Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials
Stephen Cameron Luis Silvestre Stanley Snelson



We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.

[NO \title GIVEN]
    [NO \author GIVEN]
    December 30, 2023
======================






§ INTRODUCTION


We consider the spatially inhomogeneous Landau equation, a kinetic model from plasma physics that describes the evolution of a particle density f(t,x,v)≥ 0 in phase space (see, for example, <cit.>). It is written in divergence form as

    ∂_t f + v·∇_x f = ∇_v·[a̅(t,x,v)∇_v f] + b̅(t,x,v)·∇_v f + c̅(t,x,v) f,

where t∈ [0,T_0], x∈^d, and v∈^d. The coefficients a̅(t,x,v)∈^d× d, b̅(t,x,v) ∈^d, and c̅(t,x,v)∈ are given by

    a̅(t,x,v)    := a_d,γ∫_^d( I - w/|w|⊗w/|w|) |w|^γ + 2 f(t,x,v-w)  w,
    b̅(t,x,v)    := b_d,γ∫_^d |w|^γ w f(t,x,v-w) w,
    c̅(t,x,v)    := c_d,γ∫_^d |w|^γ f(t,x,v-w) w,

where γ is a parameter in [-d,∞), and a_d,γ, b_d,γ, and c_d,γ are constants. When γ = -d, the formula for c̅ must be replaced by c̅ = c_d,γ f. Equation (<ref>) arises as the limit of the Boltzmann equation as grazing collisions predominate, i.e. as the angular singularity approaches 2 (see the discussion in <cit.>). The case d=3, γ=-3, corresponds to particles interacting by Coulomb potentials in small scales. The case γ∈ [-d,0) is known as soft potentials, γ = 0 is known as Maxwell molecules, and γ >0 hard potentials. In this paper, we focus on moderately soft potentials, which is the case γ∈ (-2,0).

We assume that the mass density, energy density, and entropy density are bounded above, and the mass density is bounded below, uniformly in t and x:

    0<m_0≤∫_^d f(t,x,v) v ≤ M_0,
    ∫_^d |v|^2 f(t,x,v) v ≤ E_0,
    ∫_^d f(t,x,v) log f(t,x,v)  v ≤ H_0.

In the space homogeneous case, because of the conservation of mass and energy, and the monotonicity of the entropy, it is not necessary to make the assumptions (<ref>), (<ref>) and (<ref>). It would suffice to require the initial data to have finite mass, energy and entropy. It is currently unclear whether these hydrodynamic quanitites will stay under control for large times and away from equilibrium in the space inhomogeneous case. Thus, at this point, it is simply an assumption we make.

We now state our main results. Our first theorem makes no further assumption on the initial data f_in:^2d→ beyond what is required for a weak solution to exist in [0,T_0].


Let γ∈ (-2,0]. If f:[0,T_0]×^2d→ is a bounded weak solution of (<ref>) satisfying (<ref>), (<ref>), and (<ref>), then there exists K_0>0 such that f satisfies

    f(t,x,v) ≤ K_0 (1+t^-d/2) (1+|v|)^-1,

for all (t,x,v)∈ [0,T_0]×^2d. The constant K_0 depends on d, γ, m_0, M_0, E_0, and H_0.


Note that even though we work with a bounded weak solution f, none of the constants in our estimates depend on f_L^∞. Note also that our estimate does not depend on T_0. We use a definition of weak solution for which the estimates in <cit.> apply, since that is the main tool in our proofs.

We will show in Theorem <ref> that an estimate of the form (<ref>) cannot hold with a power of (1+|v|) less than -(d+2), which also implies there is no a priori exponential decay. On the other hand, if f_in satisfies a Gaussian upper bound in the velocity variable, this bound is propagated:

Let f: [0,T_0]×^2d→ be a bounded weak solution of the Landau equation (<ref>) such that f_in(x,v) ≤ C_0 e^-α|v|^2, for some C_0>0 and a sufficiently small α>0. Then

    f(t,x,v) ≤ C_1e^-α|v|^2,

where C_1 depends on C_0, α, d, γ, m_0, M_0, E_0 and H_0. The value of α must be smaller than some α_0>0 that depends on γ, d, m_0, M_0, E_0 and H_0.



This estimate is also independent of T_0. As a consequence of Theorem <ref>, we will show in Theorem <ref> that in this regime, f is uniformly Hölder continuous on [t_0,T_0]×^2d for any t_0∈ (0,T_0). 

