/v^0 (v^0)^k dv dC_u(t).


When k=0, we drop the dependance in k of the energy norm. For instance, 𝔼^0_N[g] is denoted by 𝔼_N[g].



The following energy estimates hold.


Let g and H be such that T_F(g)=H. Then, assuming that g and H are sufficiently regular, we have for all N ∈ℕ and for all t ∈ [0,T[, 


     𝔼_N[g](t)-2𝔼_N[g](0)  ≲∑_|β| ≤N ∫_0^t 1/v^0 Z^β H _L^1_x,v(s)ds    

 
    +∑_l|γ|+|δ| ≤ N 
     |δ| ≤ N-1 ∫_0^t ℒ_Z^γ(F)(v/v^0,∇_v Z^δ(g) ) _L^1_x,v(s) ds

and 

     𝔼^2_N[g](t)-2𝔼^2_N[g](0) ≲∑_|β| ≤N ∫_0^t v^0 Z^β H _L^1_x,v(s)+ v^iF_i0 Z^β g _L^1_x,v (s) ds    

 
    + ∑_l|γ|+|δ| ≤ N 
     |δ| ≤ N-1 ∫_0^t ℒ_Z^γ(F)(v,∇_v Z^δ(g))v^0 _L^1_x,v(s) ds,

with Z^δ∈ℙ_0^|δ|, Z^β ∈ℙ_0^|β| and Z^γ∈𝕂^|γ|.





The first estimate follows from Corollary <ref>, Remark <ref> and Proposition <ref>, applied to Z^β g for |β| ≤ N. For the second one, apply the same results to (v^0)^2Z^β g and note that

    T_F ( (v^0)^2 ) = F ( v, ∇_v (v^0)^2 ) = 2v^μ v^i F_μ i=-2v^μ v^0 F_μ 0 = -2 v^i v^0 F_i0.




Assuming enough decay on the data, similar inequalities holds for 𝔼^k_N[g].

We also have an energy estimates which implies the weights transported by the flow. 



Let g and H be two sufficiently regular functions such that T_F(g)=H. For all N ∈ℕ, m ∈{0,1 } and t ∈ [0,T[, we have


     𝔼_N,1,m[g](t)-2𝔼_N,1,m[g](0) ≲∑_z ∈𝐤_m ∑_|β| ≤N ∫_0^t z/v^0 Z^β H _L^1_x,v(s)ds     

 
    +∑_z ∈𝐤_m∑_|β| ≤ N∫_0^t  F(v/v^0, ∇_v z ) Z^β g _L^1_x,v(s)ds

  
    +∑_z ∈𝐤_m∑_l|γ|+|δ| ≤ N 
     |δ| ≤ N-1 ∫_0^t   z ℒ_Z^γ(F)(v/v^0,∇_v Z^δ g ) _L^1_x,v(s)ds

and
 
     𝔼^2_N,1,m[g](t) - 2𝔼^2_N,1,m[g](0) ≲∑_z ∈𝐤_m∑_|β| ≤N ∫_0^t v^0zZ^β H _L^1_x,v(s)ds    

 
    + ∑_z ∈𝐤_m∑_|β| ≤ N∫_0^t zv^iF_i0Z^β g _L^1_x,v(s)+ v^0 F(v, ∇_v z) Z^β g _L^1_x,v(s) ds
 
 
    + ∑_z ∈𝐤_m∑_l |γ|+|δ| ≤ N 
     |δ| ≤ N-1 ∫_0^t   v^0z ℒ_Z^γ(F)(v,∇_v Z^δ g) _L^1_x,v(s)ds ,

with Z^δ∈ℙ_0^|δ|, Z^β ∈ℙ_0^|β| and Z^γ∈𝕂^|γ|.



Note that, for z ∈𝐤_m and according to equations (<ref>) and (<ref>),

    T_F( z Z^β g  )= zT_F(  Z^β g  )+ T_F(z) Z^β g =zT_F(  Z^β g  )+ F(v, ∇_v z) Z^β g.

The remaining of the proof is then similar to the one of Proposition <ref>.





 §.§ Energy estimates for the wave equation


Recall that a potential A in the Lorenz gauge satisfies the wave equation <ref>. In order to bound its L^2 norm, we recall here a classical energy estimates for the wave equation  using the vector field K_0. We mostly follow <cit.>, Chapter II.

During this subsection, we consider u :[0,T[ ×^n → a smooth function such that

    u_L^2(^n)(0)+∑_Z ∈𝕂Zu _L^2(^n)(0) < + ∞.

We also introduce its energy momentum tensor

    T_μν[u]= ∂_μ u ∂_νu-1/2η_μνη^σρ∂_σ u ∂_ρ u.


Since K_0 is merely a conformal Killing vector field and as T_μν[u] is not traceless, T_μν[u] K_0^ν is not divergence free when □ u =0. In fact

    ∇^μ (T_μν[u] K_0^ν) = □ u K_0 u+1/2T_μν[u] π^μν,

with

    π^μν= ∂^μK_0^ν+∂^νK^μ_0.

Since K_0 is a conformal vector field of conformal factor 4t, π^μν=4 tη^μν. So


    ∇^μ (T_μν[u] K_0^ν) = □ u K_0 u+(1-n)t ∂^μ u ∂_μ u.

The equality

    t∂^μ u ∂_μ u=∂_μ (tu ∂^μ u)-∂_μ(t)u∂^μ u-tu □ u=∂^μ(tu∂_μ u- 1/2u^2 ∂_μ t)-tu □ u,

suggests us to introduce the 1-form

    P_μ =T_μν[u] K^ν_0+(n-1)tu∂_μ u- n-1/2u^2 ∂_μ t.


Applying the divergence theorem on [0,t] ×^n to T_μ0[u] and P_μ, we obtain



∀ t ∈ [0,T[,

    ∑_μ=0^n ∂_μ u _L^2(Σ_t)≤∑_μ=0^n ∂_μ u _L^2(Σ_0)+∫_0^t ∫_Σ_s |□ u | d Σ_s ds

and

    ∫_Σ_t P_0 d Σ_t ≤∫_Σ_0 P_0 d Σ_0+∫_0^t ∫_Σ_s |□ u| |K_0 u+(n-1)tu|d Σ_s ds.




The first thing to verify is that ∫_Σ_t P_0 d Σ_t can be compared with the L^2 norm of u (and of its derivatives). 




We suppose that n ≥ 3. We have, for all t ∈ [0,T[,

    ∑_|β| ≤ 1 Z^β u ^2_L^2(^n)(t) ≲∫_Σ_t  P_0 d Σ_t ≲∑_|β| ≤ 1 Z^β u ^2_L^2(^n)(t).






Let us first remark that

    P_0=1/2(1+|x|^2+t^2)|∇_t,x u|^2+2tx^i ∂_i u ∂_t u+(n-1)tu∂_t u- n-1/2u^2 .

Moreover,

    (1+|x|^2+t^2)|∇_t,x u|^2+4tx^i ∂_i u ∂_t u=|∇_t,x u|^2+|Su|^2+∑_0 ≤μ < ν≤ n |Ω_μν u |^2

together with

    ∫_^n 2tu∂_t u dx = ∫_^n 2uSu-x^i∂_i( u^2) dx= ∫_^n 2uSu+nu^2dx

gives

    ∫_Σ_t P_0 d Σ_t = 1/2∫_Σ_t |∇_t,x u|^2+|Su+(n-1)u|^2+∑_0 ≤μ < ν≤ n |Ω_μν u |^2 d Σ_t.


This proves the second inequality and reduces the first one to 

    u^2_L^2(^n)(t)+Su ^2_L^2(^n)(t) ≲∫_Σ_t  P_0 d Σ_t.


In order to transform ∫_^n 2tu∂_t u d x in an alternative expression, we remark that

    2u ∂_t u = 2u1/rΩ_0r u-t/r^2x^i ∂_i (u^2), withΩ_0r=x^i/rΩ_0i.


So, by integration by parts,

    ∫_^n 2tu∂_t u dx=∫_^n( 2t/ru Ω_0r u+(n-2)t^2/r^2u^2 )dx.

Combined with equation (<ref>), we get

     ∫_^n ( 2(n-1)tu∂_t u- (n-1)u^2 )dx =2n-3/2∫_^n ( 2uSu+nu^2 )dx    

 
    +1/2∫_^n( 2t/ru Ω_0r u+(n-2)t^2/r^2u^2 ) dx-(n-1)∫_^n u^2 dx.


It then comes that

    2∫_Σ_t P_0 d x = ∫_Σ_t |Su|^2+22n-3/2uSu+2n^2-5n+2/2u^2 d x

 
    +∫_Σ_t |∇_t,x u|^2+∑_μ < ν |Ω_μν u |^2-|Ω_0ru|^2 +|Ω_0ru|^2+t/ru Ω_0r u+(n-2)t^2/2r^2u^2 dx.

The second integral on the right hand side of (<ref>) is nonnegative since

    |Ω_0ru|^2 =|x^i/rΩ_0iu|^2 ≤∑_i=1^n |Ω_0iu|^2

and

    |Ω_0ru|^2+t/ru Ω_0r u+(n-2)t^2/2r^2u^2=(Ω_0ru+t/2ru)^2+(2n-5)t^2/4r^2u^2.

Consequently,

    u^2_L^2(^n)(t)+Su ^2_L^2(^n)(t) ≲∫_Σ_t  P_0 d Σ_t

comes from

    |Su|^2+22n-3/2uSu+2n^2-5n+2/2u^2=( Su+ 2n-3/2u )^2+2n-5/4u^2

and from

     |Su|^2+22n-3/2uSu+2n^2-5n+2/2u^2 =    

 
    ( 2n-3/√(4n^2-10n+4)Su+ (n^2-5/2n+1)^1/2u )^2+2n-5/4n^2-10n+4|Su|^2.





We also proved that 
    ∫_Σ_tt^2/r^2u^2 d Σ_t ≲∫_Σ_t P_0 d Σ_t.



Finally, we obtain the expected estimate.


We have, for all t ∈ [0,T[,

    ∑_|β| ≤ 1 Z^β u ^2_L^2(^n)(t) ≲∑_|β| ≤ 1 Z^β u ^2_L^2(^n)(0)+ ∑_|β| ≤ 1∫_0^t ∫_Σ_s |Z^β u| |τ_+ □ u | dx ds,

with Z^β∈𝕂 if |β|=1, leading to, for all t ∈ [0,T[,



    ∑_|β| ≤ 1 Z^β u _L^2(^n)(t) ≲∑_|β| ≤ 1 Z^β u _L^2(^n)(0)+  ∫_0^t  τ_+ □ u _L^2(^n)  ds.








We have, according to Propositions <ref> and <ref>,

     ∑_|β| ≤1  Z^β u  ^2_L^2(^n)(t) ≲∑_|β| ≤1  Z^β u ^2_L^2(^n)(0)    

 
    + ∫_0^t ∫_Σ_s |□ u| |K_0 u+(n-1)tu|dx ds.

The result then follows from Remark <ref>, which gives us

    |K_0 u| ≲τ_+^2 |Lu|+ τ_-^2 |L u| ≲τ_+ ∑_Z ∈𝕂 |Zu|.




We now apply this to the electromagnetic potential in the Lorenz gauge. Since we will need to estimate S(ℒ_Z^β(A) ) _L^2(^n) in order to bound the energy of the electromagnetic field F (see Proposition <ref> below), we consider the following norms.



Let A be a sufficiently regular 1-form defined on [0,T[ ×^n. We define, for N ∈ℕ and all t ∈ [0,T[,

    ℰ_N[A](t)= ∑_μ=0^n ∑_|β| ≤ 1∑_|γ| ≤ N Z^β( ℒ_Z^γ (A)_μ) ^2_L^2(^n)(t).







Note that

    ∑_μ=0^n ∑_|β| ≤ N+1Z^β A_μ_L^2(^n)^2 ≲ℰ_N[A] ≲∑_μ=0^n ∑_|β| ≤ N+1Z^β A_μ_L^2(^n)^2.

We work with ℰ_N[A] as we will apply Proposition <ref> to ℒ_Z^β(A)_μ.


Using Proposition <ref>, we get the following result.



Let N ∈ℕ and A_μ be a sufficiently regular 1-form, defined on [0,T[ ×^n, such that ℰ_N[A](0)< + ∞. Then, ∀ t ∈ [0,T[,

    √(ℰ_N[A])(t) ≲√(ℰ_N[A])(0)+∑_μ=0^n ∑_|γ| ≤ N∫_0^t  τ_+ □ℒ_Z^γ (A)_μ_L^2(Σ_s)  ds.







 §.§ Energy estimates for the Maxwell equations


We prove three conservation laws for the Maxwell equations, using each time a different multiplier (∂_t, K_0 or S). In the study of the massive case, we will mostly use the one associated to the Morawetz vector field.

For the remaining of this section, we consider a 2-form G and a current J, sufficiently regular and defined on [0,T[, such that



    {[                ∇^μ G_μν                     J_ν; ∇^μ^*  G_μλ_1 ... λ_n-2                      0.;                         ].


The following lemmas hold.




We have, for all 0 ≤ν≤ n,

    ∇^μ T[G]_μν = G_νρ J^ρ.







According to Proposition <ref>,


    G_μρ∇^μG_ν^ρ    =     G^μρ∇_μ G_νρ
        =    1/2 G^μρ (∇_μ G_νρ-∇_ρ G_νμ) 
        =    1/2 G^μρ∇_ν G_μρ
        =    1/4∇_ν (G^μρ G_μρ).


So,

    ∇^μ T[G]_μν = ∇^μ (G_μρ)G_ν^ρ+1/4∇_ν (G^μρ G_μρ)-1/4η_μν∇^μ (G^σρ G_σρ)=G_νρ J^ρ.






We have, denoting by (α, α, ρ, σ) the null decomposition of G,

    T[G]_L L=|α|^2,  T[G]_L L=|α|^2 and T[G]_L L=|ρ|^2+|σ|^2.








  §.§.§ Using ∂_t as a multiplier


As we use here the multiplier ∂_t, we work with T[G]_μ 0. Applying the divergence theorem to T[G]_μ 0 on [0,t] ×^n and V_u(t), we obtain the following result.



For all t ∈ [0,T[,


     ∫_Σ_t |α|^2+|α|^2+2|ρ|^2+2|σ|^2 dx =      

      ∫_Σ_0 |α|^2+|α|^2+2|ρ|^2+2|σ|^2 dx+4∫_0^t ∫_Σ_s G_0 μ J^μ  dx ds    

and

     √(2) sup_u ≤t ∫_C_u(t) |α|^2+|ρ|^2+|σ|^2 dC_u(t) ≤    

     ∫_Σ_0 |α|^2+|α|^2+2|ρ|^2+2|σ|^2 dx+4∫_0^t ∫_Σ_s |G_0 μ J^μ|  dx ds.    




This explains the introduction of the following norms.


Let N ∈ℕ. We define, for t ∈ [0,T[,

     ℰ^0[G](t)= ∫_Σ_t( |α|^2+|α|^2+2|ρ|^2+2|σ|^2 )dx    

    +sup_u ≤t ∫_C_u(t) ( |α|^2+|ρ|^2+|σ|^2 ) dC_u(t)    

 and

    ℰ_N^0[G](t)= ∑_|β| ≤ Nℰ^0_N[ℒ_Z^β(G)](t),

with Z^β∈𝕂^|β|.


Using the previous energy identities and commutation formula of Proposition <ref>, we obtain


For all N∈ℕ and all t ∈ [0,T[, we have

    ℰ^0_N[F](t)-2ℰ_N^0[F](0) ≲∑_|β|,|γ| ≤ N∫_0^t ∫_Σ_s |e^k ℒ_Z^β(F)_0 μ J(Z^γ f_k)^μ)|dxds,

with Z^β∈𝕂^|β| and Z^γ∈^|γ|.




  §.§.§ Using K_0 as a multiplier


As T[G] is not traceless in dimension n ≥ 4, ∇^μ(T[G]_μνK_0^ν) does not necessarily vanishes when G solves the free Maxwell equations. We then consider, in the spirit of what is done for the wave equation, for A a sufficiently regular potential of G in the Lorenz gauge, the current

    P_μ=T[G]_μνK^ν+(n-3)(tA_β∂_μA^β-1/2∂_μ(t)A_βA^β-tA_β∂^β A_μ+∂^β(t)A_βA_μ).


In order to establish an energy estimate for the electromagnetic field, we compute the divergence of P_μ.



We have


    ∇^μ P_μ = G_μνK_0^ν J^μ+(n-3)tA_β□ A^β.





We have


    ∇^μ ( T[G]_μνK^ν_0)= ∇^μ (T[G]_μν) K^ν_0+T[G]_μν∇^μK^ν_0.


Since T[G] is symmetric, 


    T[G]_μν∇^μK^ν_0 = 1/2 T[G]_μνπ^μν,

with π^μν=∇^μK_0^ν+∇^νK_0^μ. As K_0 is a conformal vector field (of conformal factor 4t), we have

    π_μν =4t η_μν .

Thus, 

    T[G]_μν∇^μK^ν_0=2t T(G)_μ^μ=3-n/2tG_σρ G^σρ.



Now, according to Lemma <ref>, we obtain that

    ∇^μ ( T[G]_μνK^ν_0)= G_νρK^ν_0 J^ρ+3-n/2tG_σρ G^σρ.

We now compute the divergence of

    (n-3) ( tA_β∂_μA^β-1/2∂_μ(t)A_βA^β-tA_β∂^β A_μ+∂^β(t)A_βA_μ).

First,

    ∇^μ( tA_β∂_μA^β) = -A_β∂_0A^β+t∂^μ A_β∂_μ A^β+tA_β□ A^β.

Secondly,

    ∇^μ( 1/2∂_μ(t)A_βA^β) =   A_β∂^0  A^β.

We also have, using in particular that in Lorenz gauge ∂^μ A_μ=0,

    ∇^μ( tA_β∂^β A_μ)     =    - A_β∂^β A_0+t∂^μ (A_β)∂^β A_μ+tA_β∂^β∂^μ A_μ
        =     - A_β∂^β A_0+t∂^μ (A_β)∂^β A_μ .

Finally

    ∇^μ( ∂^β(t)A_βA_μ)     =    -∂^μ(A_0)A_μ -A_0∂^μ A_μ
        =     -∂^μ(A_0)A_μ .

Hence,

     (n-3) ∇^μ( tA_β∂_μA^β-1/2∂_μ(t)A_βA^β-tA_β∂^β A_μ-∂^β(t)A_βA_μ ) =    

 
    (n-3)tA_β□ A^β+(n-3)t(∂_μA_β∂^μA^β-∂_μA_β∂^βA^μ).

And, since G_μν=∂_μ A_ν-∂_ν A_μ,

    1/2G_μβG^μβ=∂_μA_β∂^μA^β-∂_μA_β∂^βA^μ,

which gives us the result.







We are now ready to prove the following energy estimate.



For all t ∈ [0,T[,

    ∫_Σ_tτ_+^2 |α|^2+τ_-^2| α|^2+(τ_+^2+τ_-^2)(|ρ|^2+|σ|^2) d Σ_t+(n-3)^2 ∑_μ=0^n  A_μ^2_L^2(Σ_t)≤
 
    ∫_Σ_0 (1+r^2)(|α|^2+|α|^2+|ρ|^2+|σ|^2) d Σ_0 +4∫_0^t ∫_Σ_s |K_0^ν G_νμ J^μ | dx ds
 
    +(n-3)∑_μ=0^n   S A_μ^2_L^2(Σ_t) +4(n-3)∫_0^t ∫_Σ_s s|A_μ□ A^μ| dx ds.





In order to apply the divergence theorem to P_μ in [0,t] ×ℝ^n, we transform

    ∫_ℝ^n( t A_β∂_t A^β-1/2A_βA^β-tA_β∂^β A_0-A_0^2 ) dx.

On the one hand, let us notice that

    -1/2A_βA^β-A_0^2=-1/2∑_β=0^n A_β^2.

On the other hand,

    -t∫_ℝ^n A_β∂^β A_0 dx=-t∫_ℝ^n A_0 ∂^0 A_0 dx+t∫_ℝ^n∂^j( A_j) A_0 dx=t∂_t∫_ℝ^n A_0^2dx,

since ∂^μ A_μ=0 in the Lorenz gauge.
As 

    t∫_ℝ^n A_β∂_t A^β dx = t/2∂_t ∫_ℝ^n A_β A^β dx,

we finally obtain that

    ∫_ℝ^n( t A_β∂_t A^β-1/2A_βA^β-tA_β∂^β A_0-A_0^2 ) dx=1/2∑_β=0^n (t ∂_t-1)A_β_L^2(ℝ^n)^2 .



The divergence theorem applied to P_μ in [0,t] ×ℝ^n gives, using Lemma <ref> and <ref>,


     ∫_Σ_t τ_+^2 |α|^2+τ_-^2| α|^2+(τ_+^2+τ_-^2)(|ρ|^2+|σ|^2) dx  ≤4∫_0^t ∫_Σ_s |∇^μ P_μ | dx ds    

 
    + ∫_Σ_0 (1+r^2)(|α|^2+|α|^2+2|ρ|^2+2|σ|^2) dx + 2(n-3)∑_μ=0^n ( ∫_Σ_t (1-t∂_t) A_μ^2 dx ) .

It only remains to use the last lemma and the inequality

    -t∂_t ∫_Σ_t A_μ^2 dx ≤1-n/2 A_μ^2_L^2(Σ_t)+1/2 SA_μ^2_L^2(Σ_t)

which ensues from (<ref>).



This estimate justifies the introduction of the following norms.



Let G be a 2-form defined on [0,T[ and N ∈ℕ. We define, for all t ∈ [0,T[,


    ℰ[G](t) = ∫_Σ_tτ_+^2 |α(G)|^2+τ_-^2|α(G)|^2+(τ_+^2+τ_-^2)(|ρ (G) |^2+|σ (G) |^2) d x

and

    ℰ_N[G](t)= ∑_|β| ≤ Nℰ[ℒ_Z^βG](t),

with Z^β∈𝕂^|β|.



We then deduce, using Propositions <ref>, <ref> and Lemma <ref>, an energy estimate for the electromagnetic field F.



Let A be a sufficiently regular potential in the Lorenz gauge of F. We have, for all N ∈ℕ and all t ∈ [0,T[,


    ℰ_N[F](t) - ℰ_N[F](0)-(n-3)∑_|κ| ≤N ∑_μ=0^n   S ℒ_Z^κ(A)_μ ^2_L^2(Σ_t) ≲   


    + ∑_|β|,|γ| ≤ N∫_0^t ∫_Σ_s |e^kK_0^νℒ_Z^β(F)_νμ J^μ(Z^γf_k) | dx ds

 
    + ∑_|κ| ≤ N∫_0^t ∫_Σ_s s| ℒ_Z^κ(A)_μ□ℒ_Z^κ(A)^μ| dx ds,
 
with Z^β∈𝕂^|β|, Z^κ∈𝕂^|κ| and Z^γ∈ℙ_0^|γ|.




  §.§.§ Using S as a multiplier


The main difference with the previous case comes from the fact that the scaling is not a timelike vector field. Because of that we are not able to estimate all the null components of the electromagnetic field with this energy estimate. We start by introducing, for A a potential of G satisfying the Lorenz gauge,

    Q_μ=T(G)_μν S^ν+n-3/2(A_β∂_μ A^β-A_β∂^β A_μ ).

As the potential A satisfies the Lorenz gauge and since the conformal factor of the scaling is 2, we have


    ∇^μ Q_μ = G_μν S^ν J^μ+n-3/2A_β□ A^β.

We can now state the energy estimate.



For all t ∈ [0,T[,

    ∫_Σ_t(t+r)|α|^2+(t-r)|α|^2+2t(|ρ|^2+|σ|^2)dx+(n-3)∂_t∑_β=0^n A_β_L^2(Σ_t)^2  =
 
    ∫_Σ_0 r(|α|^2-|α|^2)dx+(n-3)∂_t∑_β=0^n A_β_L^2(Σ_0)^2+4 ∫_0^t ∫_Σ_s∇^μ Q_μ dxds.






Note first that we proved, during the proof of Proposition <ref> (see Equations (<ref>) and (<ref>)), 
    ∫_^n A_β∂_0 A^β-A_β∂^β A_0dx=∂_t/2∑_β=0^n A_β_L^2(^n)^2 .

It then remains to apply the divergence theorem to Q_μ on [0,T] ×^n (recall that 2S=(t+r)L+(t-r)L).



Note that (t-r)|α|^2 is not necessarily non negative, which invites us to transform the equality in the following estimate. 


For all t ∈ [0,T],

    ∫_Σ_t(1+|t-r|)|α|^2 dx ≤∫_Σ_0(1+r)(|α|^2+|α|^2)+2|ρ|^2+2|σ|^2 dx    

 
    +(n-3)(n+2)ℰ_0[A](0)+2/1+t(ℰ[F](t)+(n-3)(n+2)/2ℰ_0[A](t))
 
    +4∫_0^t ∫_Σ_s |G_0 μJ^μ|+|S^ν G_μν J^μ | dx ds+2(n-3) ∫_0^t ∫_Σ_s |A_μ□ A^μ |dx ds.





Adding the energy identities of Propositions <ref> and <ref>, we can obtain, 


    ∫_Σ_t(1+|t-r|)|α|^2 dx ≤∫_Σ_0(1+r)(|α|^2+|α|^2)+2|ρ|^2+2|σ|^2 dx    

 
    +∫_Σ_t(t+r)|α|^2+2t(|ρ|^2+|σ|^2)dx+4∫_0^t ∫_Σ_s |G_0 μ J^μ| +|∇^μ Q_μ | dx ds
 
 
    + (n-3)|∂_t∑_β=0^n (A_β_L^2(^n)^2(0)- A_β_L^2(^n)^2(t))|.

The result then ensues from the three following inequalities. Using Definition <ref>, one has 

    (1+t)∫_Σ_t(t+r)|α|^2+2t(|ρ|^2+|σ|^2)dx ≤ 2ℰ[F](t).

According to (<ref>), we have

    ∫_0^t ∫_Σ_s |∇^μ Q_μ | dx ds ≤∫_0^t ∫_Σ_s |S^ν G_μν J^μ | +(n-3)/2  |A_μ□ A^μ |dx ds.

Finally, Equation (<ref>) gives us

    (1+t)|∂_t A_μ_L^2(Σ_s)^2 | ≤S A_μ_L^2(Σ_s)^2+ ∂_t A_μ_L^2(Σ_s)^2+(n+2)A_μ_L^2(Σ_s)^2.



Let us introduce the following norms.


We define, for N ∈ℕ and t ∈ [0,T[,

    ℰ_N^S[F](t)=∑_l Z^β∈𝕂^|β|
    |β| ≤ N ∫_Σ_tτ_- |α(ℒ_Z^β(F)) |^2 dx.




Commuting the equation satisfied by the electromagnetic field F and using the previous energy estimate, we get the following proposition (see the commutation formulas of Proposition <ref> and Lemma <ref>).



Let A a sufficiently regular potential of the the electromagnetic field F in the Lorenz gauge. Then, for N ∈ℕ, we have, for all t ∈ [0,T[,

    ℰ_N^S[F](t) -ℰ_N[F](0) ≲ℰ_N[A](0)+1/1+t(ℰ_N[A](t) +ℰ_N[F](t))    


    +∑_|β|,|γ| ≤ N |e^k| ∫_0^t ∫_Σ_s | ℒ_Z^β(F)_0 μ J^μ(Z^γf_k) |+ |S^νℒ_Z^β(F)_νμ J^μ(Z^γf_k) | dx ds

 
    +∑_|β| ≤ N∫_0^t ∫_Σ_s |ℒ_Z^β(A)_μ□ℒ_Z^β(A)^μ |dx ds,
 

with Z^β∈𝕂^|β| and Z^γ∈^|γ|. 



Later, we will have, in the 4 dimensional massless case, a strong loss on ℰ_N[F] which will lead to a poor pointwise decay estimate on |α|. With this inequality, we will avoid the τ_+-loss and we will have an extra τ_--decay (which is not given by Proposition <ref>).




§ SOME TECHNICAL RESULTS




 §.§ An integral estimate


The following lemma is useful so as to estimate a quantity like

    ∫_0^t ∫_ℝ^n |u(s,x)| ∫_v |f(s,x,v)| dv dx ds,

where we already have a bound on u(s,.)_L^2 and a pointwise decay estimate on ∫_v |f(s,x,v)| dv.



Let m ∈ℕ^* and let a, b ∈ℝ, such that a+b >m and b ≠ 1. Then

    ∃ C_a,b,m >0, ∀ t ∈ℝ_+ ∫_0^+ ∞r^m-1/τ_+^a τ_-^bdr ≤ C_a,b,m1+t^b-1/1+t^a+b-m .





A proof of this estimate can be found in <cit.>, Appendix B.




 §.§ The null coordinates of ∇_v f


Let f : [0,T[ ×ℝ^n ×ℝ^n be a smooth function. We designate by (( ∇_v f )^L, ( ∇_v f )^L,  ( ∇_v f )^B,...) the null components of ∇_v f. Later, we will have to transform the v-derivatives in combinations of ℙ_0-derivatives. If we only use the relation 

    v^0∂_v^k = Ω_0k-t∂_k-x^k∂_t,

 we get that

    | ( ∇_v f )^L |, | ( ∇_v f )^L|, | ( ∇_v f )^B | ≤τ_+/v^0∑_Z∈ℙ_0  |Z f |,

which will not be good enough to close the energy estimates (for the Vlasov-Maxwell system).

We then use the following lemma.



Let f be a smooth function. We have

    | ( ∇_v f )^L |, | ( ∇_v f )^L| ≤τ_-/v^0∑_Z∈ℙ_0  |Z f |.








Since ( ∇_v f )^0=0 (by definition),

    ( ∇_v f )^L=x^i/r∂_v^i f.

Now, we use ∂_v^i=1/v^0(Ω_0i-t∂_i-x_i∂_t). As

    x^i/rv^0(t∂_i+x_i∂_t)=1/v^0(t∂_r+r∂_t)=1/v^0(S+(r-t)L),
 we have

    ( ∇_v f )^L = x^i /rv^0Ω_0i f-1/v^0S f +t-r/v^0Lf .

It only remains to notice that ( ∇_v f )^L=-( ∇_v f )^L, since ( ∇_v f )^0=0.




We are now interested in ( ∇_v  f )^B. During the study of the Vlasov equation, each time that (<ref>) is not sufficient to close the estimates, (∇_v f )^B is multiplied by v^L, which reflects the null structure of the system. This leads us to study v^L( ∇_v f )^B.



For 1 ≤ i ≤ n, we have 
    2v^Lx^i/r=v^0x^i/r-v^i+z_ijx^j/r^2,

where z_μν=x^νv^μ-x^μv^ν.





If μ≠ν, z_μν/v^0∈𝐤_1 and if μ = ν, then z_μμ =0.





For simplicity, we take i=1. We have


    2v^Lx^1/r    =    x^1v^0/r-x^1/r^2x_iv^i
        =    x^1v^0/r-v^1+z_1jx^j/r^2 .




And we obtain



Let i, j ∈ 1,n  such that i ≠ j. We have


    2v^L( x^i/r∂_v^j-x^j/r∂_v^i)=(x^i/r+z_ikx^k/v^0r^2)Ω_0j-(x^j/r+z_jkx^k/v^0r^2)Ω_0i-Ω_ij
 
    -(x^i(t-r)/r+tx^kz_ik/r^2v^0) ∂_j+(x^j(t-r)/r+ tx^kz_jk/r^2v^0) ∂_i-z_ij/v^0∂_t.






By the previous lemma,


    2v^L( x^i/r∂_v^j-x^j/r∂_v^i)= (v^0x^i/r+z_ikx^k/r^2) ∂_v^j-(v^0x^j/r+z_jkx^k/r^2)∂_v^i
 
    -Ω_ij+x^i∂_j-x^j ∂_i.

Now, using the relation v^0∂_v^k = Ω_0k-t∂_k-x^k∂_t, we have


    2v^L( x^i/r∂_v^j-x^j/r∂_v^i) = ( x^i/r+ z_ikx^k/v^0r^2)Ω_0j-( x^j/r+ z_jkx^k/v^0r^2)Ω_0i-Ω_ij
 
    +x^i∂_j-x^j ∂_i-t/v^0(v^0x^i/r+z_ikx^k/r^2)∂_j+t/v^0(v^0x^j/r+ z_jkx^k/r^2)∂_i-z_ij/v^0∂_t.

It remains to remark that tv^0x^i/r-v^0x^i=v^0x^i/r(t-r).


The naive estimation gave us

    | v^L( ∇_v f )^B  | ≲ |x| |∂_t f |+∑_k=1^n ( |Ω_0k f| +t |∂_k f | ) ,

whereas, with this lemma and the fact that ( ∇_v f )^B is a combination with bounded coefficients of ( x^i/r∂_v^jf-x^j/r∂_v^if )_1 ≤ i < j ≤ n, we have 

    | v^L( ∇_v f )^B  | ≲∑_Z∈ℙ |Z f | + ∑_1 ≤ i < j ≤ n |z_ij|/v^0|∂_t f |+ ∑_k=1^n (τ_-+ t ∑_i=1^n |z_ki|/rv^0) |∂_k f | .

Therefore, with the last corollary, we transform a t-loss (and a |x|-loss) in a τ_--loss and a t/r-loss (thanks, among others, to the weights transported by the flow). It is particularly useful when we look for an estimate of ∫_v | v^L  ( ∇_v Z^β f )^B  | dv  _L^2_x and we already have an estimate of ∫_v v^0 | z Z^δ f | dv. We can then use Lemma <ref>. One can also transform the t/r-loss.


For 1 ≤ j ≤ n,

    | x^j(t-r)/r+ tx^kz_jk/r^2v^0| ≲τ_-∑_z ∈𝐤_1 |z|.




We obviously have τ_-^-1| x^j(t-r)/r| ≤v^0/v^0. For the second term, we need to study different cases. 

If r ≤ 1, then

    τ_-^-1|tx^kz_jk/r^2v^0| = t/τ_-x_k/rx^kv^j-x^jv^k/rv^0≲1/v^0∑_i=1^n |v^i|.

Otherwise, r ≥ 1, and

    τ_-^-1|  tx^kz_jk/r^2v^0| ≤t/τ_-rx^k/r|z_jk|/v^0.

It remains to note that if r ≤t/2, τ_- ≥t/2 and if r ≥t/2, then t/r≤ 2.


One then obtains the following result.


We have

    | v^L( ∇_v f )^B  | ≲τ_-∑_Z∈ℙ∑_z ∈𝐤_1   |zZ f | .




Note that later, in Sections <ref> and <ref>, when we will establish an estimate on ∫_v |Z^βf| dv _L^2_x, we will not be able to apply Propositions <ref> or <ref>. A vector X will contain various derivatives of f and we will split it in two vectors H+G such that

    T_F(H)=0, with H(0)=X(0), and T_F(G)=T_F(X), with G(0)=0.

Note yet that, for instance, if X_μ is ∂_μf and X_S is S(f), we have x^μ X_μ=X_S whereas we do not necessarily have x^μG_μ=G_S.



 §.§ Some Sobolev inequalities


The following results come from <cit.> and in order to be self-sufficient, we also recall their proof. We will use them to prove pointwise decay estimate for the electromagnetic field.

We first recall two classical Sobolev inequalities.



Let u : ℝ^n →ℝ be a sufficiently regular function. We have

    ∀ x ∈ℝ^n,  |u(x)| ≲∑_|β| ≤n+2/2∂^β u _L^2_y(|y-x| ≤ 1).


Let v : 𝕊^n-1→ℝ a sufficiently regular function (where 𝕊^n-1 is the unit sphere in ℝ^n). We have

    ∀ξ∈𝕊^n-1,  |v(ξ)| ≲∑_|β| ≤n/2∇_Z^β v _L^2(𝕊^n-1),

with Z^β∈𝕆^|β|.



In order to treat the interior of the light cone (or rather the domain in which |x| ≤ 1+1/2t), we will use.



Let U be a smooth tensor field defined in the Euclidian space ℝ^n. Then,

    ∀ t ∈ℝ_+, sup_|x| ≤ 1+t/2 |U(x)| ≲1/(1+t)^n/2∑_k=0^n+2/2 (1+t)^k ∇^k U _L^2 ( {|y| ≤ 3+3/4t }).






As it suffices to prove the result for each component of the tensor, we assume that U is a scalar function. Let t ∈ℝ_+ and |x| ≤ 1+ 1/2t. If t ≤ 1, then |x| ≤ 2, so, according to Lemma <ref>,

    |U(x)| ≲∑_|β| ≤n+2/2∇^β U _L^2_y(|y| ≤ 3).


Now, if t ≥ 1, we apply Lemma <ref> to y ↦ U(x+t/4y). It comes that (after a change of variables)


    |U(x)| ≲( t/4)^-n/2∑_|β| ≤n+2/2( t/4)^|β|∇^β U _L^2_y(|y-x| ≤t/4).


It remains to observe that |y-x| ≤t/4 imply |y| ≤ 1+3/4t.



For the other region (|x| ≥ 1+1/2t), we have the following inequality.


Let U be a sufficiently regular tensor field, which in particular vanishes at ∞, defined in the euclidian space ℝ^n. Then, for t ∈_+,

    ∀ x ≠ 0,  |U(x)| ≲1/|x|^n-1/2τ_-^1/2( ∫_|y| ≥ |x| |U(y)|^2_𝕆,n/2+τ_-^2|∇_∂_r U(y) |^2_𝕆,n/2 dy )^1/2.






As ∑_|β| ≤ k |∇_Z^β U|^2 ≲ |U|^2_𝕆,k, for Z^β∈𝕆^|β|, we only have to prove the result for each component of U and we can assume that U is a scalar function. 

Let x ≠ 0 such that x=r ξ, with r=|x| and ξ∈𝕊^n-1. Since ∂_r ( (√(τ_-)U)^2 ) = 2√(τ_-)U∂_r (√(τ_-)U),

    τ_-|U(r ξ)|^2 ≲ r^-(n-1)∫_r^+∞ |√(τ_-)U(λξ)|∂_r(√(τ_-) U)(λξ)| λ^n-1 d λ .

Therefore, an integration over 𝕊^n-1 and the inequality 2|ab| ≤ a^2+b^2 gives us

    U(r ξ) _L^2_ξ( 𝕊^n-1)≲ r^-n-1/2τ_-^-1/2( ∫_|y| ≥ r |U(y)|^2+τ_-^2|∂_r U(y) |^2 dy )^1/2.

As every vector field of 𝕆 commute with ∂_r, we obtain, using Lemma <ref>,


    |U(x)| ≲ r^-n-1/2τ_-^-1/2( ∫_|y| ≥ r |U(y)|_𝕆,n/2^2+τ_-^2|∂_r U(y) |_𝕆,n/2^2 dy )^1/2.






 §.§ Pointwise decay estimate for the null decomposition of the electromagnetic field


In this section, we recall some inequalities coming from <cit.> between quantites linked to the null decomposition of a 2-form (see Section <ref> for its definition) and we then prove pointwise decay estimates on it. However, we cannot adapt the method used in <cit.> to establish, in dimension 3, the optimal decay estimate on the null component α. To circomvent this difficuty, we make crucial use of an electromagnetic potential satisfying the Lorenz gauge. We first introduce some notations.


Let F be a 2-form. We define its pointwise norm |F|^# by

    |F|^#=√(τ_+^2|α|^2+τ_-^2|α|^2+(τ_-^2+τ_+^2)(|ρ|^2+|σ|^2)),

which is also equal to √(4 T[F](K_0,∂_t)). 

We also define, for 𝕃=𝕆 or 𝕃=𝕂 and k ∈ℕ,


    |F|^#_𝕃,k=√(∑_|β| ≤ k( |ℒ_Z^βF|^#)^2 ),

with Z^β∈𝕃^|β|.

Similarly, we define

    |F|=√(|α|^2+|α|^2+2(|ρ|^2+|σ|^2))

and

    |F|_𝕃,k=√(∑_l Z^β∈𝕃^|β|
     |β| ≤ k   |ℒ_Z^βF|^2 ).








By definition of |F|^#, it comes that τ_- |F| ≤ |F|^#.



We have the following inequality (cf Remark <ref>).



Let F be a 2-form and k a non-negative integer. Then

    ∀ |β| = k,  |∇^β F|^#≲τ_-^-k |F|^#_𝕂,k.






We also have, according to <cit.>.



Let F be a 2-form and (α, α, ρ, σ) its null decomposition. Then, for all k ∈ℕ,

     ∑_l=0^k∑_i+j=l τ_-^2ir^2j(|∇_L^i ∇_L^j α|^2_𝕆,k-i-j+|∇_L^i ∇_L^j α|^2_𝕆,k-i-j     

    +|∇_L^i ∇_L^j ρ|^2_𝕆,k-i-j+|∇_L^i ∇_L^j σ|^2_𝕆,k-i-j) ≲|F|_𝕂,k^2.    

and

     ∑_l=0^k∑_i+j=l τ_-^2ir^2j(τ_-^2|∇_L^i ∇_L^j α|^2_𝕆,k-i-j+r^2(|∇_L^i ∇_L^j α|^2_𝕆,k-i-j     

    +|∇_L^i ∇_L^j ρ|^2_𝕆,k-i-j+|∇_L^i ∇_L^j σ|^2_𝕆,k-i-j )) ≲( |F|^#_𝕂,k )^2.    




The first inequality is not proved in <cit.> but can be treated similarly as the second one.

The following corollary will be useful, particularly for the massless case in dimension 4, to obtain an extra decay on α away from the light cone.



Using the same notations as in the previous lemma, we have, for F a 2-form,

    |√(τ_-)α|^2_𝕆,k+τ_-^2|∇_r (√(τ_-)α)|^2_𝕆,k-1≲τ_- |F|^2_𝕂,k.





One only has to use that

    |∇_r √(τ_-)| ≤τ_-^-1/2,  2∇_r=L-L

and the previous lemma.



Let us show how to establish pointwise decay estimates on the null decomposition of the electromagnetic field with these inequalities.


Let G be a 2-form and J be a 1-form, both defined on [0,T[ ×^n, such that

    ∇^μ G_μν    =    J_ν, 
    ∇^μ^*  G_μλ_1 ... λ_n-2    =     0.

If G and J are sufficiently regular, we have, for all (t,x) ∈ [0,T[ ×^n,


    |α(G)|(t,x),  |ρ(G)|(t,x)  ,  |σ(G)|(t,x)  ≲√(ℰ_n+2/2[G](t))/τ_+^n+1/2τ_-^1/2,
 

    |α(G)|(t,x) ≲√(ℰ_n+2/2[G](t))/τ_+^n-1/2τ_-^3/2

 and

    |α(G)|(t,x) ≲√(ℰ_n+2/2[G](t)/1+t)+√(ℰ^S_n+2/2[G](t))/τ_+^n-1/2τ_-.




When we will study the massless Vlasov-Maxwell system in dimension n=4, a strong t-loss on ℰ_n+2/2[G] will lead to a strong τ_+-loss on the pointwise estimate (<ref>). Since we will not need all the τ_- decay rate of (<ref>), we will rather use (<ref>).






Let us denote the null decomposition of G by (α, α, ρ, σ). Let (t,x) ∈ [0,T[ ×^n.

First, we consider the case |x| ≤ 1+1/2t.

As

    ∫_Σ_t( |G|^#_𝕂, n+2/2)^2 dx ≲ℰ_n+2/2[G](t),


Lemma <ref> and Remark <ref> give us

    ∑_|β| ≤n+2/2∫_Σ_tτ_-^2|β|+2 |∇^β G|^2 dx ≲ℰ_n+2/2[G](t).

Moreover, 

    ∀ (t,y) ∈ [0,T[ ×ℝ^n such that |y| ≤ 3+3/4t, τ_-(t,y) ≳ 1+t .

Hence,

    ∑_|β| ≤n+2/2∫_|y| ≤ 3+ 3/4t (1+t)^2|β|+2 |∇^β G|^2 dy ≲ℰ_n+2/2[G](t).

Using Lemma <ref>, we obtain

    |G(t,x)| ≲√(ℰ_n+2/2[G](t))/(1+t)^n+2/2.


We consider now the case |x| ≥ 1+1/2t.

According to Lemma <ref>,

     ∑_l=0^1∑_i+j=l∫_|y| ≥1+ 1/2t  τ_-^2ir^2j(τ_-^2|∇_L^i ∇_L^j α|^2_𝕆,n+2/2-i-j+r^2(|∇_L^i ∇_L^j α|^2_𝕆,n+2/2-i-j     

    +|∇_L^i ∇_L^j ρ|^2_𝕆,n+2/2-i-j+|∇_L^i ∇_L^j σ|^2_𝕆,n+2/2-i-j) ) dy  ≲ℰ_n+2/2[G](t).   

Let w be either rα, rρ, rσ or τ_- α. Since ∂_r=L-L/2 and |∂_r(τ_-)| ≤ 1, we have

    ∫_|y| ≥ 1+1/2t |w|^2_𝕆,n+2/2+τ_-^2|∇_∂_r w |_𝕆,n/2^2 dy ≲ℰ_n+2/2[G](t).

Lemma <ref> then gives us

    |w(t,x)| ≲√(ℰ_n+2/2[G](t))/|x|^n-1/2τ_-^1/2.

Thus, 

    |α(t,x)|,  |ρ(t,x)|  ,  |σ(t,x)|  ≲√(ℰ_n+2/2[G](t))/|x|^n+1/2τ_-^1/2and  |α(t,x)| ≲√(ℰ_n+2/2[G](t))/|x|^n-1/2τ_-^3/2.


We now prove (<ref>). Using Corollary <ref>, we have

     ∫_|y| ≥1+1/2t |√(τ_-)α|^2_𝕆,n+2/2+τ_-^2|∇_∂_r (√(τ_-)α)|_𝕆,n/2^2 dy  ≲    


    ∑_|β| ≤n+2/2∫_Σ_tτ_-(|α(ℒ_Z^βG)|^2+|α(ℒ_Z^βG)|^2+|ρ(ℒ_Z^βG)|^2+|σ(ℒ_Z^βG)|^2 )dx.


As, by Definition <ref>,

    ∑_|β| ≤n+2/2∫_Σ_tτ_-(|α(ℒ_Z^βG)|^2+|ρ(ℒ_Z^βG)|^2+|σ(ℒ_Z^βG)|^2 )dx ≲ℰ_n+2/2[G](t)/1+t

and, by Definition <ref>

    ∑_|β| ≤n+2/2∫_Σ_tτ_-|α(ℒ_Z^βG)|^2dx ≲ℰ^S_n+2/2[G](t),

we obtain, again by Lemma <ref>

    |α(t,x)| ≲√(ℰ_n+2/2[G](t)/1+t)+√(ℰ^S_n+2/2[G](t))/|x|^n-1/2τ_-.





Our goal now is to show how to improve the decay estimate on α, in the Lorenz gauge, near the light cone (we cannot reproduce the method used by <cit.> to treat the 3d case). We start by the following lemma.


Let 𝒜 be a sufficiently regular current, defined on [0,T] ×^n, such that 

    ∂^μ𝒜_μ =0 and∀ t ∈ [0,T], ℰ_n+2/2[𝒜](t) ≤ℰ(t),

with ℰ : [0,T] →_+ an increasing function. Then

    |𝒜_L|(t,x),  |𝒜_B|(t,x) ≲ℰ(t)/τ_+^n-1/2τ_-^1/2and |𝒜_L|(t,x) ≲ℰ(t)/τ_+^n/2.




Using a classical L^2-Klainerman-Sobolev inequality, we have, ∀ |γ| ≤ 1,  1 ≤μ≤ n,  (t,x) ∈ [0,T] ×^n,


    |Z^γ𝒜_μ|(t,x) ≲√(ℰ_n+2/2[𝒜](t))/τ_+^n-1/2τ_-^1/2.

We then have

    |(Z^γ𝒜)_L|,  | (Z^γ𝒜)_L|,  |(Z^γ𝒜)_B| ≲√(ℰ(t))/τ_+^n-1/2τ_-^1/2.


It then remains to improve the decay estimate on 𝒜_L near the light cone. Since, ∂^μ𝒜_μ=0,

    (∇^L 𝒜)_L+(∇^L𝒜)_L+(∇^B 𝒜)_B=0.

So, as ∇_L L =0,

    -∇_L𝒜_L-(∇_L𝒜)_L+(∇^B 𝒜)_B =0.


If r ≤t/2 or r ≥t/2 and t ≤ 1, the result comes from (<ref>). For the remaining case, r ≥t/2 and t ≥ 1, note first that

    |L (𝒜_L) | ≲√(ℰ(t))/τ_+^n-1/2τ_-^1/2r≲√(ℰ(t))/τ_+^n+1/2τ_-^1/2.

Indeed, using Remark <ref> and (<ref>), we have

    |(∇_L𝒜)_L|(t,x),  | (∇^B 𝒜)_B|(t,x) ≲√(ℰ(t))/τ_+^n-1/2τ_-^1/2r

so that (using (<ref>)), L (𝒜_L) satisfies also this decay rate. As for a sufficiently regular function g, 
    g(t,r)=g(0,t+r)+∫_u=-t-r^t-rL(g) du,
 and since ℰ is a increasing function, we have

    |𝒜_L|(t,r)    ≤    |𝒜_L|(0,t+r)+∫_u=-t-r^t-r|L (𝒜_L)| du 
       ≲   √(ℰ(0))/τ_+^n/2+√(ℰ(t))/τ_+^n+1/2∫_u=-t-r^t-rτ_-^-1/2 du 
       ≲   √(ℰ(t))/τ_+^n/2.



Finally, we obtain the following pointwise decay on α.


Let G and J be a sufficiently regular 2-form and 1-form (respectively), defined on [0,T] ×^n, such that

    ∇^μ G_μν    =      J_ν, 
    ∇^μ^*  G_μλ_1 ... λ_n-2    =     0.
 
 Let A be a potential of G in the Lorenz gauge such that ℰ_n+4/2[A](t) ≤ℰ(t). We suppose that
 
    |J|(t,x) ≲θ(t)/τ_+^n-1τ_-
 and that ℰ and θ are increasing functions. Then, 
 
    ∀ (t,x) ∈ [0,T] ×ℝ^n,  |α(G)|(t,x) ≲√(ℰ(t))/τ_+^n+2/2+θ(t)log(τ_-)/τ_+^n-1.



the functions ℰ and θ will later be of the form t ↦ (1+t)^a or t ↦log^k(1+t).



We consider a spherical variable B. We have

    α_B(G)     =     (∂_μ A_ν-∂_ν A_μ)_BL
        =     e_B(A)_L-L(A)_B 
        =    ℒ_e_B(A)_L-1/rA_B-L(A)_B,

since e_B(A)_L=ℒ_e_B(A)_L-1/rA_B. Indeed, as e_B can be written as a linear combination of rescaled rotations (namely Ω_ij/r), we only have to prove

    Ω_ij(A)_L=ℒ_Ω_ij(A)_L-1/rA_Ω_ijfor all 1 ≤ i < j ≤ n.

Consider for instance Ω_12. As Ω_12=x^1∂_2-x^2 ∂_1, we have

    ℒ_Ω_12(A)_1=Ω_12 A_1+A_2, ℒ_Ω_12(A)_2=Ω_12 A_2-A_1
 and 
    ℒ_Ω_12(A)_k=Ω_12 A_k if k ≥ 3,

so that

    ℒ_Ω_12(A)_L=Ω_12(A)_L+1/rA_Ω_12.