Note that under some formal asymptotic regime, the hydrodynamic quantities of the inhomogeneous Landau equations converge to solutions of the compressible Euler equation <cit.>, which is known to develop singularities in finite time. Should we expect singularities to develop in finite time for the inhomogeneous Landau equation as well? That question seems to be out of reach with current techniques. A more realistic project is to prove that the solutions stay smooth for as long as the hydrodynamic quantities stay under control (as in (<ref>), (<ref>) and (<ref>)). The results in this paper are an important step forward in that program.



 §.§ Related work
 It was established in <cit.> that solutions to (<ref>) become C^∞ smooth in all three variables conditionally to the solution being away from vacuum, bounded in H^8 (in the d=3 case) and having infinitely many finite moments. It would be convenient to extend this conditional regularity result to have less stringent assumptions. In particular, the assumptions (<ref>), (<ref>) and (<ref>) are a much weaker assumption, which is also in terms of physically relevant hydrodynamic quantities. In <cit.>, the authors show how their local Hölder continuity result for linear kinetic equations with rough coefficients can be applied to solutions of the Landau equation provided that (<ref>), (<ref>) and (<ref>) hold and in addition the solution f is assumed to be bounded. While we also assume boundedness of f, our results do not quantitatively rely on this and in addition tell us some information about the decay for large velocities.



The local estimates for parabolic kinetic equations with rough coefficients play an important role in this work. Local L^∞ estimates were obtained in <cit.> using Moser iteration, and local Hölder estimates were proven in <cit.> using a weak Poincaré inequality. A new proof was given in <cit.> using a version of De Giorgi's method.


Classical solutions for (<ref>) have so far only been constructed in a close-to-equilibrium setting: see the work of Guo <cit.> and Mouhot-Neumann <cit.>. A suitable notion of weak solution, for general initial data, was constructed by Alexandre-Villani <cit.>.

The global L^∞ estimate we prove in Theorem <ref> is similar to an estimate in <cit.> for the Boltzmann equation. The techniques in the proof are completely different. The propagation of Gaussian bounds that we give in Theorem <ref> is reminiscent of the result in <cit.>. That result is for the space-homogeneous Boltzmann equation with cut-off, which is in some sense the opposite of the Landau equation in terms of the angular singularity in the cross section.

In order to keep track of the constants for parabolic regularization estimates (as in <cit.>) for large velocities, we describe a change of variables in Lemma <ref>. This change of variables may be useful in other contexts. It is related to one mentioned in the appendix of <cit.> for the Boltzmann equation.

For the homogeneous Landau equation, which arises when f is assumed to be independent of x in (<ref>), the theory is more developed. The C^∞ smoothing is established for hard potentials in <cit.> and for Maxwell molecules in <cit.>, under the assumption that the initial data has finite mass and energy. Propagation of L^p estimates in the case of moderately soft potentials was shown in <cit.> and <cit.>. Global upper bounds in a weighted L^1_t(L^3_v) space were established in <cit.>, even for γ=-3, as a consequence of entropy dissipation. Global L^∞ bounds that do not depend on f_in and that do not degenerate as t→∞ were derived in <cit.> for moderately soft potentials, and this result also implies C^2 smoothing by standard parabolic regularity theory. 

Note that in the space homogeneous case our assumptions (<ref>), (<ref>) and (<ref>) hold for all t>0 provided that the initial data has finite mass, energy and entropy. Both Theorems <ref> and <ref> are new results even in the space homogeneous case. The previous results for soft potentials do not address the decay of the solution for large velocities.






 §.§ Organization of the paper
 In Section <ref>, we establish precise bounds on the coefficients a̅, b̅, and c̅ in (<ref>). In Section <ref>, we derive the local estimates we will use to prove Theorem <ref>, starting from the Harnack estimate of <cit.>. Section <ref> contains the proof of Theorem <ref> and a propagating lower bound that implies the exponent of (1+|v|) in (<ref>) cannot be arbitrarily high. In Section <ref>, we prove Theorem <ref> and the Hölder estimate, Theorem <ref>. In Appendix <ref>, we derive a convenient maximum principle for kinetic Fokker-Planck equations.



 §.§ Notation
 We say a constant is universal if it depends only on d, γ, m_0, M_0, E_0, and H_0. The notation A≲ B means that A≤ CB for a universal constant C, and A ≈ B means that A≲ B and B≲ A. We will let z=(t,x,v) denote a point in _+×^d×^d. For any z_0=(t_0,x_0,v_0), define the Galilean transformation

    𝒮_z_0(t,x,v) := (t_0+t, x_0 + x +tv_0,v_0+v).

We also have

    𝒮_z_0^-1(t,x,v) := (t-t_0, x - x_0 -(t-t_0)v_0,v-v_0).