As 4τ_- ≥τ_+ if t ≥ 2r or t+r ≤ 2, we only have to consider the case[When 4τ_- ≥τ_+, the result comes from Proposition <ref> or from |∂ A| ≲τ_+^-n-1/2τ_-^-3/2∑_Z ∈𝕂 |Z A| (cf Remark <ref>).] where 2r ≥  t and t+r ≥ 2, so that 3r ≥τ_+.
Recall from Lemma <ref> that 

    ∀Ω∈𝕆, ∂^μℒ_Ω A_μ=0.
 So, using Lemma <ref> and that e_B can be written as a linear combination of rescaled rotations, we have

    |ℒ_e_B(A)_L|(t,x) ≲√(ℰ(t))/rτ_+^n/2≲√(ℰ(t))/τ_+^n+2/2.


For the remaining term, rewritting the wave equation (<ref>) satisfied by A in null coordinates, we have, for 0 ≤μ≤ n,

    -LL A_μ +∇^C ∇_C A_μ + 1/r L A_μ-1/rL A_μ = J_μ.

Hence

    L((L+1/r)  A_μ)= ∇^C ∇_C A_μ + 1/r L A_μ+L( 1/r) A_μ-J_μ.

Now, note that, using a classical L^2 Klainerman-Sobolev inequality and Remark <ref>,

    |L( 1/r) A_μ|, | ∇^C ∇_C A_μ| ≲√(ℰ(t))/r^2 τ_+^n-1/2τ_-^1/2,   |1/r L A_μ| ≲√(ℰ(t))/rτ_+^n+1/2τ_-^1/2, |J^μ| ≲θ(t)/τ_+^n-1τ_-

so that, as 3r ≥τ_+,

    |L((L+1/r)  A_μ) | ≲√(ℰ(t))/τ_+^n+3/2τ_-^1/2+θ(t)/τ_+^n-1τ_-.

Hence, as for a sufficiently regular function g, 
    g(t,r)=g(0,t+r)+∫_u=-t-r^t-rL(g) du,

we have (using that ℰ and θ are increasing functions)

    | ( L+1/r)  A_μ|(t,x)    ≲   √(ℰ(0))/τ_+^n+2/2+√(ℰ(t))/τ_+^n+3/2∫_-t-r^t-r1/τ_-^1/2du+θ(t)/τ_+^n-1∫_-t-r^t-r1/τ_-du 
       ≲   √(ℰ(t))/τ_+^n+2/2+θ(t)log(τ_-)/τ_+^n-1,

implying

    | 1/r  A_B+L(A)_B  |(t,x) ≲√(ℰ(t))/τ_+^n+2/2+θ(t)log(τ_-)/τ_+^n-1.



In the context of the Vlasov-Maxwell system, using the null component v^B of the velocity vector, we have a better pointwise estimate on the component J_B of the source term, as J_B is a linear combination of the terms ∫_v v^B/v^0Z^β f dv. Since the dimension n is such that n ≥ 4, we do not need this extra decay (and we then worked with the Cartesian components of the source term in the proof of the previous proposition).




 §.§ A Grönwall inequality


Later, when we will study the velocity support of the scalar field in the massless case, we will need the following variant of Grönwall's lemma.


Let T>0, f and g two continuous nonnegatives functions defined on [0,T] and C ≥ 0. If 
    ∀ t ∈ [0,T],  f(t) ≤ C+2∫_0^t g(s) √(f(s)) ds,

then 
    ∀ t ∈ [0,T],  f(t) ≤( √(C)+∫_0^t g(s) ds )^2.






First, we suppose that C >0. Let F: t ↦ C+2∫_0^t g(s) √(f(s)) ds. We have

    F'(t) ≤ 2g(t)√(F(t)).

Since C>0, F is nonnegative and we can divide by 2√(F(t)). Integrating the above, we obtain

    √(F(t))≤√(C)+∫_0^t g(s) ds,

which implies the result. If C=0, then, for all ϵ >0,

    ∀ t ∈ [0,T],  f(t) ≤ϵ+2∫_0^t g(s) √(f(s)) ds.

It only remains to apply the inequality in the case C ≠ 0 and let ϵ tends to zero.





§ DECAY ESTIMATE FOR THE MASSIVE CASE


Recall that, as we study massive particles in this section, v^0=√(1+|v|^2). We will use the commutation vector fields of  and the weights of 𝐤_1 preserved by the operator T_1 (see Subsections <ref> and <ref> for their definitions). We fix for all this section a sufficiently regular 2-form F defined on [0,T^*[ ×^n and we recall that we defined T_F as the operator

    T_F : g ↦ v^μ∂_μ g + F(v, ∇_v g)

and that ∇_v g= (0, ∂_v^1 g, ..., ∂_v^n g). The main result of this section is the following estimate.


Let T^*>0 and f : [0, T^*[ ×^n_x ×^n_v → be a sufficiently regular function such that

    ∑_z ∈𝐤_1∑_|β| ≤ n∫_Σ_0∫_^n_v| z Z^β f | dv dx   < + ∞,

 Then, for all (t,x) ∈ [0,T^*[ ×^n,

    ∫_^n_v |f(t,x,v)| dv/(v^0)^2≲1/τ_+^n∑_z ∈𝐤_1∑_|β| ≤ n(       ∫_Σ_0∫_^n_v| z Z^β f | dv dx 
            + ∫_0^t∫_Σ_s∫_^n_v| T_F (  z Z^β f ) | dv/v^0 dxds ).




Compared to Theorem 8 in <cit.>, the advantage is that the L^1 norms on the right hand side are taken on {t}×^n (or {0 }×^n) and not on a hyperboloid. On the other hand, our estimate is not a pure Sobolev inequality (we applied the operator T_F to Z^β f to establish it).



To simplify the notation, we took the mass to be 1, but the estimate is true as long as the mass is strictly positive (the constant hidden in ≲ is however proportional to 1/m^2).



As we will need an estimate on ∫_v |f| dv in this article, we will apply Theorem <ref> to (v^0)^2f. Note that, since T_1((v^0)^2 z)=0, the spacetime integral given by Theorem <ref> can be bounded, in that case, by

    ∫_0^t∫_Σ_s∫_^n_v|v^0z T_F (Z^β f ) |+| v^0 F(v, ∇_v z ) Z^β f |+|zv^i F_i0Z^β f | dv dxds.

One can then use commutation formula of Proposition <ref> in order to compute T_F (Z^β f).


The proof is based on a partition of the spacetime. In the interior (|x| ≤t/2) and the exterior (t ≤ |x|) of the light cone, the proof relies on the Klainerman-Sobolev inequality of Theorem <ref>. In the exterior region, the lack of decay is compensated by using the weights (x^i-tv^i/v^0) ∈𝐤_1 defined in Section <ref>. For the remaining region, we work on subsets of ^n+1 composed of a piece of an hyperboloid and a piece of a slice t=constant as <cit.> for the Klein-Gordon equation, mixing what is usually done for such problems.






[color=gray!35] 
(0,0)
– plot [domain=0:3] (, sqrt(*+7))
–(7,4)–(7,0)
–cycle;
[color=black!100] (3,2) node[scale=1.5]D_a(T);
(0,4) node[scale=1.5,left]T;
(0,0)–(0,sqrt(7));
[-Straight Barb[angle'=60,scale=3.5]] (0,sqrt(7))–(0,5);
(3,4)–(7,4);
(0,0)–(7,0) node[scale=1.5,right]Σ_0;
[domain=0:3] plot (, sqrt(*+7));
(-0.1,4)–(0,4);
(0,sqrt(7)) node[scale=1.5,left]a;
(-0.1,sqrt(7))–(0,sqrt(7));
(0,-0.5) node[scale=1.5]r=0;
[align=center,font=, yshift=-1em] (title) 
    at (current bounding box.south)
    The set D_a(T) and its boundary;




 §.§ Sobolev inequalities


We start by a Sobolev inequality independent of time.



Let g : ^n_x ×^n_v → be a sufficiently regular function. Then, for all x ∈^n,


    |x|^n∫_v ∈^n |g(x,v)| dv ≲∑_l | β| ≤ n-1 
     j ≤ 1 ∫_v ∈^n (r ∂_r)^j(|Ω^β g|)(y,v) dv _L^1(|y| ≤ |x|),

where Ω^β∈𝕆^|β|.


During the proof of this lemma, we will use many time the following one dimensional Sobolev inequality. For w ∈ W^1,1, we have, for all a ∈ and all δ≥η >0,

    |w(a)| ≤ C_η (w(y)_L^1(a- δ≤ y ≤ a)+w'(y)_L^1(a- δ≤ y ≤ a)),

with C_η a positive constant depending only on η.


As there is nothing to prove when x = 0, we suppose x ≠ 0. We start by introducing spherical coordinates. A point y ∈^n has for coordinates (r,θ), with r=|y| and θ∈𝕊^n-1. We denote by (|x|,ω) the spherical coordinates of x and by (θ_1,...,θ_n-1) a local coordinate map in a neighbourhood of ω∈𝕊^n-1 (by the symmetry of the sphere, we can suppose that the θ_i take their values in an interval of a size independent of ω). Let h be the function defined by h(r,θ,v)=g(|x|rθ,v). By a one dimensional Sobolev inequality,

     ∫_v ∈^n |h(1,ω,v)| dv  ≲∫_θ_1  |∫_v ∈^n |h|(1,ω_1+θ_1,ω_2,...,ω_n,v)dv |    


    + |∂_θ_1∫_v ∈^n |h|(1,ω_1+θ_1,ω_2,...,ω_n,v) dv | d θ_1 .

As ∂_θ_1 is a linear combination of the rotation vector fields, Remark <ref> gives us

     |∂_θ_1∫_v ∈^n |h|(1,ω_1+θ_1,ω_2,...,ω_n,v) dv | ≲   
 

    ∑_Ω∈𝕆∫_v ∈^n |Ωh|(1,ω_1+θ_1,ω_2,...,ω_n,v) dv.

Thus,

    ∫_v ∈^n |h(1,ω,v)| dv ≲∑_lΩ^β∈𝕆^|β|
     |β| ≤ 1 ∫_θ_1∫_v ∈^n |Ω^βh|(1,ω_1+θ_1,ω_2,...,ω_n,v)dv  d θ_1 .

Using the same argument for the variables θ_2, ..., θ_n-2 and θ_n-1, it comes

    ∫_v ∈^n |h(1,ω,v)| dv ≲∑_lΩ^β∈𝕆^|β|
     |β| ≤ n-1 ∫_θ∈𝕊^n-1∫_v ∈^n |Ω^βh|(1,θ,v)dv  d θ.


The one dimensional Sobolev inequality, applied this time to the first variable, gives 


    ∫_v ∈^n |h(1,ω,v)| dv ≲∑_j ≤ 1∑_lΩ^β∈𝕆^|β|
     |β| ≤ n-1 ∫_1/2^1 | ∂^j_r ∫_θ∈𝕊^n-1∫_v ∈^n |Ω^βh|(r,θ,v)dv  d θ| dr.

Hence, as 1/2≤ r,

     ∫_v ∈^n |h(1,ω,v)| dv ≲   

 
    ∑_j ≤ 1∑_lΩ^β∈𝕆^|β|
     |β| ≤ n-1 ∫_1/2^1 ∫_θ∈𝕊^n-1| (r∂_r)^j  ∫_v ∈^n |Ω^βh|(r,θ,v)dv | d θ  r^n-1dr,

which implies

    ∫_v ∈^n |g(x,v)| dv ≲∑_j ≤ 1, | β| ≤ n-1∫_v ∈^n (r ∂_r)^j(|Ω^β (g(|x|y,v))|) dv _L^1(|y| ≤ 1),

It only remains to remark that, as r∂_r and Ω are homogeneous vector fields, 

    (r ∂_r)^j(|Ω^β(g(|x|y,v) )|)=(r ∂_r)^j(|Ω^β g|)(|x|y,v)

and to make the change of variables y'=|x|y.

We are now able to prove the following time dependent Sobolev inequality.



Let g : [0,T^*[ ×^n_x ×^n_v → a sufficiently regular function. For all (t,x) ∈ [0,T^*[ ×^n such that |x| ≤ t, we have

    |x|^n∫_v ∈^n |g(t,x,v)| dv ≲∑_|β| ≤ nZ^β g (√(|y|^2+a^2),y,v) _L^1(|y| ≤ |x|)L^1_v  ,

with a^2=t^2-|x|^2 and Z^β∈^|β| (more precisely the vector fields involved are either rotations or Lorentz boosts).




Let (t,x) ∈ [0,T^*[ ×^n such that |x| ≤ t and a^2=t^2-|x|^2. We apply the previous lemma to (y,v) ↦ g(√(|y|^2+a^2),y,v) to get


     |x|^n∫_v ∈^n |g(t,x,v)| dv ≲   

 
    ∑_j ≤ 1, | β| ≤ n-1∫_v ∈^n (r ∂_r)^j(|Ω^β g|(√(|y|^2+a^2),y,v)) dv _L^1(|y| ≤ |x|),

where we used that

    Ω^β( g(√(|y|^2+a^2),y,v) ) = Ω^β( g ) (√(|y|^2+a^2),y,v) ,

since Ω(√(|y|^2+a^2))=0 for all Ω∈𝕆.

Now, we remark that

     r ∂_r(|Ω^β g|(√(|y|^2+a^2),y,v))=    

 
    Ω^β g(√(|y|^2+a^2),y,v)/|Ω^β g|(√(|y|^2+a^2),y,v)y^i/√(|y|^2+a^2)Ω_0iΩ^βg(√(|y|^2+a^2),y,v).

Note also that, droping the dependance in (√(|y|^2+a^2),y,v) of the functions considered,

    ∫_v ∈^nΩ^β g/|Ω^β g|y^i/√(|y|^2+a^2)v^0∂_v^iΩ^βgdv=-∫_v ∈^nv^i/v^0y_i/√(|y|^2+a^2)|Ω^βg| dv.

It then comes that

     |x|^n∫_v ∈^n |g(t,x,v)| dv ≲   

 
    ∑_| β| ≤ n( 1+|y|/√(|y|^2+a^2))  ∫_v ∈^n |Z^β g|(√(|y|^2+a^2),y,v) dv _L^1(|y| ≤ |x|),

which allows us to deduce the result.




 §.§ An energy estimate


Before starting the proof of Theorem <ref>, we establish the following lemma, which combined with our last Sobolev inequality, will give us the expected decay on the velocity average of the Vlasov field for a spacetime region.



Let a>0, T ∈ ]a,T^*[ and g: [0,T^*[ ×^3_x ×^3_v → be a sufficiently regular function. Then,

    ∫_|y| ≤√(T^2-a^2)∫_^n_v |g(√(|y|^2+a^2),y,v)| dv/(v^0)^2 dy    ≤    2∫_Σ_0∫_^n_v |g| dx dv 
           +2∫_0^T ∫_Σ_s∫_^n_v| T_F(g) |dv/v^0 dxds.




We use again the vector field N^μ(g):=∫_^n_v g v^μ/v^0 dv and recall from (<ref>) that

    ∂_μ N^μ(|g|)= ∫_v ∈^ng/|g|T_F(g)/v^0-g/|g|F( v/v^0,∇_v g ) dv= ∫_v ∈^ng/|g|T_F(g) dv/v^0.

We now introduce the following subset of _+ ×^n :

    D_a(T)= {(s,y) ∈ℝ_+ ×ℝ^n  /  a^2 ≥ s^2-|y|^2,  0≤ s ≤ T } .

Denoting by ν is the outward pointing unit normal field to ∂ D_a(T), the divergence theorem (in W^1,1, for the euclidian space ^n+1) gives us

    ∫_∂ D_a(T)ν_μ N^μ(|g|) d ∂ D_a(T) = ∫_D_a(T)∫_v ∈^ng/|g| T_F(g) dv/v^0dy d D_a(T).

The boundary term is equal to 

     ∫_|y| ≤√(T^2-a^2) ∫_v ∈^n ν_μ v^μ/v^0|g|(√(|y|^2+a^2),y,v) dv d λ(y)    

 
    +g_L^1_x(|y| ≥√(T^2-a^2))L^1_v(T)-g_L^1_xL^1_v(0),

where dλ(y) is the surface measure on the hyperboloid {s^2-|y|^2=a^2 }. More precisely, on this hyperboloid[Here, ^ty denotes the transpose of y.],

    d λ(y)= √( det( I_n+1/|y|^2+a^2 ^tyy ) ) dy=√(2|y|^2+a^2/|y|^2+a^2)
 and 
    ν(y)=1/√(2|y|^2+a^2)(√(|y|^2+a^2),-y) .

We then deduce, as D_a(T) ⊂ [0,T] ×^n,

    ∫_|y| ≤√(T^2-a^2)∫_^n_v(√(|y|^2+a^2)v^0- y_iv^i )|g|(√(|y|^2+a^2),y,v) dvdy/v^0√(|y|^2+a^2)≤
 
    g_L^1_xL^1_v(0) +∫_0^T ∫_Σ_s∫_^n_v| T_F(g) | dv/v^0.

Finally, note that for s=√(|y|^2+a^2)≥ |y|, 
    sv^0-y_iv^i/s≥sv^0-|y||v|/s≥ s(v^0-|v|)(v^0+|v|)/s(v^0+|v|)≥1/2v^0 .

The result follows from a combination of the last two inequalities.



The lemma is also valid on the cone s=|y|, which means that the result is true for a=0, but we already knew it with Proposition <ref>. 




 §.§ Proof of Theorem <ref>


We consider a partition of the spacetime into four regions.


  * The bounded region, t+|x| ≤ 2, where a standard Sobolev inequality gives the result.

  * The interior of the light cone, where |x| ≤t/2.

  * The exterior of the light cone, where t ≤ |x| and |x| ≥ 1.

  * The remaining region where t/2≤ |x| ≤ t and t ≥ 1.






  §.§.§ The interior of the light cone




Let (t,x) ∈ [0,T^*[ ×^n such that |x| ≤t/2. Thus, τ_- ≥1/3τ_+ and the Klainerman-Sobolev inequality of Theorem <ref> gives

    ∫_v ∈^n |f(t,x,v)| dv ≲1/τ_+^n∑_|β| ≤ nZ^β f _L^1_x,v (t).

It only remains to apply Proposition <ref>, which gives us

    Z^β f _L^1_x,v (t) ≲Z^β f _L^1_x,v (0)+ ∫_0^t ∫_Σ_s∫_^n_v| T_F ( Z^β f ) | dv/v^0 dx ds.



  §.§.§ The exterior of the light cone



We use (x^i-tv^i/v^0) ∈𝐤_1, for 1 ≤ i ≤ n, which are solutions to the homogeneous relativistic transport equation. Let (t,x) ∈ [0,T^*[ ×^n such that t ≤ |x| and |x| ≥ 1. By Theorem <ref>, we have

    ∫_^n_v| x^i-tv^i/v^0| |f|(t,x,v) dv  ≲1/τ_+^n-1∑_z ∈𝐤_1∑_|β| ≤ nz Z^β f_L^1_x,v(t).

Since |xv^0-tv| ≥ v^0|x|-t|v| ≥|x|/2v^0, it comes

    |x|  ∫_v |f|(t,x,v) dv/(v^0)^2   ≲   ∫_v | x-tv/v^0| |f|(t,x,v) dv  
       ≲   ∑_i=1^n ∫_v | x^i-tv^i/v^0||f|(t,x,v) dv .

Hence, using that |x| ≳τ_+ (recall that |x| ≥ 1 and t ≤ |x| in the region studied) and applying Proposition <ref>, we finally obtain

    ∫_v  |f|(t,x,v) dv/(v^0)^2   ≲   1/|x|τ_+^n-1∑_z ∈𝐤_1∑_|β| ≤ nz Z^β f_L^1_x,v(t) 
       ≲   1/τ_+^n∑_  |β| ≤ n 
     z ∈𝐤_1  z Z^β f _L^1_x,v (0)  + ∫_0^t ∫_Σ_s∫_v| T_F (z Z^β f ) | dv/v^0 dx ds.



  §.§.§ The remaining region 


Let (t,x) ∈ [0,T^*[ ×^n such that t/2≤ |x| ≤ t and t ≥ 1.
We start by applying Lemma <ref> to Z^β f, for all |β| ≤ n, with T=t and a^2=t^2-|x|^2. We have


     ∑_|β| ≤n ∫_|y| ≤|x| ∫_^n_v |Z^β f(√(|y|^2+a^2),y,v)| dv/(v^0)^2 dy ≲   

 
    ∑_|β| ≤ n( ∫_Σ_0∫_^n_v| Z^βf | dv dx+∫_0^t ∫_Σ_s∫_^n_v T_F(Z^βf) dv/v^0 dx ds ).

As |Z^γ( (v^0)^-2) | ≲ (v^0)^-2, Lemma <ref> applied to g=(v^0)^-2 f allows us to bound by below the left hand side of the previous inequality by

    |x|^n ∫_v |f|(t,x,v) dv/(v^0)^2 .

The result follows from |x|^n ≳τ_+^n (as |x| ≥t/2≥1/2).



 §.§ Improved decay for the derivatives of the velocity averages


Let us introduce the following vector fields.


For 1 ≤ i ≤ n and 1 ≤ k,l ≤ n, with k ≠ l, we consider

    X_i=v^i/v^0∂_t+∂_i and Y_kl=v^k/v^0∂_l-v^l/v^0∂_k.





The vector fields 1/v^0T_1, X_i and Y_kl are good derivatives (as the derivates tangential to the light cone L and e_B, see Remark <ref>), which means that if W denotes one of them, we have, for a smooth function f,

    |∫_v Wf dv| ≲1/τ_+(∑_Z∈| ∫_v Z f dv |+∑_z ∈𝐤_1∫_v |z| |∇_t,x f| dv)




For T_1, we remark that

    tT_1=v^0S+(tv^i-x^iv^0)∂_i,  rT_1=tT_1+(r-t)T_1

and that 
    | (r-t) ∫_v ∂ f dv | ≲∑_Z∈𝕂| ∫_v Z f dv | ≲∑_lZ^β∈^|β|
     |β| ≤ 1 | ∫_v Z^β f dv |.
 For X_i, that ensues from

    tv^0X_i=v^0 Ω_0i+(tv^i-x^iv^0)∂_t and rX_i=tX_i +(r-t)X_i.

For Y_kl, that follows from

    tv^0Y_kl=v^0Ω_kl+(tv^k-x^kv^0)∂_l-(tv^l-x^lv^0)∂_k and rY_kl=tY_kl +(r-t)Y_kl.



Finally, let us show how we can obtain extra decay on ∂∫_v f dv if f solves an equation such as T_F(f)=0.

Let f : [0,T] ×^n_x ×^n_v → be a function such that

    ∀Z∈,  z ∈𝐤_1, | ∫_v (v^0)^2z Z fdv| ≲τ_+^-n .

Then, for all 0 ≤μ≤ n,

    | ∂_μ∫_v f dv | ≲τ_+^-n-1.




As 
    T_1=v^μ∂_μ=v^0 ∂_t +v^iX_i-|v|^2/v^0∂_t=v^iX_i+1/v^0∂_t,

we have

    ∂_t=v^0T_1-v^0v^iX_i.

Similarly

    ∂_i=(v^0)^2X_i-v^iT_1-v^0v^kY_ki.




We can prove a similar proposition for derivatives of higher orders.






§ THE MASSIVE VLASOV-MAXWELL EQUATIONS




 §.§ Global existence for small data


The aim of this section is to prove Theorem <ref>. We suppose that the dimension n is at least 4 and we consider the massive Vlasov-Maxwell system (<ref>)-(<ref>) with at least two species, so that K ≥ 2. For simplicty, we suppose that m_k=1 for all 1 ≤ k ≤ K.

To simplify the notation, we denote during this chapter the energy norm 𝔼^k_M,q,1, introduced previously in Definition <ref>, by 𝔼^k_M,q . We also introduce the function χ defined on _+ by

    χ ( s )= log^3(3+s)if n=4 andχ(s) = 1 if n ≥ 5.
 

This is a more precise version of Theorem <ref>.



Let n ≥ 4, K ≥ 2 and N ≥5/2n+1. Let (f_0,F_0) be an initial data set for the massive Vlasov-Maxwell system. Let (f,F) be the unique classical solution to the system and let A be a potential in the Lorenz gauge. There exists ϵ >0 such that[A smallness condition on F, which implies ℰ_N[A](0) ≤ϵ, is given in Proposition <ref>.], if 

     ℰ_N[A](0) ≤ϵ,  ℰ_N[F](0) ≤ϵ   

and if, for all 1 ≤ k ≤ K,

     𝔼^2_N+n,1[f_k](0) ≤ϵ,    
 
then (f,F) exists globally in time and verifies the following estimates. 


  * Energy bounds for A, F and f_k: ∀ 1 ≤ k ≤ K and ∀ t ∈ℝ_+,

     ℰ_N[A](t) ≲ϵχ(t),  ℰ_N[F](t) ≲ϵχ(t),     

     𝔼^2_N[f_k](t) ≲ϵ and   𝔼^2_N,1[f_k](t) ≲ϵχ^1/6(t).    



  * Pointwise decay for the null decomposition of ℒ_Z^β(F): ∀ |β| ≤ N-n, (t,x) ∈ℝ_+ ×ℝ^n,

     |α( L_Z^β F)| ≲√(ϵ) √(χ(t))τ_+^-n+2/2,  |α ( L_Z^β F)| ≲√(ϵ) √(χ(t))τ_+^-n-1/2τ_-^-3/2,     

     |ρ(L_Z^β F)| ≲√(ϵ) √(χ(t))τ_+^-n+1/2τ_-^-1/2,  |σ(L_Z^β F)| ≲√(ϵ) √(χ(t))τ_+^-n+1/2τ_-^-1/2.    