For any r>0 and z_0 = (t_0,x_0,v_0), let

    Q_r(z_0) := (t_0-r^2,t_0] ×{x : |x-x_0 - (t-t_0) v_0| < r^3 }× B_r(v_0),

and Q_r = Q_r(0,0,0). The shift 𝒮_z_0 and the scaling of Q_r correspond to the symmetries of the left-hand side of (<ref>). We will sometimes write ∂_i or ∂_ij, and these will always refer to differentiation in v.



§ THE COEFFICIENTS OF THE LANDAU EQUATION


In this section we review various estimates of the coefficients a̅, b̅ and c̅ in (<ref>). In calculating these upper and lower bounds, the dependence of f on t and x is irrelevent, so in this section we will write f(v) and a̅(v), etc.


Let γ∈ [-2, 0), and assume f satisfies (<ref>), (<ref>), and (<ref>). Then there exist constants c and C depending on d, γ, m_0, M_0, E_0, and H_0, such that for unit vectors e∈^d,

    a̅_ij(v) e_i e_j ≥ c (1+|v|)^γ,     e∈𝕊^d-1, 
    
    (1+|v|)^γ+2,    e· v = 0,





and

    a̅_ij(v) e_i e_j ≤ C (1+|v|)^γ+2,    e∈𝕊^d-1,
    
    (1+|v|)^γ,      e· v = |v|,

where a̅_ij(v) is defined by (<ref>).



The lower bounds (<ref>) are proven in <cit.>. For the upper bounds, the formula (<ref>) implies

    a̅_ij(v)e_i e_j    = a_d,γ∫_^d(1 - (w· e/|w|)^2)|w|^γ+2 f(v-w) w 
       ≲∫_^d |w|^γ+2 f(v-w) w
       = ∫_^d |v-z|^γ+2 f(z) z
       ≲∫_^d (|v|^γ+2 + |z|^γ+2)f(z) z
       ≲  M_0(1+|v|^γ+2) +E_0,

since 0≤γ+2≤ 2.











The above bound is valid for all e∈𝕊^d-1. If e is parallel to v, then

    ∫_^d(1 - (w· e/|w|)^2)|w|^γ+2 f(v-w) w     = ∫_^d(1 - ((v-z)· e/|v-z|)^2)|v-z|^γ+2 f(z) z
       =  ∫_^d(|v-z|^2 - (|v|-z· e)^2)|v-z|^γ f(z) z
       =  ∫_^d(|z|^2 - (z· e)^2)|v-z|^γ f(z) z
       = ∫_^d |z|^2sin^2θ |v-z|^γ f(z) z,

where θ is the angle between v and z. Let R = |v|/2. If z∈ B_R(v), then |sinθ| ≤ |v-z|/|v|, and

    ∫_B_R(v) |z|^2sin^2θ |v-z|^γ f(z) z    ≤∫_B_R(v) |z|^2 |v|^-2  |v-z|^γ+2 f(z) z
       ≤|v|^γ/2^γ+2∫_B_R(v)|z|^2f(z) z ≲ E_0|v|^γ.

If |v-z| ≥ R=|v|/2, then |v-z|^γ≲ |v|^γ, and we have

    ∫_^d∖ B_R(v) |z|^2sin^2θ |v-z|^γ f(z) z    ≲  |v|^γ∫_^d∖ B_R(v) |z|^2 f(z) z ≲ E_0 |v|^γ.



In the proof of Theorem <ref>, we will need to keep track of how the bounds on b̅ and c̅ in the next two lemmas depend on the local L^∞ norm of f. In Lemma <ref> and Lemma <ref>, f_L^∞(A) means f(t,x,·)_L^∞(A) for any set A⊆^d.


Let f satisfy (<ref>), (<ref>), and (<ref>). Then c̅(v) defined by (<ref>) satisfies

    c̅(v) ≲ (1+|v|)^γ(1+f_L^∞(B_ρ(v)))^-γ/d,    -2dd+2≤γ < 0,
    
    (1+|v|)^-2-2γ/d(1+f_L^∞(B_ρ(v)))^-γ/d,    -d < γ < -2dd+2,

where the constants 
depend on d, γ, M_0, and E_0, and

    ρ =  1,    |v|<2,
    
    |v|^-2/d,    |v|≥ 2.









Assume first |v|≥ 2. Let r := |v|^-2/d (1+f_L^∞(B_ρ(v)))^-1/d < ρ. Consider

    I_1 = ∫_B_r |w|^γ     f(v-w) w,    I_2 = ∫_B_|v|/2∖ B_r |w|^γ f(v-w) w,
    
          I_3 = ∫_^d ∖ B_|v|/2 |w|^γ f(v-w) w.

We have

    I_1 ≲f_L^∞(B_ρ(v))  r^d+γ≲ |v|^-2-2γ/df_L^∞(B_ρ(v))^-γ/d .