  * Pointwise decay for ∫_v ∈ℝ^n  | Z^β f_k | dv: 
    ∀ |β| ≤ N-3n+2/2,  (t,x) ∈ℝ_+ ×ℝ^n, ∫_v ∈ℝ^n   | Z^β f_k | dv ≲ϵ/τ_+^n.


  * Pointwise decay for ∫_v ∈ℝ^n   | Z^β f_k |(v^0)^2 dv and ∫_v ∈ℝ^n   |zZ^β f_k | (v^0)^2dv: 

    ∀ |β| ≤ N-n,  (t,x) ∈ℝ_+ ×ℝ^n, ∫_v ∈ℝ^n  | Z^β f _k|(v^0)^2 dv ≲ϵ/τ_+^n-1τ_-,


    ∀ |β| ≤ N- 3n+2/2,  z ∈𝐤_1,  (t,x) ∈ℝ_+ ×ℝ^n,
 
    ∫_v ∈ℝ^n  |z Z^β f_k | (v^0)^2dv ≲ϵ/τ_+^n-1τ_-.


  * L^2 estimates on ∫_v ∈ℝ^n  | Z^β f_k | dv:

    ∀ |β| ≤ N,  t ∈ℝ_+, ∫_v ∈ℝ^n   | Z^β f_k | dv _L^2(Σ_t)≲ϵχ^1/6(t)/(1+t)^n/2.





In dimension 4, if N ≥ 14, we can take χ(t)=log^2 (3+t) and avoid the log^1/2(3+t)-loss on the L^2 estimate on ∫_v |Z^β f_k| dv.




 §.§ Structure and beginning of the proof


Let (f_0,F_0) be an initial data set satisfying the assumptions of Theorem <ref>. By a standard local well-posedness argument, there exists a unique maximal solution (f,F) of the massive Vlasov-Maxwell system defined on [0,T^*[, with T^* ∈_+^* ∪{+ ∞}.  

We consider the following bootstrap assumptions. Let T be the largest time such that, ∀ 1 ≤ k ≤ K and ∀ t ∈ [0,T],


    ℰ_N[F](t) ≤ 2Cϵχ(t), ℰ_N^S[F](t) ≤ 2Cϵ,


    𝔼^2_N[f_k](t) ≤ 4ϵ, 𝔼^2_N-n+2/2,1[f_k](t) ≤ 4ϵand𝔼^2_N,1[f_k](t) ≤ 4ϵχ^1/6(t),


where C and C are positive constants which will be specified during the proof. Note that by continuity, T>0. We now present our strategy to improve these bootstrap assumptions.



  * First, using the bootstrap assumptions, we obtain decay estimates for the null decomposition of F (and its Lie derivatives) and for velocity averages of derivatives of f_k.


  * Next, we improve the bounds on the Vlasov fields energies by means of the energy estimates proved in Propositions <ref> and <ref>. To bound the right hand side in these energy estimates, we make fundamental use of the null structure of the system and the pointwise decay estimates on ρ, σ, α, α and ∫_v ∈ℝ^n   |z Z^β f_k | dv.


  * Then, using Theorem <ref>, we improve the decay estimate on ∫_v  |Z^β f_k | dv near the light cone.


  * In order to improve the estimates on the electromagnetic field energies, we establish an L^2_x estimate for the velocity averages of the Vlasov fields (and its derivatives). For this purpose, we follow <cit.> and we rewrite all the transport equations as an inhomogeneous system of transport equations. The velocity averages of the homogeneous part of the solution verify strong pointwise decay estimates (we use particularly the control that we have at our disposal on the initial data of f, for derivatives of order N+n or less). The inhomegeneous part is decomposed into a product of an integrable function and a pointwise decaying function which gives us the expected estimate.


  * Finally, we bound the energy of the electromagnetic potential (which satisfy the Lorenz gauge) and we improve the estimates on the electromagnetic field energies with the energy estimates for the Maxwell equations (Propositions <ref> and <ref>). We use again the null decomposition of F (and its Lie derivatives), which, combined by the estimates on τ_+ ∫_ℝ^n   | Z^β f_k | dv _L^2_x, gives us the improvement.





 §.§ Step 1: Decay estimates



Using the bootstrap assumption on ℰ_N[F] and Proposition <ref>, one immediately obtains the following pointwise decay estimates on the electromagnetic field.



For all t ∈ [0,T], |β| ≤ N- n+2/2, we have

    |α ( ℒ_Z^β F)| ≲√(ϵ)√(χ(t))τ_+^-n+1/2τ_-^-1/2,  |α ( ℒ_Z^α F)| ≲√(ϵ)√(χ(t))τ_+^-n-1/2τ_-^-3/2,


    |ρ (ℒ_Z^α F)| ≲√(ϵ)√(χ(t))τ_+^-n+1/2τ_-^-1/2,  |σ (ℒ_Z^α F)| ≲√(ϵ)√(χ(t))τ_+^-n+1/2τ_-^-1/2.






We will improve later the decay estimate on α(ℒ_Z^β F), for |β| ≤ N-n, near the light cone (see Section <ref>).



The pointwise decay estimates on the velocity averages of the Vlasov fields are given by Klainerman-Sobolev inequalities and the bootstrap assumptions on the f_k energy norms. Using Theorem <ref>, we have that ∀|β| ≤ N-n, (t,x) ∈ [0,T] ×^n, 1 ≤ k ≤ K


    ∫_^n |Z^β f_k|(v^0)^2 dv ≲𝔼^2_N[f_k](t)/τ_+^n-1τ_-≲ϵ/τ_+^n-1τ_-.


In the same spirit, using Corollary <ref>, we have that ∀ |β| ≤ N-3n+2/2, z  ∈𝐤_1, (t,x) ∈ [0,T] ×^n,


    ∫_^n |zZ^β f_k| (v^0)^2dv ≲𝔼^2_N-n+2/2,1[f_k](t)/τ_+^n-1τ_-≲ϵ/τ_+^n-1τ_-,





 §.§ Step 2: Improving the energy estimates for the transport equation


We fix, for this section, 1 ≤ k ≤ K. According to Proposition <ref>, 𝔼^2_N[f_k] ≤ 3 ϵ on [0,T], for ϵ small enough, would follow if we prove

    ∫_0^t ∫_Σ_s∫_v ∈^n |ℒ_Z^β_1(F)(v,∇_v Z^β_2f_k)| v^0dv dx ds ≲ϵ^3/2,

for all |β_1|+|β_2| ≤ N, with |β_2| ≤ N-1, and
 
    ∫_0^t ∫_Σ_s∫_v ∈^n |v^iF_i0Z^β f_k|   dvdxds ≲ϵ^3/2,

 for all |β| ≤ N. The second integral is easy to bound. Using Proposition <ref> and the bootstrap assumption on 𝔼^2_N[f_k], we have
 
    ∫_0^t ∫_Σ_s∫_^n |v^iF_i0Z^β f_k|   dvdxds    ≲   ∫_0^t F _L^∞(Σ_s)𝔼^2_N[f_k](s) ds 
       ≲   ϵ^3/2.

Similarly, according to Proposition <ref>, 𝔼^2_N,1[f_k] ≤ 3 ϵχ^1/6 (t) on [0,T], for ϵ small enough, would follow if we prove

    ∫_0^t ∫_Σ_s∫_v ∈^n |zv^0ℒ_Z^β_1(F)(v,∇_v Z^β_2f_k)| dv dx ds ≲ϵ^3/2χ^1/6 (t),

for all z ∈𝐤_1 and |β_1|+|β_2| ≤ N, with |β_2| ≤ N-1,

    ∫_0^t ∫_Σ_s∫_v ∈^n |v^0F(v,∇_v z)Z^β f_k| dvdxds ≲ϵ^3/2χ^1/6 (t),

for all z ∈𝐤_1, |β| ≤ N and

    ∫_0^t ∫_Σ_s∫_v ∈^n |zv^iF_i0Z^β f_k|   dvdxds ≲ϵ^3/2,

for all |β| ≤ N. Again, the last integral is easy to bound.


We fix |β_1|+|β_2| ≤ N (with |β_2| ≤ N-1), |β| ≤ N and z ∈𝐤_1. We denote respectively ρ(ℒ_Z^β_1(F)), σ(ℒ_Z^β_1(F)), α(ℒ_Z^β_1(F)) and α(ℒ_Z^β_1(F)) by ρ, σ, α and α. We denote also Z^β_2f_k by g and Z^β f_k by h. To unify the study of the remaining integrals, we introduce b, which could be equal to 0 or 1, z_0=v^0 and z_b=v^0z. The null decomposition of ℒ_Z^β_1(F)(v,∇_v g) (for (<ref>) and (<ref>)) or F(v,∇_v z) (for (<ref>)) brings us to control the integral of the following terms.

The good terms 


    |z_bv^Lρ( ∇_v g )^L|, |v^0v^L hρ(F) ( ∇_v z )^L|,



    |z_bv^L ρ( ∇_v g )^L|, |v^0v^L h ρ(F) ( ∇_v z )^L|,



    |z_bv^B σ_BD( ∇_v g )^D|, |v^0v^B h σ(F)_BD( ∇_v z )^D|,



    |z_bv^L α_B ( ∇_v g )^B|, |v^0v^L h α(F)_B ( ∇_v z )^B |,



    |z_bv^B α_B ( ∇_v g )^L|, |v^0v^B h α(F)_B ( ∇_v z )^L|,


and the bad terms 


    |z_bv^Lα_B ( ∇_v g )^B|, |v^0v^L  hα(F)_B ( ∇_v z )^B|,



    |z_bv^B α_B ( ∇_v g )^L|, |v^0v^B h α(F)_B ( ∇_v z )^L|.


The study of 𝔼^2_N[f_k] corresponds to b=0 and, in this case, we only have to estimate the spacetime integral of each of the first terms of (<ref>)-(<ref>). The study of 𝔼^2_N,1[f_k] corresponds to b=1. For both of them, when |β_1| ≤ N-n+2/2 we can use the pointwise decay estimates on the electromagnetic field given by Proposition <ref>. When |β_1| > N-n+2/2, |β_2| ≤ N-3n+2/2 (since N ≥5/2n+1), and we can then use the pointwise estimates (<ref>) and (<ref>) on the velocity averages of the Vlasov field.

For the part where |β_1| ≤ N-n+2/2, our proof leads also to 𝔼_N-n+2/2,1^2[f_k] ≤ 3 ϵ, for ϵ small enough, on [0,T].



To simplify the argument we will sometimes denote 𝔼^2_N[f_k] by 𝔼^2_N,0[f_k].






  §.§.§ Estimating the v derivatives


To deal with the v derivatives of the Vlasov field, which do not commute with the relativistic transport operator, we recall (<ref>)


    | ( ∇_v ψ)^L|, | (∇_v ψ)^L|, | ( ∇_v ψ)^B | ≲τ_+/v^0∑_Z∈ |Zψ|.

We will also use

    | ( ∇_v ψ)^L|, | ( ∇_v ψ)^L| ≲τ_-/v^0∑_Z∈|Zψ|

and

    | v^L( ∇_v ψ)^B  | ≲τ_-∑_Z∈ℙ∑_z ∈𝐤_1  |zZψ | ,

which come from Lemma <ref> and Proposition <ref>. In order to reutilize certain estimates of this section, we will not use inequalities (<ref>) and (<ref>) in the case where we have a pointwise estimate on the electromagnetic field. We make this choice because we do not identify such null structures in the equations studied in Section <ref>, where we will make similar computations as in Subsection <ref>.




  §.§.§ If |β_1| ≤ N-n+2/2



We start by treating the good terms. We use ζ to denote α, ρ or σ. Thus, according to Proposition <ref>,

    |ζ| ≲√(ϵ)√(χ(t))/τ_+^n+1/2τ_-^1/2.

Using (<ref>), we can bound by ∑_Z∈τ_+|ζ||z_bZg| each first term of (<ref>)-(<ref>) so that their integrals on [0,t] ×ℝ^n_x ×ℝ^n_v are bounded by


    ∑_Z∈ℙ_0∫_0^t ∫_Σ_sτ_+ |ζ| ∫_v |z_bZg| dv dx ds.

It remains to notice that 
    ∫_0^t ∫_Σ_sτ_+ |ζ| ∫_v |z_bZg| dv dx ds ≲∫_0^t √(ϵ)log^3/2 (3+s)/(1+s)^3/2𝔼^1_N,b[f_k](s) ds ≲ϵ^3/2,

since 𝔼^1_N,b[f_k](s) ≤𝔼^2_N,1[f_k](s) ≤  4 ϵlog^1/2(3+s) for all s ∈ [0,T]. Similarly, each second term of (<ref>)-(<ref>) is bounded by ∑_Z∈v^0τ_+|ζ(F)| |h| |Z(z)| and, using Lemma <ref>, their integral on [0,t] ×ℝ^n_x ×ℝ^n_v are bounded by

    ∑_z' ∈𝐤_1∫_0^t ∫_Σ_sτ_+ |ζ(F)| ∫_v v^0|z'h| dv dxds.

Using the pointwise estimate on ζ and the bootstrap assumption (<ref>), one has

    ∫_0^t ∫_Σ_sτ_+ |ζ(F)| ∫_v v^0|z'h| dv dxds ≲∫_0^t √(ϵ)log^3/2 (3+s)/(1+s)^3/2𝔼^1_N,1[f_k](s) ds ≲ϵ^3/2.

We now study the bad terms. Recall that, according to Proposition <ref>,

    |α| ≲√(ϵ)√(χ(t))/τ_+^n-1/2τ_-^3/2.

Let us denote [0,t] ×_x^n ×_v^n by X_t and dvdxds by dX_t. Then, using (<ref>), ∫_C_u(t)∫_v v^L|z_b Z g| dv d C_u(t)  ≤𝔼^2_N,b[f_k](t) and the bootstrap assumption (<ref>), we have,

    ∫_X_t |z_bv^Lα_B ( ∇_v g )^B|dX_t    ≲   ∑_Z∈ℙ_0∫_0^t ∫_Σ_sτ_+ |α| ∫_v v^L/v^0|z_b Z g| dvdxds 
       ≲   ∑_Z∈ℙ_0∫_u=-∞^t∫_C_u(t)τ_+|α|∫_v v^L|z_b Z g| dv dC_u(t) du 
       ≲   ∑_Z∈ℙ_0∫_u=-∞^t1/τ_-^3/2∫_C_u(t)∫_v v^L| z_bZ g| dv dC_u(t) du 
       ≲   ϵ^1/2𝔼^2_N,b[f_k](t)∫_u=-∞^+∞1/τ_-^3/2 du 
       ≲   ϵ^3/2if b=0, ϵ^3/2χ^1/6(t) if b=1.

Finally, for the first term of (<ref>), we use successively (<ref>), the inequality |v^B| ≲ v^0 v^L (which ensues from Proposition <ref>) as well as the bootstrap assumptions (<ref>) to get


    ∫_X_t | z_bv^B α_B ( ∇_v g )^L|dX_t    ≲   ∑_Z∈ℙ_0∫_0^t ∫_Σ_s∫_v |α| v^0 v^Lτ_+/v^0|z_b  Z g| dvdxds 
       ≲   ∑_Z∈∫_u=-∞^t∫_C_u(t)τ_+|α| ∫_v v^L |z_b Z g| dv d C_u(t) du 
       ≲   ϵ^1/2∫_u=-∞^tτ_-^-3/2𝔼^2_N,b[f_k](t) du 
       ≲   ϵ^1/2𝔼^2_N,b[f_k](t).

The integrals of the second terms of (<ref>) and (<ref>) are treated similarly. For instance, as ∑_Z∈ |Z(z)| ≲∑_z' ∈𝐤_1 |z'|, we have


    ∫_X_t |v^0 v^B hα_B ( ∇_v z )^L|dX_t    ≲   ∑_Z∈ℙ_0∫_0^t ∫_Σ_sτ_+ |α| ∫_v √(v^Lv^L) |h  Z (z)| dvdxds 
       ≲   ∑_z' ∈𝐤_1∫_-∞^t∫_C_u(t)τ_+ |α| ∫_v v^Lv^0 |z' h| dv d C_u(t) du 
       ≲   ϵ^1/2𝔼^2_N,1[f_k](t).




  §.§.§ If |β_1| > N-n+2/2


In this case we cannot use Proposition <ref> anymore. As |β_2| ≤n/2, we can however use the pointwise estimates on the velocity averages of v^0z_bZ^βg given by (<ref>). This time, we only have to bound the first terms of (<ref>)-(<ref>).

Again, we start by studying the good terms. Let us denote again α, ρ or σ by ζ . Then, according to Definition <ref>, for all s ∈ [0,T],


    ∫_Σ_sτ_+^2 |ζ|^2 dx ≤ℰ_N[F](s) ≲ϵχ(s).

Recall that the integral of each first term of (<ref>)-(<ref>) can be bounded by (<ref>). Using the Cauchy-Schwarz inequality and 
    ∫_v | z_bZ g| dv ^2_L^2(Σ_s)≲ϵ^2 ∫_0^+∞r^n-1/τ_+^2n-2τ_-^2dr ≲ϵ^2 (1+s)^-(n-1),

which comes from (<ref>) and Lemma <ref>, we have 

    ∫_0^t ∫_Σ_sτ_+ |ζ| ∫_v |z_b Z g| dv dx ds    ≲   ∫_0^t τ_+ ζ_L^2(Σ_s)∫_v |z_b Z g| dv _L^2(Σ_s) ds 
       ≲   ϵ^3/2.


In order to close the estimates for the bad terms, we use (<ref>) or (<ref>). The integral of the first term of (<ref>) is then bounded by

    ∑_Z∈ℙ_0∫_0^ + ∞τ_- α_L^2(Σ_s)∫_v |v^B/v^0z_bZ g | dv _L^2(Σ_s) ds.
 
Now, using (<ref>) and Lemma <ref>, we have

    ∫_v |v^B/v^0z_bZ g| dv _L^2(Σ_s)≲∫_v |z_bZ g| dv _L^2(Σ_s)≲ϵ (1+s)^-3/2.

Since τ_- α^2_L^2(Σ_s)≤ℰ_N[F](s) ≲ϵχ(t),

    ∫_0^t ∫_Σ_s∫_v |z_bv^B α_B ( ∇_v g )^L|dvdxds ≲ϵ^3/2.

For the remaining term, z_b v^Lα_B(∇_v g )^B, we treat the two cases separately. First, if b=0, ∫_0^t ∫_Σ_s∫_v v^0 v^L |α_B(∇_v g )^B|dvdxds is bounded by


    ∑_Z∈ℙ_0∑_z' ∈𝐤_1∫_0^ + ∞τ_- α_L^2(Σ_s)∫_v v^0|z'Zg| dv _L^2(Σ_s)ds .
 
Now, using (<ref>) and Lemma <ref>, we have


    ∫_v v^0|z'Zg| dv ^2_L^2(Σ_s)≲ϵ^2 (1+s)^-(n-1).

Hence, as τ_- α^2_L^2(Σ_s)≤ℰ_N[F](s) ≲ϵχ(t), we obtain

    ∫_0^t ∫_Σ_s∫_v v^0 v^L |α_B(∇_v g )^B|dvdxds ≲ϵ^3/2.


Finally, if b=1, we have, by (<ref>), 

     ∫_0^t ∫_Σ_s ∫_v  v^0v^L |zα_B(∇_v g )^B|dvdxds ≲   


    ∑_Z∈∫_0^t ∫_Σ_sτ_-^1/2/(1+s)^n-3/2|α|τ_+(1+s)^n-3/2/τ_-^1/2∫_v v^L| zZ g | dv dx ds.


By the Cauchy-Schwarz inequality (in (s,x)), the right-hand side of the previous inequality is bounded by

    ( ∫_0^t √(τ_-) |α| ^2_L^2(Σ_s)/(1+s)^n-3 ds )^1/2∑_Z∈( ∫_0^t ∫_Σ_sτ_+^n-1/τ_-( ∫_v v^L |z Zg| dv )^2 dx ds )^1/2.

By the bootstrap assumption[Note that if we used the bound on τ_- α_L^2(Σ_s) we would have in 4d an extra loss on 𝔼^2_N,1[f_k] which would lead to a (1+t)^η-loss for the electromagnetic energy.] (<ref>), √(τ_-) |α| ^2_L^2(Σ_s)≲ϵ, so
 
    ∫_0^t √(τ_-) |α| ^2_L^2(Σ_s)/(1+s)^n-3 ds ≲ϵχ^1/3(t).

The second factor of (<ref>) is bounded by ϵ^2. Indeed, as, by (<ref>),

    τ_+^n-1/τ_-( ∫_v v^L |z Zg| dv )^2 ≲ϵ/τ_-^2∫_v v^L |z Zg| dv,

we have

    ∫_0^t ∫_Σ_sτ_+^n-1/τ_-( ∫_v v^L |z Zg| dv )^2 dx ds    ≲   ϵ∫_-∞^t τ_-^-2∫_C_u(t)∫_v v^L|z Zg| dv d C_u(t)du 
       ≲   ϵ∫_u=-∞^t τ_-^-2𝔼^2_N-n+2/2,1[f_k](t) du 
       ≲   ϵ^2,

since 𝔼^2_N-n+2/2,1[f_k](t) ≤ 4 ϵ by the bootstrap assumption (<ref>). Thus

    ∫_0^t ∫_Σ_s∫_v  v^0v^L |zα_B(∇_v g )^B|dvdxds ≲ϵ^2 χ^1/6(t).

This concludes the improvement of the bootstrap assumption (<ref>).



 §.§ Step 3: Improved decay estimates for velocity averages

In this section, we improve the pointwise decay estimate on ∫_v |Z^β f_k| dv near the lightcone.

We have, for all 1 ≤ k ≤ K,

    ∀ (t,x) ∈ [0,T] ×^n,  |β| ≤ N- 3n+2/2, ∫_v ∈ℝ^n |Z^β f_k| dv ≲ϵ/τ_+^n


and

    ∀ (t,x) ∈ [0,T] ×^n,  |β| ≤ N-n, ∫_v ∈ℝ^n |Z^β f_k| dv ≲ϵχ^1/6(t)/τ_+^n.




This ensues from Theorem <ref>, Remark <ref> and the estimations made in Section <ref>. The loss for the derivatives of higher order is linked to the loss on 𝔼^2_N,1[f_k].





 §.§ Step 4: L^2 estimates for the velocity averages


In view of commutation formula of Propositions <ref> and the energy estimates of Propositions <ref>, <ref>, we need to prove enough decay on τ_+ ∫_ℝ^n   |Z^β f_k | dv _L^2_x for all |β| ≤ N. The goal of this section is to prove the following proposition.


We have, for all 1 ≤ k ≤ K, |β| ≤ N and for all t ∈ [0,T],

    τ_+ ∫_ℝ^n |Z^β f_k | dv _L^2(Σ_t)≲ϵχ^1/6(t)/(1+t)^n/2-1 .

The log^1/2(3+t)-loss (specific to the dimension 4) can be removed for |β| ≤ N-3n+2/2 or improved in a log^1/4(3+t)-loss for |β| ≥ N-n+1.


Note that if |β| ≤ N-n, that ensues from Proposition <ref> and Lemma <ref>. For the higher order derivatives, we follow the strategy used in <cit.>, in Section 4.5.7, to prove similar L^2 estimates. Let[If n =4 and N ≥ 14, we can take 8 ≤ M ≤ N-6 and avoid the log^1/2(3+t)-loss for all derivatives.] 1 ≤ k ≤ K and M ∈ℕ such that 3n+4/2≤ M ≤ N-n+1. Let I_1 and I_2 be two sets defined as 

    I_1= {βmulti-index /  M ≤ |β| ≤ N }and I_2= {βmulti-index /  |β| ≤ M-1 }.

We consider an ordering on I_i, for 1 ≤ i ≤ 2, so that I_i={β_i,1,...,β_i,|I_i|} and two vector valued fields X and Y, of respective length |I_1| and |I_2|, such that

    X^j=Z^β_1,jf_k and Y^j=Z^β_2,j f_k.



There exists three matrices valued functions A_1 :[0,T] ×^n →𝔐_|I_1|(), A_2 :[0,T] ×^n →𝔐_|I_2|() and B :[0,T] ×^n →𝔐_|I_1|,|I_2|() such that

    T_F(X)+A_1X=B Y , and T_F(Y)=A_2Y.

If 1 ≤ j ≤ I_1, A_1 and B are such that T_F(X^j) is a linear combination of 
    v^μ/v^0ℒ_Z^γ_1(F)_μ mX^β_1,q,  tv^μ/v^0ℒ_Z^γ_1(F)_μ mX^β_1,q, v^μ/v^0ℒ_Z^γ_1(F)_μ ix^iX^β_1,q,


    v^μ/v^0ℒ_Z^γ_2(F)_μ mY^β_2,l,  tv^μ/v^0ℒ_Z^γ_2(F)_μ mY^β_2,landv^μ/v^0ℒ_Z^γ_2(F)_μ ix^iY^β_2,l,

with |γ_1| ≤ N-3n+2/2, |γ_2| ≤ N, 1 ≤ m ≤ n, 1 ≤ q ≤ |I_1| and 1 ≤ l ≤ |I_2|.
Similarly, if 1 ≤ j ≤ I_2, A_2 is such that T_F(Y^j) is a linear combination of

    v^μ/v^0ℒ_Z^γ(F)_μ mY^β_2,l,  tv^μ/v^0ℒ_Z^γ(F)_μ mY^β_2,landv^μ/v^0ℒ_Z^γ(F)_μ ix^iY^β_2,l,

with |γ| ≤ N-n, 1 ≤ m ≤ n and 1 ≤ l ≤ |I_2|. Note also, using Proposition <ref>, that

    ∫_v |Y|_∞ dv ≲ϵχ^1/6(t)/τ_+^n.
 





Let |β| ≤ N. According to commutation formula of Lemma <ref>, T_F(Z^β f_k) is a linear combination of terms such as ℒ_Z^γ(F)(v,∇_v Z^δ(f_k)), with |γ| + |δ| ≤ |β| and |δ| ≤ |β|-1. Replacing each ∂_v^iZ^δ f_k by 1/v^0(Ω_0iZ^β f_k-t∂_iZ^β f_k-x^i∂_tZ^β f_k ), the matrices naturally appear.



Now, we split X in G+H where G is the solution of the homogeneous system and H is the solution to the inhomogeneous system,

    {[ T_F(H)+A H=0 ,  H(0,.,.)=X(0,.,.),;        T_F(G)+AG=BY ,  G(0,.,.)=0. ].


The goal now is to prove L^2 estimates on the velocity averages of H and G.



  §.§.§ The homogeneous part


We start by the following commutation formula.



Let 1   ≤ i ≤ |I_1| and consider Z^δ∈ℙ_0^|δ|, with |δ| ≤ n. Then, T_F(Z^δH^i) can be written as a linear combination of terms of the form

    ℒ_Z^γ(F)(v,W),

where W is such that 

    ∀ 0 ≤μ≤ n,  |W^μ| ≲τ_+/v^0∑_|θ| ≤ n∑_ q=1^|I_1| |Z^θH^q|,

and where |γ| ≤ N-n+2/2, so that the electromagnetic field can be estimated pointwise.



The proof is similar to the ones of Lemma <ref> and Corollary <ref>.



We introduce the energy 𝔼[H] of H.

    𝔼[H]=∑_i=1^|I_1|𝔼^2_n[H^i]+ 𝔼^2_n,1[H^i]

and we have the following lemma.


If ϵ is small enough, we have, for all t ∈ [0,T],

    𝔼[H](t) ≤ 8ϵand∀ 1 ≤ i ≤ |I_1|, ∫_v τ_+ |H^i| dv _L^2(Σ_t)≲ϵ/(1+t)^n-2/2.




We follow here what we have done in Section <ref>. Since 𝔼[H](0) ≤3/2𝔼^2_N+n[f_k](0) + 3/2𝔼^2_N+n,1[f_k](0) ≤ 3ϵ for ϵ small enough, there exists 0 < T≤ T such that 

    ∀ t ∈ [0,T], 𝔼[H](t) ≤ 8 ϵ.

To improve this bootstrap assumption, for ϵ small enough, we only have to use the previous lemma and to follow Section <ref> (as we always estimated |(∇_v w)^L|, |(∇_v w)^L| and |(∇_v w)^B| by τ_+/v^0∑_Z∈ |Zw|). We can then take T=T and obtain, as in Section <ref>, that

    ∀ 1 ≤ i ≤ |I_1|,  (t,x) ∈ [0,T] ×ℝ^n, ∫_v ∈ℝ^n |H^i(t,x,v)| dv ≲ϵ/τ_+^n.

The L^2 estimate then ensues from Lemma <ref>.




  §.§.§ The inhomogeneous part


Let us introduce K, the solution of T_F(K)+A_1K+KA_2=B which verifies K(0,.,.)=0, and the function


    |KKY|_∞ = ∑_l 1 ≤ i ≤ |I_1| 
     1 ≤ j,p ≤ |I_2|  | K^j_i|^2 |Y_p|.


KY and G are solutions of the same system,

    T_F(KY)=T_F(K)Y+KT_F(Y)    =     BY-A_1KY-KA_2Y+KA_2Y 
        =     BY-A_1KY.

As KY(0,.,.)=0 and G(0,.,.)=0, it comes that KY=G. For 1 ≤ i ≤ |I_1| and 1 ≤ j,p ≤ |I_2|, | K^j_i|^2 Y_q sastifies the equation

    T_F( |K^j_i|^2 Y_p) = |K^j_i |^2(A_2)^q_pY_q-2((A_1)^q_i K^j_q +K^q_i (A_2)^j_q ) K^j_i Y_p+2B^j_iK^j_iY_p,


which will allow us to estimate 
    𝔼[|KKY|_∞]:=𝔼^0_0[|KKY|_∞].
 

We will then be able to bound τ_+ ∫_v ∈ℝ^n |G| dv _L^2(Σ_t) thanks to the estimates on ∫_v ∈ℝ^n |Y| dv and 𝔼[|KKY|_∞].


We have, 

    ∀ t ∈ [0,T], 𝔼[|KKY|_∞] ≤ϵ.




We use again the continuity method. Let T_0 be the largest time such that 𝔼[|KKY|_∞]≤ 2ϵ for all t ∈ [0,T_0] and let us prove, with the energy estimate of Proposition <ref>, that for ϵ small enough, 𝔼[|KKY|_∞] ≤ϵ on [0,T_0]. Let t ∈ [0,T_0].

As for the estimate of 𝔼[H] in the proof of Lemma <ref>, we have

    ∫_0^t ∫_Σ_s∫_v 1/v^0| |K^j_i |^2(A_2)^q_pY_q-2((A_1)^q_i K^j_q +K^q_i (A_2)^j_q ) K^j_i Y_p |dvdxds ≲ϵ^3/2.

Next, we need to estimate the following integral,

    ∫_0^t∫_Σ_s∫_v 1/v^0|B^j_iK^j_iY_p| dv dx .

The components of the matrix B involve terms in which the electromagnetic field has too many derivatives to be estimated pointwise. Indeed, recall from Lemma <ref> that 

    |B^j_iK^j_iY_p| ≲∑_m=1^n∑_|γ| ≤ Nτ_+|v^μ/v^0ℒ_Z^γ(F)_μ mK^j_iY_p|.

We fix |γ| and we denote the null decomposition of ℒ_Z^γ(F) by (α, α, ρ, σ). In order to bound (<ref>), we bound the integral of the five following terms, given by the null decomposition of the velocity vector v and ℒ_Z^γ(F).


  * The good terms

    τ_+|α||KY|/v^0, τ_+|ρ||KY|/v^0andτ_+|σ||KY|/v^0.


  * The bad terms

    τ_+v^L/(v^0)^2|α||KY| andτ_+ |v^B|/(v^0)^2|α||KY|.



We start by bounding the integral on Σ_s ×^n_v of the good terms. We use ζ to denote either α, ρ or σ. Using twice the Cauchy-Schwarz inequality (in x and then in v), we have

    ∫_Σ_s∫_v τ_+ |ζ| |KY|/v^0 dvdx    ≲   τ_+ |ζ| _L^2(Σ_s)( ∫_Σ_s(∫_v |KY| dv )^2 dx )^1/2
       ≲   √(ℰ_N[F](s))( ∫_Σ_s∫_v |Y| dv ∫_v |KKY|_∞ dv dx )^1/2
       ≲   √(ℰ_N[F](s))∫_v |Y| dv _L^∞(Σ_s)^1/2𝔼[|KKY|_∞]^1/2.


Using the bootstrap assumptions, on ℰ_N[F] and 𝔼[|KKY|_∞], and the pointwise estimate ∫_v |Y| dv ≲ϵlog^1/2(3+t)τ_+^-n given in Lemma <ref>, we obtain

    ∫_0^t ∫_Σ_s∫_v τ_+ |ζ| |KY|/v^0 dvdx ds ≲∫_0^t ϵ^3/2log^2 (3+t)/(1+s)^2ds ≲ϵ^3/2.


To unify the study of the bad terms, we use v to denote v^L or v^B. Using the Cauchy-Schwarz inequality (in (s,x)), the integral on [0,t] ×Σ_s ×^n_v of a bad term is bounded by 

    ( ∫_0^t ∫_Σ_sτ_-^2 |α|^2/(1+s)^3/2dxds  ∫_0^t ∫_Σ_sτ_+^2 (1+s)^3/2/τ_-^2(∫_v |v/v^0| |KY|dv )^2 dxds )^1/2.

As τ_- |α| ^2_L^2(Σ_s)≲ϵlog^3(3+t), we have

    ∫_0^t ∫_Σ_sτ_-^2 |α|^2/(1+s)^3/2dxds ≲ϵ.

For the second factor of the product in (<ref>), we first note that, by the Cauchy-Schwarz inequality, 

    (∫_v |v/v^0| |KY|dv )^2 ≤∫_v |Y|dv ∫_v |v/v^0|^2|KKY|_∞ dv.

Now, recall from Proposition <ref> that |v^B| ≲√(v^L v^L) so that |v/v^0|^2 ≲v^L/v^0. Using the pointwise decay estimate ∫_v |Y|dv ≲ϵlog^1/2(3+t)τ_+^-n, it comes

    (∫_v |v/v^0| |KY|dv )^2 ≤ϵlog^1/2(3+t)/τ_+^n∫_v v^L/v^0|KKY|_∞ dv.

As ∫_C_u(t)∫_v v^L/v^0 |KKY|_∞ dC_u(t) dv ≤𝔼[|KKY|_∞](t) ≤ 2ϵ, we obtain

    ∫_0^t ∫_Σ_sτ_+^2 (1+s)^3/2/τ_-^2(∫_v |v/v^0| |KY|dv )^2 dxds ≲ϵ^2 ∫_u=-∞^+∞τ_-^-2 du ≲ϵ^2.

Hence,

    ∫_0^t ∫_Σ_s∫_v τ_+ |v|/(v^0)^2|α| |KY| dv dx ds ≲ϵ^3/2

and the energy estimate of Proposition <ref> gives that, for ϵ small enough, 𝔼[|KKY|_∞] ≤ϵ on [0,T_0].



A naive estimation of the bad terms in the previous lemma would lead to a (1+t)^η-loss which would affect the electromagnetic energy.


We are now able to prove the expected L^2 estimate on ∫_v |G| dv.


If ϵ is small enough, we have,

    ∀ t ∈ [0,T],  1 ≤ i ≤ |I_1|, ∫_v τ_+ |G^i| dv _L^2(Σ_t)≲ϵχ^1/12(t)/(1+t)^n-2/2.




Let 1 ≤ i ≤ |I_1|. The Cauchy-Schwarz inequality (in v) gives us

    τ_+∫_v |G^i| dv _L^2(Σ_t)≲∑_j=1^|I_2|τ_+^2 ∫_v |Y_j| dv ∫_v |(K_i^j)^2 Y_j| dv _L^1(Σ_t)^1/2.

Thus, using once again that ∫_v |Y_j|dv ≲ϵχ^1/6(t) τ_+^-n, we obtain

    τ_+ ∫_v | G^i| dv _L^2(Σ_t)≲ϵχ^1/12(t)/(1+t)^n/2-1.


We can now conclude this section.


As mentionned earlier, for |β| ≤ M-1, the estimate ensues from Proposition <ref> and Lemma <ref>. If M ≤ |β| ≤ N, as there exists 1 ≤ i ≤ |I_1| such that Z^β f_k=H^i+G^i, we have

    τ_+∫_v |Z^β f_k| dv _L^2(Σ_t)≤τ_+∫_v |H^i| dv _L^2(Σ_t)+τ_+∫_v |G^i| dv _L^2(Σ_t).

It then remains to use Lemmas <ref> and <ref>.




 §.§ Step 5: Improvement of the electromagnetic field energy estimates




  §.§.§ The bound on the potential energy


According to the energy estimate given by Proposition <ref> and the commutation formula of Proposition <ref>, we have, for all t ∈ [0,T],


    √(ℰ_N[A])(t) ≲√(ℰ_N[A])(0)+∑_|γ| ≤ N∫_0^t |e^k|τ_+ ∫_^n |Z^γ f_k| dv_L^2(Σ_s) ds.



Using the L^2 decay estimate of Proposition <ref> and ℰ_N[A](0) ≤ϵ, we obtain, for ϵ small enough and if the constant C is large enough, that

    ∀ t ∈ [0,T], ℰ_N[A](t) ≤C/2(n-3)ϵlog^3(3+t)if n=4

and

    ∀ t ∈ [0,T], ℰ_N[A](t) ≤C/2(n-3)ϵif n ≥ 5.


We are now able, using Proposition <ref>, to improve the pointwise decay estimate on α.

    ∀ |β| ≤ N-n,  (t,x) ∈ [0,T] ×^n,  |α(ℒ_Z^β(F)|(t,x) ≲√(ϵ)√(χ(t))/τ_+^n+2/2.




  §.§.§ Improvement of the electromagnetic field energy estimates


Recall from Proposition <ref> that

    ℰ_N[F](t) ≤ℰ_N[F](0)+(n-3)ℰ_N[A](t)+φ(t),

where φ(t) is a linear combination of terms such that

    ∫_0^t ∫_Σ_s  | K_0^νℒ_Z^β (F)_μν J(Z^γ f_k)^μ| dx ds and∫_0^t ∫_Σ_s s |ℒ_Z^δ A_μ□ℒ_Z^δ A^μ | dx ds,

 with |β|, |γ|, |δ| ≤ N and 1 ≤ k ≤ K. Then, if we could prove that each integrals of (<ref>) is bounded by ϵ^3/2χ(t), we would have, for ϵ small enough and if the constant C is large enough, ℰ_N [F] ≤ C ϵχ(t) on [0,T] since 𝔼_N[F](0) ≤ϵ and (n-3)ℰ_N[A](t) ≤C/2ϵχ(t).

We start by bounding the integrals involving the potential. Using Proposition <ref> and the Cauchy-Schwarz inequality, we have, for |δ| ≤ N,


     ∫_0^t ∫_Σ_s s |ℒ_Z^δ A_μ □ℒ_Z^δ A^μ | dx ds ≲   

 
    ∑_k=1^K ∑_|γ| ≤ |δ|∫_0^t √(ℰ_N[A](s))τ_+∫_v |Z^γ f_k| dv _L^2(Σ_s) ds.

Using the L^2 estimate of Proposition <ref> and that ℰ_N[A](s) ≲ϵχ(s), it comes

    ∑_|δ| ≤ N∫_0^t ∫_Σ_s s |ℒ_Z^δ A_μ□ℒ_Z^δ A^μ | dx ds    ≲   ϵ^3/2∫_0^t log^2(3+s)/(1+s)^n-2/2 ds 
       ≲   ϵ^3/2χ(t).


In order to estimate the remaining integrals of (<ref>), we express K_0^νℒ_Z^β (F)_μν J(Z^γ f_k)^μ in null coordinates. Dropping the dependance in ℒ_Z^β(F) or Z^γ f_k, this gives us the four following terms :


    τ_+^2 ρ J^L, τ_-^2 ρ J^L, τ_+^2 α_B J^B,  andτ_-^2 α_B J^B.


As 
    J^L=∫_v v^L/v^0Z^γ f_k dv,  J^L=∫_v v^L/v^0Z^γ f_k dv and J^B=∫_v v^B/v^0Z^γ f_k dv,

we have,

    |J^L|,  |J^L|,  |J^B| ≲∫_v ∈^n |Z^γ f_k| dv.

The integrals (on [0,T] ×ℝ^n_x ×ℝ^n_v) of each of the four terms of (<ref>) are then bounded, using the Cauchy-Schwarz inequality, by


    ∫_0^t √(ℰ_N[F])(s) τ_+ ∫_v |Z^γ f_k| dv _L^2(Σ_s) ds.

By Proposition <ref> and the bootstrap assumption (<ref>),

    ∫_0^t √(ℰ_N[F])(s) τ_+ ∫_v |Z^γ f_k| dv _L^2(Σ_s) ds    ≲   ∫_0^t √(ϵχ(s))ϵlog^1/2(3+s)/(1+s)^n-2/2 ds 
       ≲   ϵ^3/2χ(t).


Hence, ℰ_N[F](t) ≤ Cϵχ(t) for all t ∈ [0,T] if ϵ is small enough. 

We can prove in the same way, using in particular the energy estimate of Proposition <ref> and

    ∫_v |Z^β f_k |dv _L^2(Σ_t)≤1/1+tτ_+ ∫_v |Z^β f_k |dv _L^2(Σ_t),

that ℰ_N^S[F] ≤Cϵ on [0,T] if ϵ is small enough and the constant C is large enough. 
We then improve the bootstrap assumption (<ref>).



§ THE MASSLESS VLASOV-MAXWELL EQUATIONS



 §.§ Global existence for small data


The aim of this section is to prove Theorem <ref>. We then consider the massless Vlasov-Maxwell system (<ref>)-(<ref>), with at least two species[We recall that we take K ≥ 2 since we suppose that the initial energy ℰ[F] is finite, which implies that the plasma is electrically neutral (see Remark <ref> for more details).], in dimension n ≥ 4. This means that K ≥ 2 and m_k=0 for all 1 ≤ k ≤ K.

To simplify the notation, we denote, during this chapter, 𝔼^0_M[f] by 𝔼_M[f] and 𝔼^0_M,1,0 by 𝔼_M,1. In view of Definition <ref> and 1 ∈𝐤_0, we have 
    𝔼_M[f] ≤𝔼_M,1[f].
 We introduce the functions χ, defined on _+ by

    χ ( s )= 1+s if n=4, χ ( s )= log^2(3+s)if n=5 andχ(s) = 1 if n ≥ 6,
 
and log^*, defined on _+ by

    log^*=logif n=4 andlog^*= 1 if n ≥ 5.


We give a more precise version of Theorem <ref>.


Let n ≥ 4, K ≥ 2, N ≥ 6n+2 if n is even and N ≥ 6n+3 is n is odd, 0 < η < 1/2 if n=4 and η=0 if n ≥ 5 and R>0. Let (f_0,F_0) be an initial data set for the massless Vlasov-Maxwell system. Let (f,F) be the unique classical solution to the system and let A be a potential in the Lorenz gauge. There exists ϵ >0 such that[We recall that a smallness condition on F, which implies ℰ_N[A](0) ≤ϵ, is given in Proposition <ref>.], if 


     ℰ_N[A](0) ≤ϵ,  ℰ_N[F](0) ≤ϵ   


and if, for all 1 ≤ k ≤ K,

      supp(f_0k) ⊂{(x,v) ∈ℝ^n_x ×ℝ^n_v ∖{ 0 }  /  |v| ≥R } ,    


     𝔼_N+n,1[f_k](0) ≤ϵ,     
 


then (f,F) exists globally in time and verifies the following estimates.



  * Vanishing property for small velocities : for all 1 ≤ k ≤ K,

    supp (f_k) ⊂{ (t,x,v) ∈ℝ_+ ×ℝ^n_x ×ℝ^n_v ∖{ 0 } /  |v| ≥R/2} .




  * Energy bounds for F and f_k : ∀ 1 ≤ k ≤ K and ∀ t ∈ℝ_+,


     ℰ_N[F](t) ≲ϵχ(t)(1+t)^η,  ℰ_N-2n[F](t) ≲ϵχ(t),     

     𝔼_N[f_k](t) ≲ϵlog^* (3+t),  𝔼_N-n,1[f_k](t) ≲ϵ.    




  * Pointwise decay for the null decomposition of ℒ_Z^β(F) : ∀ |β| ≤ N-5n+4/2, (t,x) ∈ℝ_+ ×ℝ^n,

     |α( ℒ_Z^β F)| ≲√(ϵχ(t)) τ_+^-n+2/2,  |α ( ℒ_Z^β F)| ≲√(ϵχ(t)) τ_+^-n-1/2τ_-^-3/2,     
 
     |ρ(ℒ_Z^β F)| ≲√(ϵχ(t)) τ_+^-n+1/2τ_-^-1/2,  |σ(ℒ_Z^β F)| ≲√(ϵχ(t)) τ_+^-n+1/2τ_-^-1/2    

and

    |α ( ℒ_Z^β F)| ≲√(ϵ)τ_+^-n-1/2τ_-^-1.



  * Pointwise decay for ∫_v ∈ℝ^n ∖{0 } | zZ^β f_k | dv : 
    ∀ |β| ≤ N-2n,  z ∈𝐤_0,  (t,x) ∈ℝ_+ ×ℝ^n, ∫_v   | zZ^β f_k |dv ≲ϵ/τ_+^n-1τ_-.


  * L^2 estimates on ∫_v ∈ℝ^n ∖{0 } |Z^β f_k | dv :

    ∀ |β| ≤ N,  t ∈ℝ_+, ∫_v ∈ℝ^n ∖{0 } | Z^β f_k | dv _L^2(Σ_t)≲ϵ/(1+t)^n-1-η/2.



  * Energy bound for a potential A satisfying the Lorenz gauge :

    ∀ t ∈ℝ_+, ℰ_N[A](t) ≲ϵχ(t)(1+t)^ηandℰ_N-2n[A](t) ≲ϵχ(t).








 §.§ Structure and beginning of the proof


Let (f_0,F_0) be an initial data set satisfying the assumptions of Theorem <ref>. By a standard local well-posedness argument, there exists a unique maximal solution (f,F) of the massless Vlasov-Maxwell system defined on [0,T^*[, with T^* ∈_+^* ∪{+ ∞}. 

We consider the following bootstrap assumptions. Let T be the largest time such that, ∀ 1 ≤ k ≤ K and ∀ t ∈ [0,T],


    ℰ_N[F](t) ≤ 2Cϵχ(t)(1+t)^η, ℰ_N-2n[F](t) ≤ 2Cϵχ(t),

 
    ℰ^0_N[F](t) ≤ 4ϵ, ℰ_N^S[F](t) ≤ 2Cϵ(1+t)^η, ℰ_N-2n^S[F](t) ≤ 2C,


    ℰ_N[A](t) ≤ 2Cϵχ(t)(1+t)^η, ℰ_N-2n[A](t) ≤ 2Cϵχ(t),


    𝔼_N[f_k](t) ≤ 4log^*(3+t), and𝔼_N-n,1[f_k](t) ≤ 4ϵ,


where C and C are positive constants which will be specified during the proof. Note that by continuity, T>0. We now present our strategy to improve these bootstrap assumptions.



  * First, using the bootstrap assumptions, we obtain decay estimates for the null decomposition of F (and its Lie derivatives) and for velocity averages of derivatives of f_k.


  * Then, we prove that 0 is not in the closure of the Vlasov fields v-support. This follows from the study of the characteristics of the transport equation.


  * Next, we improve the bounds on the Vlasov fields energies by means of the energy estimates proved in Propositions <ref> and <ref>. To bound the right hand side in these energy estimates, we make fundamental use of the null structure of the system and the pointwise decay estimates on ρ, σ, α, α and ∫_ℝ^n ∖{ 0 } |z Z^β f_k | dv.


  * In order to improve the estimates on the electromagnetic field energies, we establish an L^2_x estimate for the velocity averages of the Vlasov fields (and its derivatives). For this purpose, we follow <cit.> and we rewrite all the transport equations as an inhomogeneous system of transport equations. The velocity averages of the homogeneous part of the solution verify strong pointwise decay (we use particularly the control that we have at our disposal on the initial data of f_k, for derivatives of order N+n or less). The inhomegeneous part is decomposed into a product of an integrable function and a pointwise decaying function which gives us the expected estimate.


  * Finally, we improve the estimates on the energies of the electromagnetic potential and the electromagnetic field, with the energy estimate for the Maxwell equations (using in particular Propositions <ref> and <ref>). We use the null decomposition of J(Z^γ f_k)^μℒ_Z^β(F)_μνK_0^ν, which, combined with L^2_x estimates on quantities such as ∫_ℝ^n   | Z^γ f_k | dv, gives us the improvement.





 §.§ Step 1: Decay estimates


By the Klainerman-Sobolev inequality of Theorem <ref> and the bootstrap assumption (<ref>), we  have ∀ |β| ≤ N-n, (t,x) ∈ [0,T[ ×^n, 1 ≤ k ≤ K,

    ∫_v |Z^β f_k| dv ≲𝔼_N[f_k](t)/τ_+^n-1τ_-≲ϵlog^*(3+t)/τ_+^n-1τ_-.


In the same spirit[Note that the pointwise decay estimate (<ref>) implies (<ref>) for the lower order derivatives, taking z=1. ], using Corollary <ref>, we have ∀ |β| ≤ N-2n, z  ∈𝐤_0, (t,x) ∈ [0,T] ×^n,


    ∫_v | zZ^β f_k | dv ≲𝔼_N-n,1[f_k](t)/τ_+^n-1τ_-≲ϵ/τ_+^n-1τ_-.


We have improved decay estimates for the null components of the current M_μ:= ∫_v v_μ/v^0Z^β f_kdv. For all |β| ≤ N-2n, we have


    ∫_v v^L/v^0| Z^β f_k| dv ≲ϵ/τ_+^n τ_-,


    ∫_v v^L/v^0| Z^β f_k| dv ≲ϵ/τ_+^n-1τ_-^2

and

    ∫_v |v^B|/v^0| Z^β f_k| dv ≲ϵ/τ_+^n τ_-.


This results from (see Proposition <ref>)

    v^L/v^0≲1/τ_+∑_z ∈𝐤_0|z|, v^L/v^0≲1/τ_-∑_z ∈𝐤_0|z| and|v^B/v^0| ≲1/τ_+∑_z ∈𝐤_0|z|.


Using the bootstrap assumptions (<ref>), (<ref>), (<ref>), Propositions <ref>, <ref> and the pointwise decay estimate (<ref>), we obtain.



For all t ∈ [0,T], |β| ≤ N- n, we have

     |α( ℒ_Z^β F)| ≲√(ϵχ(t)(1+t)^η)/ τ_+^n+2/2,  |α ( ℒ_Z^β F)| ≲√(ϵχ(t)(1+t)^η)/ τ_+^n-1/2τ_-^3/2,     
 
     |ρ(ℒ_Z^β F)| ≲√(ϵχ(t)(1+t)^η)/ τ_+^n+1/2τ_-^1/2,  |σ(ℒ_Z^β F)| ≲√(ϵχ(t)(1+t)^η)/ τ_+^n+1/2τ_-^1/2    


and

    |α ( ℒ_Z^γ F)| ≲√(ϵ(1+t)^η)/τ_+^n-1/2τ_-.



For all t ∈ [0,T], |β| ≤ N-5n+2/2, we have

     |α ( ℒ_Z^γ F)| ≲√(ϵ) τ_+^-n-1/2τ_-^-1,  |α ( ℒ_Z^β F)| ≲√(ϵχ(t)) τ_+^-n-1/2τ_-^-3/2,     
 
     |ρ(ℒ_Z^β F)| ≲√(ϵχ(t)) τ_+^-n+1/2τ_-^-1/2,  |σ(ℒ_Z^β F)| ≲√(ϵχ(t)) τ_+^-n+1/2τ_-^-1/2.    


Finally, for all t ∈ [0,T], |β| ≤ N-5n+4/2,

    |α ( ℒ_Z^γ F)| ≲√(ϵχ(t))τ_+^-n+2/2.






The alternative estimate on α is useful to avoid a τ_+-loss when n ≤ 5 and is particularly used in Section <ref>.



We also have pointwise decay estimates if |β| ≤ N-n+2/2 but the one on α is worse near the light cone (see Proposition <ref>).




 §.§ Step 2: the Vlasov fields vanishes for small velocities


We recall that 

    ∀ 1 ≤ i ≤ n,  E^i = F_0i ,

and that the transports equation of the Vlasov-Maxwell system can be rewritten 

    v^μ∂_μ f_k +  v^0E^i∂_v^i f_k+v^jF_j^i ∂_v^i f_k =0.

We now fix 1 ≤ k ≤ K and we prove, under the bootstrap assumption, that if f_k(t,x,v) ≠ 0, with (t,x,v) ∈ [0,T] ×ℝ^n ×ℝ^n ∖{ 0 }, then |v| ≥R/2. During the argument, we will use various constants and we will all call them C for simplicity. These constants will not depend on ϵ or on T.

Let x ∈ℝ^n and |v| ≥ R. Let (X,V) be the characteristics of the transport equation such that (X(0),V(0))=(x,v). In particular


    ∀1 ≤ i ≤ n, dV^i/ds=E^i(s,X)+V^j/V^0 F_ji(s,X).


It follows that


    d(|V|^2)/ds= 2<E(s,X),V >.


So,


    |V(t) |^2=|v|^2+2∫_0^t <E(s,X(s)),V(s) > ds.


We denote |V(s)|^2 by g(s). By the Cauchy-Schwarz inequality, we have


    g(t) ≤ |v|^2+2∫_0^t |E(s,X(s)) | √(g(s)) ds.


We now use a Grönwall inequality (Lemma <ref>) and |E(s,X(s))| ≤C √(ϵ)/(1+s)^n-1/2 (which come from Proposition <ref>) to obtain 
    g(s) ≤(|v|+∫_0^t C √(ϵ) ds/(1+s)^3/2)^2.


Thus,

    |V(s)| ≤ |v| + C √(ϵ).


Returning to (<ref>), we obtain


    |V(s) |^2 ≥ |v|^2-2∫_0^t |E(s,X(s)) | |V(s) | ds.


Therefore, using again the pointwise estimate on E,


    |V(s) |^2 ≥ |v|^2-2C√(ϵ)(|v|+C√(ϵ)).


Finally, 
    |V(s)|^2 ≥ |v|(|v|-C√(ϵ))-Cϵ≥1/4|v |^2,

if ϵ is sufficiently small so that C ϵ≤R/4 and C √(ϵ)≤R/2.

Then, if (x,v) is such that |v| ≥ R, (X,V) is well defined on [0,T] (X is also bounded since | d X/ds| =1) and |V| ≥R/2. Consequently, we obtain.



    supp(f_k | [0,T]) ⊂{ (t,x,v) ∈ [0,T] ×ℝ^n ×ℝ^n ∖{ 0 } /  |v| ≥R/2} .



In the remainder, we will then be able to use inequalities like

    1/v^0|f_k(t,x,v)| ≲ |f_k(t,x,v)| .

Sometimes, we will abusively use inequalities such that 
    1/v^0∑_z ∈𝐤_0 |z| ≲∑_z ∈𝐤_0 |z|

because these quantities are always multiplied by Z^β f_k.



 §.§ Step 3: Improving the Energy estimates for the transport equations


We fix for all this section 1 ≤ k ≤ K. According to Proposition <ref>, 𝔼_N[f_k] ≤ 3ϵlog^* (3+t) on [0,T], for ϵ small enough, follows from

  
    ∫_0^t ∫_Σ_s∫_v |ℒ_Z^β_1(F)(v/v^0,∇_v Z^β_2f_k )| dv dx ds ≲ϵ^3/2log^* (3+t),

for all |β_1|+|β_2| ≤ N, with |β_2 | ≤ N-1.

Similarly, according to Proposition <ref>, 𝔼_N-n,1[f_k] ≤ 3 ϵ on [0,T], for ϵ small enough, follows from

  
    ∫_0^t ∫_Σ_s∫_v |z||ℒ_Z^β_1(F)(v/v^0,∇_v Z^β_2f_k ) | dv dx ds ≲ϵ^3/2,
 and

    ∫_0^t ∫_Σ_s∫_v | F(v/v^0,∇_v z )Z^βf_k| dv dx ds ≲ϵ^3/2,

for all z ∈𝐤_0, |β_1|+|β_2| ≤ N-n (with |β_2 | ≤ N-n-1) and |β| ≤ N-n.

To unify the study of 𝔼_N[f_k] and 𝔼_N-n,1[f_k], we consider b, which could be equal to 0 or to 1, N_0=N and N_1=N-n. Now, we fix z ∈𝐤_0, |β_1|+|β_2| ≤ N_b (with |β_2| ≤ N_b-1) and |β| ≤ N-n. We denote ρ(ℒ_Z^β_1(F)), σ(ℒ_Z^β_1(F)), α(ℒ_Z^β_1(F)) and α(ℒ_Z^β_1(F)) by ρ, σ, α and α (respesctively). We also denote Z^β_2f_k by g and Z^βf_k by h. The null decomposition of ℒ_Z^β_1(F)(v,∇_v g) or F(v,∇_v z) brings us to control the integral of the following terms, with z_0=1 and z_1=z.

The terms involving L or L components of ∇_v g or ∇_v z 


    |z_bv^L/v^0ρ( ∇_v g )^L | , |hv^L/v^0ρ(F) ( ∇_v z )^L|,



    |z_bv^L/v^0ρ( ∇_v g )^L| , |h v^L/v^0ρ(F) ( ∇_v z )^L|,




    |z_bv^B/v^0α_B ( ∇_v g )^L| , |h v^B/v^0α_B(F) ( ∇_v z )^L|,



    |z_bv^B/v^0α_B ( ∇_v g )^L|, | h v^B/v^0α_B(F) ( ∇_v z )^L|.


The terms involving angular components of ∇_v g or ∇_v z


    | z_bv^L/v^0α_B ( ∇_v g )^B |, | h v^L/v^0α_B(F) ( ∇_v z )^B |,



    |z_bv^B/v^0σ_BD( ∇_v g )^D | , |h v^B/v^0σ(F)_BD( ∇_v z )^D |,



    |z_bv^L/v^0α_B ( ∇_v g )^B |, | h v^L/v^0α_B(F) ( ∇_v z )^B |.


The study of 𝔼_N[f_k] corresponds to b=0. In this case, we only have to estimate the spacetime integral of each of the first terms of (<ref>)-(<ref>), but we need to consider two cases. When |β_1| ≤ N-n we can use the pointwise decay estimates on the electromagnetic field given by Proposition <ref>. When |β_1| > N-n, |β_2| ≤ N-2n (since N ≥ 6n+2), and we can then use the pointwise decay estimates on the velocity averages of the Vlasov field given in Section <ref>.

In the study of 𝔼_N-n,1[f_k] (which corresponds to b=1 and where z can be any weights of 𝐤_0), we can always use a pointwise estimate on the electromagnetic field (as |β_1| ≤ N-n), but we need to estimate the spacetime integral of all the terms of (<ref>)-(<ref>).



To simplify the argument we will sometimes denote 𝔼_N[f_k] by 𝔼_N_0,0[f_k] and 𝔼_N-n,1[f_k] by 𝔼_N_1,1[f_k].





  §.§.§ Estimating the v derivatives


In order to eliminate the v derivatives, we use, as in Section <ref>,

    | ∇_v w | ≲τ_+/v^0∑_Z∈ |Z w|

and


    | ( ∇_v w )^L|, | ( ∇_v w )^L| ≲τ_-/v^0∑_Z∈|Zw|.




  §.§.§ If |β_1| ≤ N-n



We start by the terms involving L or L components of ∇_v g or ∇_v z. We use ζ to denote α, α or ρ. Thus, by Proposition <ref>,

    |ζ| ≲√(ϵ)(1+t)^η/2/τ_+^3/2τ_-.

The integral on [0,t] ×ℝ^n_x ×( ℝ^n_v ∖{0 }) of each of the first terms of (<ref>)-(<ref>) are bounded by


    ∑_Z∈ℙ_0∫_0^t ∫_Σ_sτ_- |ζ| ∫_v |z_bZg| dv dx ds,

where we use in particular (<ref>) and the fact that 1/v^0≲ 1 on the support of g.
Using the bootstrap assumption (<ref>), we obtain
 
    ∫_0^t ∫_Σ_sτ_- |ζ| ∫_v |z_bZg| dv dx ds ≲∫_0^t √(ϵ)(1+s)^-3-η/2𝔼_N_b,b[f_k](s) ds ≲ϵ^3/2.


Similarly, the integrals of each of the second terms of (<ref>)-(<ref>) are bounded by


    ∑_Z∈ℙ_0∫_0^t ∫_Σ_sτ_- |ζ(F)| ∫_v |hZ(|z|)| dv/v^0 dx ds.

Using again the bootstrap assumption (<ref>) and 1/v^0≲ 1 on the support of h, one has

    ∫_0^t ∫_Σ_sτ_- |ζ(F)| ∫_v |hZ(|z|)| dv/v^0 dx ds ≲∫_0^t √(ϵ)(1+s)^-3-η/2𝔼_N_1,1[f_k](s) ds ≲ϵ^3/2,

since |Z(|z|) | ≲∑_w ∈𝐤_0 |w| by Lemma <ref>.
  

We now study the remaining terms. Using (<ref>), the pointwise decay estimates of Proposition <ref>, that 1/v^0≲ 1 on the support of g and the bootstrap assumption (<ref>), we have

    ∫_0^t∫_Σ_s∫_v  |z_b v^L/v^0α_B ( ∇_v g )^B| dvdxds    ≲   ∑_Z∈∫_0^t ∫_Σ_sτ_+ |α| ∫_v  |z_bZ  g | dv dx ds 
       ≲   √(ϵ)∫_0^t (1+s)^-3-η/2𝔼_N_b,b[f_k](s)ds 
       ≲   ϵ^3/2.


The second term of (<ref>) can be treated similarly. For the second term of (<ref>) (as the first one can be treated in a similar way), we have, using (<ref>), Lemma <ref> and |v^B| ≲√(v^Lv^L) (which comes from Proposition <ref>),


     ∫_0^t ∫_Σ_s ∫_v |h v^B/v^0σ(F)_BD( ∇_v |z| )^D | dv dx ds  ≲    


    ∑_z' ∈𝐤_0∫_0^t ∫_Σ_sτ_+ |σ(F)|∫_v √(v^Lv^L)/(v^0)^2 |z'| |h| dv dx ds .

Since, by Proposition <ref>, |σ(F)| ≲√(ϵ)τ_+^-2τ_-^-1/2(1+t)^η/2, one has, using the Cauchy-Schwarz inequality (in (s,x,v)), that the right hand side of the previous inequality is bounded by the product of

    ∑_z' ∈𝐤_0( √(ϵ)∫_0^t (1+s)^-3-η/2∫_Σ_s∫_v v^L/(v^0)^3|z'| |h| dvdxds )^1/2
 with 
    ∑_z' ∈𝐤_0(  √(ϵ)∫_u=-∞^tτ_-^-3-η/2∫_C_u(t)∫_v v^L/v^0|z'| |h| dvdC_u(t)du )^1/2.

The first factor is bounded by ϵ^3/4 since v^L/(v^0)^3≲ 1 on the support of h and ∫_Σ_s∫_v|z'| |h| dvdx ≤ 4 ϵ by the bootstrap assumption <ref>. The same is true for the second factor since ∫_C_u(t)∫_v v^L/v^0|z'| |h| dvdC_u(t) ≤𝔼_N-n,1[f_k](t) ≤ 4 ϵ, still by the bootstrap assumption (<ref>).

Finally, let us treat, for instance, the first term of (<ref>). Using the same ingredients as before, namely (<ref>), that 1/v^0≲ 1 on the support of g and the bootstrap assumption (<ref>), we have,



    ∫_0^t ∫_Σ_s∫_v |z_bv^L/v^0α_B ( ∇_v g )^B|dsdxdv     ≲   ∑_Z∈ℙ_0∫_0^t ∫_Σ_sτ_+ |α| ∫_v v^L/(v^0)^2|z_b Z g| dvdxds 
       ≲   ∫_u=-∞^tτ_+|α| _L^∞(C_u(t))𝔼_N_b,b[f_k](t) du  
       ≲   ϵ^3/2∫_-∞^+∞1/τ_-^3-η/2 du(δ_1,b+ δ_0,blog^* (3+t)).





If we used (<ref>) instead of (<ref>) to estimate (<ref>) and (<ref>), it would give us a (1+t)^η-loss on the energies (as in the proof of Lemma <ref> below). The weight v^B could be used to avoid this loss in (<ref>).







  §.§.§ If |β_1| > N-n



We study again the integrals of the first terms of (<ref>)-(<ref>), but this time when |β_1| > N- n, so that |β_2| ≤ N-2n, and z_b=1. We then use the pointwise estimate on the velocity averages of the Vlasov fields. This time, we study the terms involving α, ρ and σ together[Note that except for (<ref>), we could bound all this terms without the log^* (1+t)-loss.] and we finish with the two terms involving α. Note that as we use the extra decay given by v^L, v^L and v^B, we cannot close the estimate for 𝔼_N,1[f_k] with our method.


Let us denote this time α, ρ or σ by ζ. Then, by the bootstrap assumption (<ref>),


    ∀ t ∈ [0,T], ∫_C_u(t) |ζ|^2 dx  ≲ϵ.



All first terms of (<ref>)-(<ref>) involving α, ρ or σ have their integral on [0,t] ×^n_x ×^n_v bounded by


    M:=∑_Z∈∫_0^t ∫_Σ_s  |ζ| τ_+∫_v |v/v^0Z g| dv dx ds,

where we used (<ref>), that 1/v^0≲ 1 on the support of g and where v denotes either v^L, v^L or v^B.

Using the pointwise decay estimate on ∫_v |v/v^0Zg| dv, given by (<ref>), (<ref>) or (<ref>), and the Cauchy-Schwarz inequality (on the u=constant integrals), we have


    M    ≲   ∫_u=-∞^t( ∫_C_u(t) |ζ|^2 dx )^1/2( ∫_C_u(t)ϵ^2/τ_+^2n-4τ_-^4 d C_u(t) )^1/2 du 
       ≲   ∫_u=-∞^tϵ^3/2/τ_-^2| ∫_u=0^2t-ur^n-1/τ_+^2n-4 d u|^1/2 du 
       ≲   ϵ^3/2∫_u=-∞^t log^* (1+2t-u)/τ_-^2 du 
       ≲   ϵ^3/2log^*(1+t) ∫_u=-∞^+∞τ_-^-3/2 du.


We now study the two remaining terms, which involve α. We start by (<ref>). Using (<ref>), we obtain that ∫_0^t ∫_Σ_s∫_v |v^B/v^0α_B ( ∇_v g )^L|dvdxds is bounded by
 
    ∑_Z∈ℙ_0∫_0^+ ∞τ_- α_L^2(Σ_s)∫_v |v^B/(v^0)^2Z g | dv _L^2(Σ_s) ds .

 Using (<ref>), Lemma <ref> and that 1/v^0≲ 1 on the support of Zg, we have 
    ∫_v |v^B/(v^0)^2Z g | dv _L^2(Σ_s)≲ϵ/(1+s)^n+1/2.
 
 By the bootstrap assumption <ref>, τ_- α_L^2(Σ_s)≲√(ϵχ(t)(1+t)^η), so

    ∫_0^t ∫_Σ_s∫_v |v^B/v^0α_B ( ∇_v g )^L|dvdxds ≲ϵ^3/2.


Finally,

     ∫_0^t ∫_Σ_s ∫_v |v^L/v^0 α_B ( ∇_v g )^B |dvdxds  ≲    


    ∑_Z∈ℙ_0∫_0^ tα_L^2(Σ_s)τ_+ ∫_v  |v^L/(v^0)^2Zg | dv _L^2(Σ_s)ds .
 
Now, using the pointwise estimates (<ref>) and Lemma <ref>, we have

    τ_+ ∫_v v^L/(v^0)^2|Zg| dv ^2_L^2(Σ_s)≲ϵ^2 ∫_0^+∞r^n-1/τ_+^2n-2τ_-^2dr ≲ϵ (1+s)^-(n-1).

As, by the bootstrap assumption (<ref>), α^2_L^2(Σ_s)≲ϵ,

    ∫_0^t ∫_Σ_s∫_v |v^L/v^0α_B ( ∇_v g )^B |dvdxds ≲ϵ^3/2.

This concludes the improvement of the bootstrap assumption (<ref>).



 §.§ Step 4: L^2 estimates for the velocity averages


As for the massive case, to close the energy estimates on the electromagnetic field, we need enough decay on quantities such as ∫_v  |Z^β f_k | dv _L^2_x for all |β| ≤ N. If |β| ≤ N-2n, strong L^2 decay estimates can already be obtained on ∫_v v^L/v^0|Z^β f_k|dv, for instance,  combining (<ref>) and Lemma <ref>.

We fix, for the remaining of this section, 1 ≤ k ≤ K. Following the strategy of <cit.> (see Section 4.5.7), for a similar problem, we introduce M ∈ℕ such that 7n+4/2≤ M ≤ N-5/2n. Let I_1 and I_2 be defined as 
    I_1= {βmulti-index /  M ≤ |β| ≤ N }and I_2= {βmulti-index /  |β| ≤ M-1 }.

We consider an ordering on I_i, for 1 ≤ i ≤ 2, so that I_i={β_i,1,...,β_i,|I_i|} and two vector valued fields X and Y, of respective length |I_1| and |I_2|, such that

    X^j=Z^β_1,jf_k and Y^j=Z^β_2,j f_k.
 


There exists three matrices valued functions A_1 :[0,T] ×^n →𝔐_|I_1|(), A_2 :[0,T] ×^n →∈𝔐_|I_2|() and B :[0,T] ×^n →∈𝔐_|I_1|,|I_2|() such that

    T_F(X)+A_1X=B Y , and T_F(Y)=A_2Y.

If 1 ≤ j ≤ I_1, A_1 and B are such that T_F(X^j) is a linear combination of 
    v^μ/v^0ℒ_Z^γ_1(F)_μ mX^β_1,q,  tv^μ/v^0ℒ_Z^γ_1(F)_μ mX^β_1,q, v^μ/v^0ℒ_Z^γ_1(F)_μ ix^iX^β_1,q,


    v^μ/v^0ℒ_Z^γ_2(F)_μ mY^β_2,l,  tv^μ/v^0ℒ_Z^γ_2(F)_μ mY^β_2,landv^μ/v^0ℒ_Z^γ_2(F)_μ ix^iY^β_2,l,

with |γ_1| ≤ N-7n+2/2, |γ_2| ≤ N, 1 ≤ m ≤ n, 1 ≤ q ≤ |I_1| and 1 ≤ l ≤ |I_2|.
Similarly, if 1 ≤ j ≤ I_2, A_2 is such that T_F(Y^j) is a linear combination of

    v^μ/v^0ℒ_Z^γ(F)_μ mY^β_2,l,  tv^μ/v^0ℒ_Z^γ(F)_μ mY^β_2,landv^μ/v^0ℒ_Z^γ(F)_μ ix^iY^β_2,l,

with |γ| ≤ N-5n+2/2, 1 ≤ m ≤ n and 1 ≤ l ≤ |I_2|. Moreover,

    ∀ z ∈𝐤_0, ∫_v |z||Y|_∞ dv ≲ϵ/τ_+^n-1τ_-.
 






Let |β| ≤ N. According to commutation formula of Lemma <ref>, T_F(Z^β f_k) is a linear combination of terms such as ℒ_Z^γ(F)(v,∇_v Z^δ(f_k)), with |γ| + |δ| ≤ |β| and |δ| ≤ |β|-1. Replacing each ∂_v^iZ^δ f_k by 1/v^0(Ω_0iZ^β f_k-t∂_iZ^β f_k-x^i∂_tZ^β f_k ), the matrices naturally appear. The decay estimates ensue from the definition of Y and (<ref>).



Now, we split X in G+H where G is the solution of the homogeneous system and H is the solution to the inhomogeneous system,

    {[ T_F(H)+A H=0 ,  H(0,.,.)=X(0,.,.),;        T_F(G)+AG=BY ,  G(0,.,.)=0. ].

We will prove below that G=KY (with K a well chosen matrix), which implies, in view of the velocity support of X and Y, that H and G vanish if |v| ≤R/2.

The goal now is to prove L^2 estimates on the velocity averages of H and G.




  §.§.§ The homogeneous part


As for the massive case, we have the following commutation formula.



Let 1   ≤ i ≤ |I_1| and consider Z^δ∈ℙ_0^|δ|, with |δ| ≤ n. Then, T_F(Z^δH^i) can be written as a linear combination of terms of the form

    ℒ_Z^γ(F)(v,W),

where W is such that 

    ∀ 0 ≤μ≤ n,  |W^μ| ≲τ_+/v^0∑_|θ| ≤ n∑_ q=1^|I_1| |Z^θH^q|,

and where |γ| ≤ N-5n+2/2, so that we can use the sharpest estimates of Proposition <ref>, except for α.


We introduce the energy 𝔼_1[H] defined by

    𝔼_1[H]=∑_ q=1^|I_1|𝔼_n,1[H^q].

Note that for ϵ small enough,

    𝔼_1[H](0) ≤ 2𝔼_N+n,1[f](0) ≤ 2 ϵ.



If ϵ is small enough, we have

    ∀ t ∈ [0,T], E_1[H](t) ≤ 6ϵ(1+t)^η/2.

Moreover, 

    ∀ 1 ≤ i ≤ |I_1|,  z ∈𝐤_0,  (t,x) ∈ [0,T]×^n, ∫_v |zH^i| dv ≲ϵ(1+t)^η/2/τ_+^n-1τ_-.





 We use again the continuity method. Since, for ϵ small enough, 𝔼_1[H](0) ≤ 2ϵ, there exists a larger time 0<T≤ T such that 

    ∀ t ∈ [0, T], 𝔼_1[H](t) ≤ 6ϵ(1+t)^η/2 .


Following the argument of Section <ref>, we almost get that for ϵ small enough, 𝔼_1[H] ≤ 5 ϵ(1+t)^η on [0,T]. In fact, using Lemma <ref>, we have that T_F(H^β) is a linear combination of terms like ℒ_Z^γ(F)(v,W), with |γ| ≤ N-5n+2/2. Thus we can use the null decomposition of the velocity vector and the electromagnetic field (and use its pointwises estimates) and then make similar computations as in Section <ref>. As we cannot use (<ref>) (the algebraic relations between S Z^β f and ∂_μZ^β f (μ∈ 0, n), for instance, are not necessarily conserved by the decomposition X=H+G), we need to reexamine the terms corresponding to (<ref>)-(<ref>). For instance, for the terms analogous to one of (<ref>), we have to prove, for z ∈𝐤_0,

    ∫_0^t ∫_Σ_s∫_v τ_+ | α| | zv^B/(v^0)^2Z^θ H^q| dv dx ds ≲ϵ^3/2(1+t)^η/2.

As |v^B| ≲√(|v^L v^L) by Proposition <ref> and as τ_+ |α| ≲√(ϵ)/τ_+^n-3/2τ_-, we have, by the Cauchy-Schwarz inequality (in (s,x,v)), that (<ref>) is bounded by the product of

    (∫_0^t ∫_Σ_sϵ/τ_+^n-3∫_v |z Z^θ H^q | dv dx ds )^1/2
 with 
    (∫_u=-∞^t1/τ_-^2∫_C_u(t)∫_vv^Lv^L/(v^0)^4|z Z^θ H^q | dv d C_u(t) du )^1/2.


The first factor is bounded by 

    ( ∫_0^t ϵ/1+s𝔼_1[H](s) ds )^1/2≲ϵ(1+t)^η/4 ,

and the other one, since v^L/(v^0)^3≲ 1 on the support of H, by

    √(𝔼_1[H](t))(∫_u=-∞^+ ∞1/τ_-^2  du)^1/2≲√(ϵ) (1+t)^η/4 .


The other terms are easier to bound. Let us study also the terms analogous to one of (<ref>), as there are also the cause of the (1+t)^η/2-loss[Note that we could use that √(τ_+τ_-)|v^B| ≲ v^0∑_z ∈𝐤_0 |z| in (<ref>) to obtain a better bound in (<ref>) for an other energy of H. On the other hand, the loss coming from (<ref>) could not be avoided with such techniques.].

    ∫_0^t ∫_Σ_s∫_v τ_+|ρ| |zv^L/(v^0)^2Z^θ H^ξ| dv dx ds    ≲   √(ϵ)∫_0^t (1+s)^-n-2/2𝔼_1[H](s)ds 
       ≲   ϵ^3/2(1+t)^η/2.

The pointwise estimate on ∫_v |z||H^i|dv then ensues from the Klainerman-Sobolev inequality of Corollary <ref>.





  §.§.§ The inhomogeneous part


As in the massive case, let us introduce K, the solution of T_F(K)+A_1K+KA_2=B which verifies K(0,.,.)=0, and the function


    |KKY|_∞ = ∑_l 1 ≤ i ≤ |I_1| 
     1 ≤ j,p ≤ |I_2|  | K^j_i|^2 |Y_p|.


KY and G are solutions of the same system,

    T_F(KY)=T_F(K)Y+KT_F(Y)    =     BY-A_1KY-KA_2Y+KA_2Y 
        =     BY-A_1KY.

As KY(0,.,.)=0 and G(0,.,.)=0, it comes that KY=G. For 1 ≤ i ≤ |I_1| and 1 ≤ j,p ≤ |I_2|, | K^j_i|^2 Y_p sastifies the equation

    T_F( |K^j_i|^2 Y_p) = |K^j_i |^2(A_2)^q_pY_q-2((A_1)^q_i K^j_q +K^q_i (A_2)^j_q ) K^j_i Y_p+2B^j_iK^j_iY_p,


which will allow us to estimate 
    𝔼[|KKY|_∞]:=𝔼_0,1[|KKY|_∞].
 

We will then be able to prove L^2 estimates for  ∫_v ∈ℝ^n |G| dv thanks to the estimates on ∫_v ∈ℝ^n |Y| dv and on 𝔼[|KKY|_∞].


We have, if ϵ is small enough,

    ∀ t ∈ [0,T], 𝔼[|KKY|_∞] ≤ϵ(1+t)^η.




Let T>0 be the largest time such that 𝔼[|KKY|_∞](t)≤ 2ϵ(1+t)^η for all t ∈ [0,T] and let us prove, with Proposition <ref>, that for ϵ small enough, 𝔼[|KKY|_∞](t) ≤ϵ(1+t)^η for all t ∈ [0,T]. It will follow that T=T.
As for the estimate of 𝔼_1[H] in the proof of Lemma <ref>,
 
    ∫_0^t ∫_Σ_s∫_v | |K^j_i |^2(A_2)^q_pY_q-2((A_1)^q_i K^j_q +K^q_i (A_2)^j_q ) K^j_i Y_p ||z|/v^0dvdxds ≲

 
    ϵ^3/2(1+t)^η

 and
 
    ∫_0^t ∫_Σ_s∫_v | F ( v/v^0, ∇_v ( |z| ) ) | |KKY|_∞ dv dx ds ≲ϵ^3/2.

Next, we need to estimate the following integral,

    ∫_0^t∫_Σ_s∫_v |z|/v^0|B^j_iK^j_iY_p| dv dx .

Recall from Lemma <ref> that 

    |B^j_iK^j_iY_p| ≲∑_m=1^n∑_|γ| ≤ Nτ_+|v^μ/v^0ℒ_Z^γ(F)_μ mK^j_iY_p|.

The components of the matrix B involve terms in which the electromagnetic field has too many derivatives to be estimated pointwise. We fix |γ| and we denote the null decomposition of ℒ_Z^γ(F) by (α, α, ρ, σ). In order to bound (<ref>), we bound the integral of the five following terms, given by the null decomposition of the velocity vector v and ℒ_Z^γ(F).


  * The terms which do not involve α

    τ_+|α||z||KY|/v^0, τ_+|ρ||z||KY|/v^0andτ_+|σ||z||KY|/v^0.


  * The terms involving α

    τ_+|α|v^L/(v^0)^2|z||KY| andτ_+ |α||v^B|/(v^0)^2|z||KY|.



We start by bounding the integral on Σ_s × (^n_v ∖{ 0 } ) of the good terms. We use ζ to denote either α, ρ or σ. Using twice the Cauchy-Schwarz inequality (in x and then in v) and that 1/v^0≲ 1 on the support of Y, we have

    ∫_Σ_s∫_v τ_+ |ζ| |zKY|/v^0 dvdx     ≲   τ_+ |ζ| _L^2(Σ_s)( ∫_Σ_s(∫_v |zKY| dv )^2 dx )^1/2
       ≲   | ℰ_N[F](s) ∫_Σ_s∫_v |zY| dv ∫_v |zKKY| dv dx |^1/2
       ≲   |ℰ_N[F](s) ∫_v |zY| dv _L^∞(Σ_s)𝔼[|KKY|_∞]|^1/2.


Using the bootstrap assumptions, on ℰ_N[F] and 𝔼[|KKY|_∞], and the pointwise decay estimate ∫_v |zY| dv ≲ϵτ_+^-n+1τ_-^-1 given in Lemma <ref>, we obtain

    ∫_0^t ∫_Σ_s∫_v τ_+ |ζ| |zKY|/v^0 dvdx ds ≲∫_0^t ϵ^3/2√(χ(t))/(1+s)^3/2-ηds ≲ϵ^3/2(1+t)^η.


As in the massive case, to unify the study of the terms involving α, we use v to denote v^L or v^B. Using the Cauchy-Schwarz inequality (in (s,x)), we have

     ∫_0^t ∫_Σ_s τ_+ |α| ∫_v |v|/(v^0)^2|z||KY| dv dx ds ≲   


    | ∫_0^t ∫_Σ_sτ_- |α|^2/(1+s)^n-3dxds  ∫_0^t ∫_Σ_sτ_+^2 (1+s)^n-3/τ_-|∫_v |vz/(v^0)^2KY|dv |^2 dxds |^1/2.

As, by the bootstrap assumption <ref>, √(τ_-) |α| ^2_L^2(Σ_s)≲ϵ (1+s)^η, we have

    ∫_0^t ∫_Σ_sτ_- |α|^2/(1+s)^n-3dxds ≲ϵ(1+t)^η.

For the second factor of the product in (<ref>), we first note that, by the Cauchy-Schwarz inequality and that 1/(v^0)^2≲ 1 on the support of Y, 

    (∫_v |vz/(v^0)^2 KY|dv )^2 ≤∫_v |zY|dv ∫_v |v/v^0|^2|z||KKY|_∞ dv.

Now, recall from Proposition <ref> that |v^B| ≲√(v^L v^L) so that |v/v^0|^2 ≲v^L/v^0. Using the pointwise estimate ∫_v |zY|dv ≲ϵτ_+^-n+1τ_-^-1, it comes

    (∫_v |vz/(v^0)^2 KY|dv )^2 ≲ϵ/τ_+^n-1τ_-∫_v v^L/v^0|z||KKY|_∞ dv.

As ∫_C_u(t)∫_v v^L/v^0|z| |KKY|_∞ dC_u(t) dv ≤𝔼[|KKY|_∞](t) ≤ 2ϵ(1+t)^η, we obtain

    ∫_0^t ∫_Σ_sτ_+^2 (1+s)^n-3/τ_-|∫_v |vz/(v^0)^2 KY|dv |^2 dxds ≲ϵ^2(1+t)^η∫_u=-∞^tτ_-^-2 du .

Hence,

    ∫_0^t ∫_Σ_s∫_v τ_+ |v|/(v^0)^2|α| |zKY| dv dx ds ≲ϵ^2(1+t)^η

and the energy estimate of Proposition <ref> gives that, for ϵ small enough, 𝔼[|KKY|_∞] ≤ϵ(1+t)^η on [0,T].




  §.§.§ The L^2 estimates


We start with the following proposition.


We have,

    ∀ |β| ≤ N,  t ∈ [0,T], ∫_v ∈ℝ^n ∖{0 } | Z^β f_k | dv _L^2(Σ_t)≲ϵ/(1+t)^n-1-η/2

and

    ∀ |β| ≤ N,  t ∈ [0,T], τ_+ ∫_v ∈ℝ^n ∖{0 } | Z^β f_k | dv _L^2(Σ_t)≲ϵ/(1+t)^n-3-η/2


We can remove the (1+t)^η/2-loss if |β| ≤ N-2n.



Let 1 ≤ k ≤ K. The first inequality ensues from the second one since 1+t ≤τ_+. If |β| ≤ N-2n, we only have to use the pointwise estimate (<ref>) and Lemma <ref>. If |β| > N-2n, recall that there exists 1 ≤ i ≤ |I_1| such that Z^β f_k=H^i+G^i. For 1 ≤ i ≤ |I_1|, Lemmas <ref> and <ref> imply 

    τ_+ ∫_v | H^i| dv _L^2(Σ_t)≲ϵ/(1+t)^n-3-η/2.
 
Moreover, as G=KY, we have, by the Cauchy-Schwarz inequality (in v),

    τ_+ ∫_v | G^i| dv _L^2(Σ_t)≤τ_+^2 ∫_v |Y|_∞dv ∫_v |K^j_i|^2|Y_j|dv _L^1(Σ_t)^1/2.

As, by Lemmas <ref> and <ref>, 
    τ_+^2∫_v |Y|_∞ dv _L^∞(Σ_t)≲ϵ/(1+t)^n-3and∫_v |K^j_i|^2|Y_j|dv _L^1(Σ_t)^1/2≤ϵ^1/2(1+t)^η/2,

we have

    τ_+ ∫_v | G^i| dv _L^2(Σ_t)≲ϵ/(1+t)^n-3-η/2.




These inequalities will not be sufficient to close the estimate on the energy ℰ^S_N-n+2/2[F] in the next section. This is why we prove the following proposition.


For all |β| ≤ N and all t ∈ [0,T], we have :


    ∫_v v^L/v^0|Z^βf_k| dv _L^2(Σ_t)   ≲   ϵ/(1+t)^n+1-η/2, 
    τ_-/τ_+∫_v v^L/v^0|Z^βf_k| dv _L^2(Σ_t)   ≲   ϵ/(1+t)^n+1-η/2, 
    ∫_v |v^B/v^0Z^βf_k| dv _L^2(Σ_t)   ≲   ϵ/(1+t)^n+1-η/2.



We can remove the (1+t)^η/2-loss if |β| ≤ N-2n.





If |β| ≤ N-2n, these inequalities are implied by the pointwise estimates (<ref>), (<ref>) and (<ref>) and Lemma <ref>. 

If |β| > N-2n, we prove in the same way that these inequalities are true if we replace Z^β f_k by H^i, with 1 ≤ i ≤ |I_1| such that Z^β f_k=H^i+G^i. It then only remains to consider G^i. Recall that by Proposition <ref> and Lemma <ref>,

    v^L/v^0≤τ_+^-1∑_z ∈𝐤_0 |z|, v^L/v^0≤τ_-^-1∑_z ∈𝐤_0 |z| and∫_v |z||Y|_∞ dv ≲ϵ/τ_+^n-1τ_-.
 Hence, using also G=KY, the Cauchy-Schwarz inequality (in v) and 𝔼[|KKY|_∞](t) ≤ 2ϵ (1+t)^η, we have


    ∫_v v^L/v^0|G^i| dv _L^2(Σ_t)   ≲   ∫_v v^L/v^0 |Y|_∞ dv ∫_v v^L/v^0|(K_i^j)^2 Y_j| dv _L^1(Σ_t)^1/2
       ≲   ∑_z ∈𝐤_0 τ_+^-2∫_v  |z||Y|_∞ dv ∫_v |z| |KKY|_∞ dv _L^1(Σ_t)^1/2
       ≲   ϵ(1+t)^-n+1-η/2,


    τ_-/τ_+∫_v v^L/v^0|G^i| dv _L^2(Σ_t)   ≲   τ_-^2/τ_+^2∫_v  v^L/v^0|Y|_∞ dv ∫_v v^L/v^0|(K_i^j)^2 Y_j| dv _L^1(Σ_t)^1/2
       ≲   ∑_z ∈𝐤_0 τ_+^-2∫_v  |z||Y|_∞ dv ∫_v |z| |KKY|_∞ dv _L^1^1/2
       ≲   ϵ(1+t)^-n+1-η/2.


The remaining estimate can be proved in a similar way, using |v^B| ≲v^0/τ_+∑_z ∈𝐤_0 |z| (see Proposition <ref>).





 §.§ Step 5: Improvement of the electromagnetic field estimates



  §.§.§ Improvement of the energies estimates for the potential


According to the energy estimate of Proposition <ref> and the commutation formula of Proposition <ref>, one has, for all t ∈ [0,T],


    √(ℰ_N[A])(t) ≲√(ℰ_N[A])(0)+∑_|γ| ≤ N∫_0^t τ_+ e^k∫_^n |Z^γ f_k| dv_L^2(^n) ds.


As ℰ_N[A](0) ≤ϵ and τ_+ e^k∫_^n |Z^γ f_k| dv _L^2(^n)≲ϵ (1+t)^-n-3-η/2 (see Proposition <ref>), we have, for ϵ small enough and if the constant C is large enough,

    ∀ t ∈ [0,T], ℰ_N[A](t) ≤C/2(n-3)ϵχ(t)(1+t)^η,

with χ such that

    χ ( s )= 1+s if n=4, χ ( s )= log^2(3+s)if n=5 andχ(s) = 1 if n ≥ 6.
 

Similarly, using (<ref>) and Lemma <ref>, we obtain

    ∀ t ∈ [0,T], ℰ_N-2n[A](t) ≤C/2(n-3)ϵχ(t).

This concludes the improvement of the bootstrap assumption (<ref>).



  §.§.§ Improvement of the estimate on ℰ^0_N[F]


Recall from Proposition <ref> that, for all t ∈ [0,T],

    ℰ^0_N[F](t) - 2ℰ^0_N[F](0) ≲∑_|β|,|γ| ≤ N |e^k|∫_0^t ∫_Σ_s  | ℒ_Z^β (F)_0 ν J(Z^γ f_k)^ν| dx ds.

As, by the Cauchy-Schwarz inequality, the bootstrap assumption (<ref>) and the L^2 estimates of Proposition <ref>,

    ∫_0^t ∫_Σ_s  | ℒ_Z^β (F)_0 ν J(Z^γ f_k)^ν| dx ds    ≲   ∫_0^t ℒ_Z^β (F)_L^2(Σ_s)J(Z^γ f_k) _L^2(Σ_s) ds 
       ≲   ∫_0^t √(ℰ^0_N[F](s))∫_v |Z f_k dv _L^2(Σ_s) ds 
       ≲   ∫_0^t ϵ^1/2ϵ(1+s)^n-1-η/2 ds 
       ≲   ϵ^3/2,

we have, for ϵ small enough, ℰ^0_N[F] ≤ 3ϵ on [0,T].




  §.§.§ Improvement of the estimates on ℰ_N[F] and ℰ_N-2n[F]


Recall from Proposition <ref> that

    ℰ_N[F](t) ≤ℰ_N[F](0)+(n-3)ℰ_N[A](t)+φ(t),

where φ(t) is a linear combination of terms such that

    ∫_0^t ∫_Σ_s  | K_0^νℒ_Z^β F_μν J(Z^γ f_k)^μ| dx ds and∫_0^t ∫_Σ_s s |ℒ_Z^δ A_μ□ℒ_Z^δ A^μ | dx ds,

 with |β|, |γ|, |δ| ≤ N and 1 ≤ k ≤ K. Then, if we could prove that each integrals of (<ref>) is bounded by ϵ^3/2χ(t)(1+t)^η, we would have, for ϵ small enough and C large enough, ℰ_N [F] ≤ C ϵχ(t)(1+t)^η on [0,T] since 𝔼_N[F](0) ≤ϵ and (n-3)ℰ_N[A](t) ≤C/2ϵχ(t)(1+t)^η.


We could estimate the integrals of (<ref>) with a better bound (the computations are similar to those done below in Section <ref>, but this would not give us a better estimate on ℰ_N[F] because of the χ(t)(1+t)^η-loss on ℰ_N[A]. 


We start by bounding the integrals involving the potential. Using Proposition <ref> and the Cauchy-Schwarz inequality, we have, for |δ| ≤ N,


     ∫_0^t ∫_Σ_s s |ℒ_Z^δ A_μ □ℒ_Z^δ A^μ | dx ds ≲   

 
    ∑_k=1^K ∑_|γ| ≤ |δ|∫_0^t √(ℰ_N[A](s))τ_+∫_v |Z^γ f_k| dv _L^2(Σ_s) ds.

Using the L^2 estimates of Proposition <ref> and that ℰ_N[A](s) ≲ϵχ(s)(1+t)^η, it comes

    ∑_|δ| ≤ N∫_0^t ∫_Σ_s s |ℒ_Z^δ A_μ□ℒ_Z^δ A^μ | dx ds    ≲   ϵ^3/2∫_0^t √(χ(s))/(1+s)^n-3/2-η ds 
       ≲   ϵ^3/2χ(t)(1+t)^η.


In order to estimate the remaining integrals of (<ref>), we express K_0^νℒ_Z^β (F)_μν J(Z^γ f_k)^μ in null coordinates. Dropping the dependance in ℒ_Z^β(F) or Z^γ f_k, this gives us the four following terms :


    τ_+^2 ρ J^L, τ_-^2 ρ J^L, τ_+^2 α_B J^B,  andτ_-^2 α_B J^B.


As 
    J^L=∫_v v^L/v^0Z^γ f_k dv,  J^L=∫_v v^L/v^0Z^γ f_k dv and J^B=∫_v v^B/v^0Z^γ f_k dv,

we have,

    |J^L|,  |J^L|,  |J^B| ≲∫_v |Z^γ f_k| dv.

The integrals (on [0,T] ×ℝ^n_x ×( ℝ^n_v ∖{0})) of each of the four terms of (<ref>) are then bounded, using the Cauchy-Schwarz inequality (in x), by


    ∫_0^t √(ℰ_N[F])(s) τ_+ ∫_v |Z^γ f_k| dv _L^2(Σ_s) ds.

By Proposition <ref> and the bootstrap assumption (<ref>),

    ∫_0^t √(ℰ_N[F])(s) τ_+ ∫_v |Z^γ f_k| dv _L^2(Σ_s) ds    ≲   ∫_0^t √(ϵχ(s))ϵ/(1+s)^n-3/2-η ds 
       ≲   ϵ^3/2χ(t)(1+t)^η.


Hence, ℰ_N[F](t) ≤ Cϵχ(t)(1+t)^η for all t ∈ [0,T] if ϵ is small enough. We can prove exactly in the same way that ℰ_N-2n[F](t) ≤ Cϵχ(t) for all t ∈ [0,T] if ϵ is small enough.

We then improve the bootstrap assumption (<ref>).



  §.§.§ Improvement of the estimates on ℰ^S_N[F] and ℰ^S_N-2n[F]


Recall from Propositions <ref> and <ref> that, for all t ∈ [0,T],

    ℰ^S_N[F](t) ≤ℰ_N[F](0)+C_n(ℰ_N[A](0)+ℰ_N[A](t)/1+t+ℰ_N[F](t)/1+t)+ψ(t),

where C_n is a positive constant and where ψ(t) is a linear combination of terms such as

    ∫_0^t ∫_Σ_s |ℒ_Z^β(F)_0 μ J^μ(Z^γf_k) |+|S^νℒ_Z^β(F)_νμ J^μ(Z^γf_k) | dx ds ,

with |β|, |γ| ≤ N and 1 ≤ k ≤ K, and 

    ∫_0^t ∫_Σ_s|ℒ_Z^β(A)_μ∫_v v^μ/v^0Z^γ f_k |dx ds,

with |β|, |γ| ≤ N and 1 ≤ k ≤ K.

Let |β|+|γ| ≤ N and 1 ≤ k ≤ K. We denote the null decomposition of ℒ_Z^β(F) by (α, α, ρ, σ), Z^γf_k by g and J(Z^γf_k) by J. Expressing ℒ_Z^β(F)_0 μ J^μ(g) and S^νℒ_Z^β(F)_νμ J^μ(g) in null components, (<ref>) would be bounded by ϵ^3/2 if


     ∫_0^t∫_Σ_s |τ_+ ρJ^L|dxds ≲ϵ^3/2,  ∫_0^t∫_Σ_s |τ_- ρJ^L|dxds ≲ϵ^3/2, 

     ∫_0^t∫_Σ_s |τ_+ αJ^B|dxds ≲ϵ^3/2  and  ∫_0^t∫_Σ_s |τ_+ α J^B|dxds ≲ϵ^3/2. 

By the Cauchy-Schwarz inequality,


    ∫_0^t ∫_Σ_sτ_+ |ρ J^L| dx ds ≲∫_0^tτ_+ ρ_L^2(Σ_s)∫_v v^L/v^0|g| dv _L^2(Σ_s) ds .


Since, by the bootstrap assumption (<ref>),  τ_+ ρ^2_L^2(Σ_s)≲ϵχ(s)(1+s)^η and, according to Proposition <ref>, ∫_v v^L/v^0|g| dv _L^2(Σ_s)≲ϵ (1+s)^-n+1-η/2, it comes that
 
    ∫_0^t ∫_Σ_sτ_+^2 |ρ J^L| dx ds ≲ϵ^3/2∫_0^t √(χ(t))/(1+s)^n+1/2-η ds ≲ϵ^3/2.

The other terms are treated similarly.

    ∫_0^t ∫_Σ_sτ_- |ρ J^L| dx ds ≲∫_0^tτ_+ ρ_L^2(Σ_s)τ_-/τ_+∫_v v^L/v^0|g| dv _L^2(Σ_s) ds ≲ϵ^3/2 ,





    ∫_0^t ∫_Σ_sτ_+ |α_B J^B| dx ds ≲∫_0^tτ_+ α_L^2(Σ_s)∫_v v^B/v^0|g| dv _L^2(Σ_s) ds ≲ϵ^3/2,




    ∫_0^t ∫_Σ_sτ_- |α_B J^B| dx ds ≲∫_0^tτ_- α_L^2(Σ_s)∫_v v^B/v^0|g| dv _L^2(Σ_s) ds ≲ϵ^3/2.


Denoting ℒ_Z^β(A) by B, (<ref>) would be bounded by ϵ^3/2(1+t)^η if we prove that


    ∫_0^t ∫_Σ_s |B_L J^L| dx ds ≲ϵ^3/2 (1+t)^η,



    ∫_0^t ∫_Σ_s |B_L J^L| dx ds ≲ϵ^3/2and∫_0^t ∫_Σ_s |B_D J^D| dx ds ≲ϵ^3/2.



Let us show (<ref>) first. Using Proposition <ref> and the bound on ℰ_N[A], we have 
    B_L^2(Σ_s)≲√(ϵχ(s)(1+t)^η)andJ^D_L^2(Σ_s)+J^L_L^2(Σ_s)≲ϵ/(1+s)^n-η/2.
 Hence, by the Cauchy-Schwarz inequality,


    ∫_0^t ∫_Σ_s |B_L J^L|+|B_DJ^D| dx ds ≲ϵ^3/2∫_0^t √(χ(s))/ (1+s)^2-η ds ≲ϵ^3/2.


For (<ref>), we have


    ∫_0^t ∫_Σ_s |B_L J^L| dx ds    ≲   ∫_0^t B_L_L^2(Σ_s)∫_v |g| dv_L^2(Σ_s) ds 
       ≲   ϵ^3/2∫_0^t √(χ(s))/(1+s)^n-1/2-ηds. 
       ≲   ϵ^3/2(1+t)^η.


Hence, if ϵ is small enough and C large enough, we have ℰ^S_N[F] ≤Cϵ (1+t)^η for all t ∈ [0,T].

In view of the above, ℰ^S_N-2n[F] ≤Cϵ on [0,T], for ϵ small enough, would follow if we improve the bound in (<ref>) from ϵ^3/2(1+t)^η to ϵ^3/2, when |β| ≤ N-n+2/2. To do this, we use a pointwise estimate on B_L and we keep J^L in L^1-norm. By Lemma <ref>, we have


    |B_L(t,x)| ≲√(ϵχ(t)(1+t)^η)/τ_+^n/2,

which implies 

    ∫_0^t ∫_Σ_s |B_L J^L| dx ds    ≲   ∫_0^t B_L_L^∞(Σ_s)g_L^1(Σ_s)ds 
       ≲   ϵ^3/2∫_0^t √(χ(s))log^*(3+s)/(1+s)^n-η/2ds ≲ϵ^3/2.


This concludes the improvement of the bootstrap assumption (<ref>).



§ NON EXISTENCE


We show in this chapter the following proposition. Let us denote (1,...,1) by u⃗ and we recall that E^i=F_0i.




Let the dimension n be such that n ≥ 2 and let χ : →_+ be a function of class C^∞ such that χ=1 on ]-∞,1] and χ=0 on [3,+∞[. We suppose also that χ is decreasing on [1,3]. Let also M ∈_+ such that M^-1= ∫_v ∈^nχ(|v|^2) dv.

The Vlasov-Maxwell system (<ref>)-(<ref>), with two species (K=2), e_1=1, e_2=-1, m_1 =0, m_2 ∈_+ and the initial data 


    E_0:x ↦ 10χ(2)^-1χ(2r^2/n)u⃗, F_0_ij=0 for all 1 ≤ i,j≤ n,


    f_01=M(div(E_0)+div(E_0)_L^∞(^n))χ( 2r^2/3n) χ(|v|^2),
 and 
    f_02=M div(E_0)_L^∞(^n)χ( 2r^2/3n) χ(|v|^2),


do not admit a C^1 local solution, provided[Note that such a function χ exists. Recall for instance the classical construction of cut-off functions] w ↦ w χ'(2w^2) is not constant on a neighborhood of 1.




Note that the initial data satisfy the constaint equations. Indeed, 

    ∫_v f_01-f_02dv=div(E_0) χ(2r^2/3n)
and x ↦χ(2r^2/3n) is equal to 1 on the support of E_0. The other ones, ∇_[iF_0_jk]=0, are obvious to check.




There is uniqueness for a such Cauchy problem in the class of the local C^1 functions. Indeed, let (f_1,f_2,F) and (g_1,g_2,G) be two such solutions on [0,T]. As f_i and g_i are the unique C^1 solution of T_(-1)^i+1F(h)=0 and T_(-1)^i+1G(h)=0 on [0,T], respectively, we obtain with the method of characteristics that they both vanish for |x| ≥3/√(2)√(n)+T. In view of the wave equations (<ref>) and (<ref>), the same is true for F and G. All the integrals considered below will then be finite. We have

    T_F(f_q-g_q)     =     (G-F)(v,∇_v g_q), 
    ∇^μ (F-G)_μν    =     e^q J(f_q-g_q), 
    ∇^μ^*  (F-G)_μα_1 ... α_n-2    =     0.

Using Propositions <ref> and <ref>, we obtain

    h(t)     :=    ∑_q=1^2 𝔼_0[f_q-g_q](t)+√(ℰ^0_0[F-G](t))
       ≲   ∫_0^t h(s)(1+ ∫_v |e^k∇_v g_k|dv _L^2(Σ_s))ds.

The Grönwall lemma gives us that h=0 on [0,T], implying (f,F)=(g,G).



The strategy of the proof of Proposition <ref> is to construct, for all T_0>0, a characteristic of the system such that its velocity part vanish in a time less than T_0. For this, we make crucial use of the colinearity of y ↦ E(t,yu⃗) and u⃗ which is a corollary of the following subsection.



 §.§ A symmetry property for the Vlasov-Maxwell system


To lighten the notations, we use x_(ij), if i ≠ j, to denote (x^1,...,x^i-1,x^j,x^i+1,...,x^j-1,x^i,x^j+1,...,x^n).



We consider the n dimensional Vlasov-Maxwell system, with K species,


    T_m_q(f_q)  +e_qv^0E^i∂_v^if_q+e_qv^i F_i^j ∂_v^j f_q     =     0 ,
    ∇^μ F_μν    =     e^q J(f_q)_ν, 
    ∇^μ^*  F_μλ_1 ... λ_n-2    =     0  ,


with the initial smooth data f_q(0,.,.)=f_0q, F(0,.)=F_0. We suppose that the initial data satisfy the following symmetry relations

    f_q0(x_(ik),v_(ik))     =     f_q0(x,v),  i ≠ j, 
     E^i_0(x_(ik))     =     E^k_0(x),  i ≠ k, 
     E^i_0(x_(kl))     =     E^i_0(x),  l ≠ i,  k ≠ i. 
     (F_kl)_0(x_(kl))     =     -(F_kl)_0(x), 
    
    (F_kl)_0(x_(ik))     =     (F_il)_0(x),  l ≠ k,i,  
     (F_kl)_0(x_(ij))     =     (F_kl)_0(x),  i ≠ k,l,  j ≠ k,l.


If there is a unique classical solution (f_1,...,f_K,F) on [0,T[, then (f_1(t,.,.),...,f_K(t,.,.),F(t,.)) satisfies also these symmetries.




To simplify the notation, we suppose that K=1, e_q=1 and we consider the transposition τ=(12). We denote (y^2,y^1,y^3,...,y^n) by y_τ, m_1 by m and f_1 by f. Let g and G be defined by 

    g(t,x,v)     :=     f(t,x_τ,v_τ), 
    
    G_02(t,x)     :=     E^1(t,x_τ), 
    
    G_01(t,x)     :=     E^2(t,x_τ),
    
    G_0k(t,x)     :=     E^k(t,x_τ),  k ≥ 3, 
    
    G_12(t,x)     :=     -F_12(t,x_τ), 
    
    G_1k(t,x)     :=     F_2k(t,x_τ),  k ≠ 1,  k ≠ 2, 
    
    G_2k(t,x)     :=     F_1k(t,x_τ),  k ≠ 1,  k ≠ 2, 
    
    G_kl(t,x)     :=     F_kl(t,x_τ),  k,l ≥ 3

and let D^k=G_0k. We want to prove that (g,G)=(f,F). By assumption, this is true for t=0 and, by uniqueness, it will be true for t<T if we can prove that (g,G) is solution to the same system as (f,F).



Propagation of symmetry for the Maxwell equations 



Let us prove first that ∇^μ G_μν= J(g)_ν. As J(h)^ν= ∫_v v^ν/v^0hdv, we have, by the change of variables v'=v_τ,

    J(g)^1(t,x)=J(f)^2(t,x_τ),  J(g)^2(t,x)=J(f)^1(t,x_τ)
 and 
    J(g)^ν(t,x)=J(f)^ν(t,x_τ) if ν≠ 1,2.

The equation ∇^i G_i0=J(g)_0 then comes from

    ∂_1 D^1(t,x)=∂_2 E^2(t,x_τ), ∂_2 D^2(t,x)=∂_1 E^1(t,x_τ) and∇_i E^i=J(f)_0.

As 

    ∇^μ G_μ 1(t,x)     =     -∂_t E^2(t,x_τ)-∂_2(F_21(t,x_τ))+∑_i=3^n∇^i(F_i 2(t,x_τ)) 
        =     -∂_t E^2(t,x_τ)-∂_1 F_21(t,x_τ)+∑_i=3^n∇^iF_i 2(t,x_τ) 
        =    ∇^μ F_μ 2(t,x_τ),

we have ∇^μ G_μ 1= J(g)_1. The equation ∇^μ G_μ 2(t,x)= J(g)_2 can be obtained similarly. The remaining equations are obtained from

    ∇^j G_j k(t,x)     =    ∂_1(F_2k(t,x_τ))+∂_2(F_1k(t,x_τ))+∑_i=3^n∇^i(F_i k(t,x_τ)) 
        =    ∂_2 F_2k(t,x_τ)+∂_1 F_1k(t,x_τ)+∑_i=3^n∇^iF_i 2(t,x_τ) 
        =    ∇^j F_j k(t,x_τ)

and ∂_t D^k(t,x)=∂_t E^k(t,x_τ), for k ≥ 3. For the other part of the Maxwell equations, recall from Proposition <ref> that it is equivalent to prove 

    ∇_[ λ G_μν ]=0.

We have

    ∇_[ 1 G_2 3 ](t,x)     =    ∂_1 ( F_13(t,x_τ))+∂_2 ( F_32(t,x_τ))-∂_3 ( F_12(t,x_τ)) 
        =    ∂_2  F_13(t,x_τ)+∂_1  F_32(t,x_τ)+∂_3 F_21(t,x_τ) 
        =    ∇_[ 2 F_1 3 ](t,x_τ) 
       =    0.

The other equations can be obtained in the same way.



Propagation of symmetry for the Vlasov equation



We have

    T_m(g)(t,x,v)     =     v^1 ∂_2 f(t,x_τ,v_τ)+v^2 ∂_1 f(t,x_τ,v_τ) + ∑_lμ=0 
    μ≠ 1 ,2  ^nv^μ∂_μ f(t,x_τ,v_τ) 
        =     T_m(f)(t,x_τ,v_τ ) .


Moreover, as

    ∂_v^1g (t,x,v)= ∂_v^2f(t,x_τ,v_τ) and∂_v^2g (t,x,v)= ∂_v^1f(t,x_τ,v_τ),
 
    D^i(t,x)∂_v^i g(t,x,v)=E^i(t,x_τ)∂_v^i f(t,x_τ,v_τ).
 Finally,

    (v^kG_k1∂_v^1 g)(t,x,v)     =    (-v^2 F_21(t,x_τ)+∑_k=3^n v^k F_k2(t,x_τ) ) ∂_v^2 f(t,x_τ,v_τ) 
        =    (v^k F_k2∂_v^2 f )(t,x_τ,v_τ),

and more generally

    (v^kG_kj∂_v^j g)(t,x,v)= (v^k F_k τ(j)∂_v^τ(j) f )(t,x_τ,v_τ).


We then deduce, 
    T_G(g)=0, as T_F(f)(t,x_τ,v_τ)=0.




The symmetries are propagated over time

 

We then proved that (g,G) satisfies the same system as (f,F). As (f,F)=(g,G) at t=0, we have, by the uniqueness of the solution, that (f,F)=(g,G) for all t ∈ [0,T[. 




More generally, from the above proof, (f,F) ↦ (g,G) maps C^1 solutions of the Vlasov-Maxwell system to C^1 solutions of the Vlasov-Maxwell system.




 §.§ Proof of Proposition <ref>


Let us suppose that the system admits a local C^1 solution on [0,T], with T>0, which is then necessarily unique. We will reduce T later if necessary, but we already assume that T ≤ 1.



  §.§.§ Some informations on the electromagnetic fields around u⃗


We start by the study of the solution around u⃗. Let us introduce M_0:=20 χ( 2 )^-1 and (B_ij)_1 ≤ i,j≤ n the 2-form defined by B_ij=F_ij.



Reducing T if necessary, we have the following properties.



  * Local bounds on the field: ∀ t∈ [0,T],

    ∀  |x| ≤√(n)+2T,   1≤ i ≤ 3,  5 ≤ E^i(t,x) ≤ M_0 ,  |∂_t E(t,x)| ≤ 1

and

    ∀ t∈ [0,T],  |x-u⃗| ≤ 2T,  |B(t,x)| ≤1/4.


  * The field is locally-Lipschitz: ∃ L>0, ∀ t∈ [0,T],  |x|,  |y| ≤√(n)+2T,

    |E(t,x)-E(t,y)|+|B(t,x)-B(t,y)| ≤ L|x-y|.


  * Specific behaviour along the u⃗-direction:

    ∀  y ∈, t ∈ [0,T],  E(t,yu⃗) =E^1(t,yu⃗)u⃗and B(t,yu⃗)=0.
 





In view of the initial data, we have B(0,u⃗)=0 and

    ∀ |y| ≤√(n),  1 ≤ i ≤ n,  10 ≤ E^i(0,y) ≤M_0/2, ∂_t E(0,y)=0.

The point 1 then ensues, taking T smaller if necessary, from the uniform continuity of the electromagnetic field on every compact subset of [0,T] ×^n. The point 2 comes from the mean value theorem, as E and B are C^1 and the point 3 follows from Proposition <ref>.





  §.§.§ The method of charateristics fails


Let us denote by (X(s,t,x,v),V(s,t,x,v)) the value at s of the characteristic, for the transport equation (<ref>) satisfied by f_1, which was equal to (x,v) at t. Let η∈ ]0,T[ and

    X_η:(s,t) ↦ X(s,t,u⃗, ηu⃗),   V_η:(s,t) ↦ V(s,t,u⃗, ηu⃗).

We now fix t ∈ [0,T[. (X_η(.,t),V_η(.,t)) is well defined on a neighborhood of t and we have, denoting v/|v| by v,



    d X_η(.,t)/ds(s)     =    V_η(s), 
    dV^j_η(.,t)/ds(s)     =     E^j(s,X_η(s,t))+V_η^i(s)  F_i^j(s,X_η(s,t)).




X_η(.,t), V_η(.,t) and E (along X_η(.,t)) stay collinear to u⃗. We have, as long as V_η stay positive, X_η(s,t)=(1+1/√(n)(s-t)) u⃗ and
    V_η(s,t)=ηu⃗+∫_t^s E^1(s',(1+1/√(n)(s-t)) u⃗) ds' u⃗.




We start by a change of coordinates. We consider an orthonormal system (u_i)_1 ≤ i ≤ n such that u_1=1/√(n)u⃗ and we denote by X^i and V^i the coordinates of X_η(.,t) and V_η(.,t) in this basis. Then, for all 1 ≤ i ≤ n,

    dX^i/ds(s) = V^i(s)/|V|(s)

and, for i ≥ 2, X^i(0)=0 and V^i(0)=0. We remark, using Proposition <ref>, that if X^i=0 for i ≥ 2, then E(s,X_η(s,t))=E^1(s,X_η(s,t))u⃗ and B(s,X_η(s,t))=0. Consider now the solution of the following system

    d r/ds    =    w/|w|, 
    d w/ds    =    √(n)E^1(s,r/√(n),...,r/√(n)),

with the initial data r(t)=√(n) and w(t)=η√(n).  The solution exists as long long as w ≠ 0 and we have 
    r(s)/√(n)=1-t-s/√(n)andw(s)/√(n) = η +∫_t^s E^1(s',1-t-s'/√(n),...,1-t-s'/√(n))) ds'.
 
By uniqueness of the solution of the system (<ref>)-(<ref>), we have 
    (X^1,..., X^n, V^1,...,V^n)=(r,0,...,0,w,,0,...,0),

which implies the result.




We now try to estimate the time when V_η vanishes.


The exists 0 < η_0 < T such that for all η∈ ]0,η_0[, there exists T_η such that if t<T_η, (X_η(.,t),V_η(.,t)) is well defined on [0,t] and if T_η≤ t <T there exists τ_η(t) ≤ t such that (X_η(.,t),V_η(.,t)) is well defined on ]t-τ_η(t),t] and

    lim_s →(t-τ_η(t))^+ V_η(s,t)=0, lim_s →(t-τ_η(t))^+ X_η(s,t)= (1-τ_η(t)/√(n)) u⃗.

Moreover, t ↦ t-τ_η(t) is in C^0([T_η,T[) ∩ C^1(]T_η,T[), vanishes at T_η, and such that

    ∀ t ∈ ]T_η,T[, 4/M_0≤∂ (t-τ_η)/∂ t(t) ≤M_0+1/5  .





We fix η∈ ]0,T[. Noting, by (<ref>), that 
    |X_η(s,t)-u⃗| ≤ |t-s| ≤ T,
 we obtain by Proposition <ref>, as X_η and u⃗ are collinear, that E(s,X_η(s,t))= E^1(s,X_η(s,t)) u⃗. Hence, if t ∈ [0,T[, only two situations can occur. Either (X_η(.,t),V_η(.,t) is well defined on [0,t], or there exists τ_η(t)<t such that

    lim_s →(t-τ_η(t) )^+ V_η(s,t)=0,

and the characteristic is well defined on ]t-τ_η(t),t].
Now, consider 
    g_η : (s,t) ↦η+∫_t^s E^1(s',(1-t-s'/√(n)) u⃗) ds'

so that, by Lemma <ref>, if t ∈ ]0,T[ and s is near to t, g_η(s,t) is equal to V^i_η(s,t) (for all 1 ≤ i ≤ n). For all t ∈ ]0,T[, g_η(.,t) stricly increases on [0,t], as E^1>0 by Proposition <ref>. As

    ∫_t^s E^1(s',(1-t-s'/√(n)) u⃗) ds'=-∫_0^t-s E^1(t-s',(1-s'/√(n)) u⃗) ds'

and

    ∂ g_η/∂ t(s,t)=-E^1(s,(1-t-s/√(n)) u⃗)-∫_0^t-s∂_t E^1(t-s',(1-s'/√(n)) u⃗) ds',

we have

    ∂ g_η/∂ t(s,t) ≤ -4,

so that g_η(s,.) is strictly decreasing on [s,T[. Moreover, by the bounds given on E^1 in Proposition <ref>, if t<η/M_0, g_η(.,t) does not vanish on [0,t] and vanishes exactly one time, in t-τ_η(t), if t ≥η/5. Then, if η is small enough, there exists t ∈ ]0,T[ such that g_η(.,t) vanishes in t-τ_η(t). Let t_1 be a such time and let t_2>t_1. We have

    0=g_η(t_1-τ_η(t_1),t_1)>g_η(t_1-τ_η(t_1),t_2),

implying the existence of t_2-τ_η(t_2) and t_1-τ_η(t_1)<t_2-τ_η(t_2), since

    g_η(t_1-τ_η(t_1),t_2)<0=g_η(t_2-τ_η(t_2),t_2).
 Hence, T_η exists[More precisely, T_η = sup{t ∈ ]0,T[  /  g_η(.,t) >0 on [0,t] }.] and t ↦ t-τ_η(t) strictly increases on [T_η,T[, vanishes in T_η and tends to zero as t → T_η. The fact that t ↦ t- τ_η(t) is in C^1(]T_η,T[) follows from the implicit function theorem, as g_η(t-τ_η(t),t)=0 and ∂ g_η/∂ s(s,t) ≥ 5. Furthermore, dropping the dependance in t of τ_η,

    ∂ (t-τ_η)/∂ t(t)=E^1(t-τ_η,(1-τ_η/√(n)) u⃗)+∫_0^τ_η∂_t E^1(t-s',(1-s'/√(n)) u⃗) ds'/E^1(t-τ_η,(1-τ_η/√(n)) u⃗),

which, by Proposition <ref>, implies the last statement.





Note that, if 0 < η < η_0, τ_η(T_η)=T_η and then g_η(0,T_η)=0. 

Later, we will use again that g_η(0,.) is strictly decreasing on [0,T[.







  §.§.§ The contradiction




We fix again η∈ ]0,η_0[. As V_η(.,t) is not defined on [0,t-τ_η(t)] if t > T_η, we cannot directly express f_1(t,u⃗,ηu⃗) in terms of f_01 by the method of the characteristics.

If t ≥ T_η, we extend X_η(.,t) and  V_η(.,t) on [0,t-τ_η(t)] by

    X_η(s,t) = (1+t-s-2τ_η(t)/√(n))u⃗and V_η(s,t) = ηu⃗+∫_t^s E(s',X_η(s',t)) ds'.



If t > t - τ_η(t), dX_η/ds(s,t)=u⃗/√(n). We extend X_η(.,t) on [0,t-τ_η(t)] in order that

    dX_η/ds(s,t)=-u⃗/√(n).

We then extend V_η(.,t) such that (<ref>) remains true on [0,t-τ_η(t)].


We have the following result.


    ∀ t ∈ [0,T[,  f_1(t,u⃗,ηu⃗)=f_01(X_η(0,t),V_η(0,t)).




If t<T_η, this follows from the method of characteristics. In order to prove the result for t ≥ T_η, we consider ϵ > 0, v_ϵ=(0,...,0,ϵ),

    X_η,t^ϵ : s ↦ X(s,t-τ_η(t),X_η(t-τ_η(t),t),v_ϵ)
 and 
    V_η,t^ϵ : t ↦ V(s,t-τ_η(t),X_η(t-τ_η(t),t),v_ϵ).

Proposition <ref> gives us that X_η,t^ϵ and V_η,t^ϵ are well defined on [0,T[. Indeed, as, by (<ref>), 

    |X_η,t^ϵ(s)-u⃗| ≤ |X_η,t^ϵ(s)-X_η(t-τ_η(t),t)| + |X_η(t-τ_η(t),t)-u⃗| ≤ 2T,

it comes

    ∀ 1 ≤ i ≤ n,  5 ≤ E^i(s,X_η,t^ϵ(s)) ≤ M_0 and |B(s,X_η,t^ϵ(s))| ≤1/4,

so that V_η,t^ϵ cannot vanish. Now, the method of characteristics gives us, for all t ∈ [0,T[,

    f_1(t,X_η,t^ϵ(t),V_η,t^ϵ(t))=f_01(X_η,t^ϵ(0),V_η,t^ϵ(0)).

Then, the result, for t ≥ T_η, follows from the continuity of f_1 and the following proposition.



We have

    lim_ϵ→ 0X_η(.,t)-X_η,t^ϵ_L^∞([0,t])+V_η(.,t)-V_η,t^ϵ_L^∞([0,t])=0.






On the one hand, as 

    ∀ v,  w ∈^n ∖{0 }, | v/|v|-w/|w|| ≤2/|w||v-w|,

we have

    |X_η(s,t)-X_η,t^ϵ(s)| ≤| ∫_t-τ_η(t)^s 2/|V_η,t^ϵ(w)||V_η(w,t)-V_η,t^ϵ(w)| dw |.


On the other hand, note first that for s < T and |x|, |y| ≤√(n)+2T, we have, by the local Lipschitz property of the electromagnetic field (<ref>),

    |E(s,x)-E(s,y)+v^i B_i(s,x)-w^iB_i(s,y)| ≤ L|x-y|+|v-w||B(s,x)|.

Then, using (<ref>), (<ref>) and the bound (<ref>) on the magnetic field,

     |V_η(s,t)-V_η,t^ϵ(s)| ≤   


    ϵ + | ∫_t-τ_η(t)^s L|X_η(w,t)-X_η,t^ϵ(w)|+1/2|V_η,t^ϵ(w)||V_η(w,t)-V_η,t^ϵ(w)| dw | .


Hence, by the Grönwall lemma, for all s ∈ [0,T[,

    |X_η(s,t)-X_η,t^ϵ(s)|+|V_η(s,t)-V_η,t^ϵ(s)| ≤ϵexp( | ∫_t-τ_η(t)^s L+5/2|V_η,t^ϵ(w)| dw | ) .


We now prove that, ∃ a>0, b>5/2 such that ∀ w ∈ [0,T[,

    |V_η,t^ϵ(w)| ≥ a ϵ+b|t-τ_η(t)-w|.


Recall that 5 ≤ E^j ≤ M_0 and |B| ≤ 1 around u⃗ (see Proposition <ref>) and

    V_η,t^ϵ,j(w)=v^j_ϵ+∫^w_t-τ_η(t) E^j(s,X_η,t^ϵ (s))+V_η,t^ϵ^i(s)B_i^j(s,X_η,t^ϵ (s))ds.

Hence, we have.



  * If w ≥ t-τ_η(t),

    V_η,t^ϵ,j(w) ≥δ_j,nϵ+(5-1)(w-t+τ_η(t))
 so that
 
    |V_η,t^ϵ(w)|^2 ≥ϵ^2+n(5-1)^2(w-t+τ_η(t))^2 ≥1/2(ϵ+4√(n)|w-t+τ_η(t)|)^2.



  * If t-τ_η(t)-ϵ/2(M_0+1)≤ w ≤ t-τ_η(t),

    V_η,t^ϵ,j(w) ≤ -(5-1)(t-τ_η(t)-w) for 1 ≤ j ≤ n-1 and V_η,t^ϵ,n(w) ≥ϵ/2,

so
  
    |V_η,t^ϵ(w)|^2    ≥   ϵ^2/4+(n-1)(5-1)^2(t-τ_η(t)-w)^2  
       ≥   1/2( ϵ/2+4√(n-1)|t-τ_η(t)-w| )^2


  * If w ≤ t-τ_η(t)-ϵ/2(M_0+1), then, for 1 ≤ j ≤ n-1,

    V_η,t^ϵ,j(w) ≤ -(5-1)(t-τ_η(t)-w) ≤ -4/3|ϵ/2(M_0+1)+2(t-τ_η(t)-w)|.

It comes,
 
    |V_η,t^ϵ(w)|^2  ≥16/9(n-1)( ϵ/2(M_0+1)+2|t-τ_η(t)-w| )^2.



Inequality (<ref>) then holds with 
    a=min(1/2√(2),2√(n-1)/3(M_0+1))and b=8/3√(n-1) .


We now prove that the right hand side of (<ref>) tends uniformly to zero in s, on [0,t]. As, by (<ref>),

    |∫_t-τ_η(t)^s 5/2|V_η,t^ϵ(w)| dw | ≤5/2blog( 1+bmax(t-τ_η(t),τ_η(t))/aϵ) ,

we have, since max(t-τ_η(t),τ_η(t)) ≤ t,

    ϵexp( | ∫_t-τ_η(t)^s 5/2|V_η^ϵ(w)| dw | ) ≤exp( 2b-5/2blog ( ϵ )+5/2blog(ϵ+bt/a) ).

We then deduce, as 2b>5, that

    lim_ϵ→ 0max_s ∈ [0,t]ϵexp( | ∫_t-τ_η(t)^s L+ 5/2|V_η^ϵ(w)| dw | )  =0,

which implies the result.




Differenciating (<ref>) in t for t<T_η gives us

    ∂_t f_1(t,u⃗,ηu⃗)=∑_i=1^n -1/√(n)∂_i f_01( (1-t/√(n)) u⃗,V_η(0,t) )
 

      +dV^i_η/dt(0,t) ∂_v^i f_01( (1-t/√(n)) u⃗,V_η(0,t) ).    
 
Doing the same for t>T_η gives


    ∂_t f_1(t,u⃗,ηu⃗) =∑_i=1^n dV^i_η/dt(0,t) ∂_v^i f_01( (1+t-2τ_η(t)/√(n)) u⃗,V_η(0,t) )    



    +1/√(n)( 2∂ (t-τ_η)/∂ t(t)-1 )∂_i f_01( (1+t-2τ_η(t)/√(n)) u⃗,V_η(0,t) ).


Recall from Proposition <ref> that t ↦∂ (t-τ_η)/∂ t(t) is defined on ]T_η,T[ and takes its values in [4/M_0,M_0+1/5]. Hence, there exists a sequence (t_n), with t_n → T_η, such that, 
    ∃ C>0, lim_t_n → T_η∂ (t-τ_η)/∂ t(t_n) = C.

Using that f_1 and f_01 are C^1 and taking the limit t_n → T_η in the two last equations, we obtain

    2C ∑_i=1^n ∂_i f_01( (1-T_η/√(n)) u⃗,0 )=0

and thus

    ∑_i=1^n ∂_i f_01( (1-T_η/√(n)) u⃗,0 )=0.

Finally, we need the following proposition.


The function η↦ T_η is defined on ]0,η_0[, strictly increasing, continuous and such that

    lim_η→ 0 T_η =0.





We recall (see Remark (<ref>)) that T_η is defined by the implicit equation

    g_η(0,T_η)=η -∫_0^T_η E^1( w, ( 1-T_η-w/√(n)) u⃗) dw=0.

Let 0<η_1<η_2<T. We have

    g_η_1(0,T_η_2)<g_η_2(0,T_η_2)=0,

so

    g_η_1(0,T_η_2)<g_η_1(0,T_η_1)=0.

Since g_η_1(0,.) strictly decreases (see again Remark (<ref>), T_η_2 > T_η_1, which means that η↦ T_η is strictly increasing. As E^1 is bounded away from 0 on the domain of integration,  T_η tends to 0 as η→ 0. The continuity ensues from the implicit function theorem.



Using Equation (<ref>) and the last proposition, we can find T^*>0 such that w ↦ f_01((1-w)u⃗,0) is constant on ]0,T^*[. However, there exists C_0 >0 and C_1 >0 such that

    f_01((1-w)u⃗,0)=C_0+C_1(1-w)χ'(2(1-w)^2)

for all 0<w<T^*, and w ↦ (1-w)χ'(2(1-w)^2) is not constant around 0.


abbrv