    I_2 ≲ r^γ |v|^-2∫_B_|v|/2 |v-w|^2 f(v-w)  w    ≲ E_0 |v|^-2-2γ/d (1+f_L^∞(B_ρ(v)))^-γ/d.









Finally, for |w|≥ |v|/2, we have |w|^γ≲ |v|^γ, and

    I_3 ≲ |v|^γ∫_^d∖ B_|v|/2f(v-w) w ≤ M_0|v|^γ.

Thus c̅(v)≲ (1+f_L^∞(B_ρ(v)))^-γ/d|v|^-2-2γ/d + |v|^γ for |v|>2.

When γ∈(-d, -2dd+2), -2-2γ/d > γ and we get

    c̅(v)≲ (1+f_L^∞(B_ρ(v)))^-γ/d|v|^-2-2γ/d.

When γ∈[-2dd+2 , 0), γ > -2-2γ/d and we get

    c̅(v)≲ (1+f_L^∞(B_ρ(v)))^-γ/d|v|^γ.

This completes the proof in the case |v| > 2.

For |v| ≤ 2, γ∈ (-d, 0], and any R∈ (0,1] we have that

    ∫_^d |w|^γ f(v-w)  w    = ∫_B_R |w|^γ f(v-w) w + ∫_^d∖ B_R |w|^γ f(v-w) w, 
       ≲ R^d+γf_L^∞(B_1(v)) + R^γ M_0.

Choosing R = (1+f_L^∞(B_1(v))  )^-1/d, we then have

    c̅(v) ≲ (R^d+γf_L^∞(B_1(v)) + R^γ M_0) ≲ (1+ f_L^∞(B_1(v)))^-γ/d,
 for |v|≤ 2, completing the proof.







Let f satisfy (<ref>), (<ref>), and (<ref>).
 Then b̅(v) defined by (<ref>) satisfies






the estimate

    |b̅(v)| ≲(1+|v|)^γ+1(1+f_L^∞(B_ρ(v)))^-(γ+1)/d,    γ∈ [-2,-1),
    
    (1+ |v|)^γ+1,    γ∈ [-1,0]

where the constants depend on d, γ, M_0, and E_0, and

    ρ =  1,    |v|<2,
    
    |v|^-2/d,    |v|≥ 2.







Taking norms, we have

    |b̅(v)| ≲∫_^d |w|^1+γf(v-w) w.




If γ∈ [-2,-1), then 0>1+γ≥ -1 ≥-2d/d+2, and the conclusion follows from Lemma <ref>. If γ∈ [-1,0], we have

    |b̅(v)|    ≲∫_^d (|v|^γ+1 + |v-w|^γ+1 ) f(v-w) w 
       ≲ |v|^γ+1 M_0 + E_0^(1+γ)/2 M_0^(1-γ)/2≲ (1+ |v|)^γ+1.






§ LOCAL ESTIMATES


In this section we refine the local estimates in <cit.> and <cit.> for linear kinetic equations with rough coefficients. Essentially, we start from their results and apply scaling techniques to improve the local L^∞ estimates.

We will need the following technical lemma. See <cit.> for the proof.

Let η(r)≥ 0 be bounded in [r_0,r_1] with r_0≥ 0. Suppose for r_0≤ r<R≤ r_1, we have

    η(r) ≤θη(R) + A/(R-r)^α + B

for some θ∈ [0,1) and A,B,α≥ 0. Then there exists c(α,θ)>0 such that for any r_0≤ r<R≤ r_1, there holds

    η(r) ≤ c(α,θ)(A/(R-r)^α + B).




If g(t,x,v)≥ 0 is a weak solution of

    ∂_t g + v·∇_x g = ∇_v ·(A∇_v g) + B ·∇_v g + s

in Q_1,
with

    0 < λ I ≤ A(t,x,v) ≤Λ I,        (t,x,v)∈ Q_1,
    
    |B(t,x,v)| ≤Λ,       (t,x,v)∈ Q_1,
    
    s∈ L^∞(Q_1),

then

    sup_Q_1/2 g ≤ C(g_L^∞_t,xL^1_v(Q_1) + s_L^∞(Q_1)),

with C depending only on d, λ, and Λ.


It is proven in <cit.> that if g(t,x,v) solves (<ref>) weakly with A, B, and s as in the statement of the proposition, then

    g_L^∞(Q_1/2)≤ C (g_L^2(Q_1) + s_L^∞(Q_1)),

with C depending on d, λ, and Λ. Since g_L^2(Q_1)≤√(ω_d)g_L_t,x^∞ L_v^2(Q_1), where ω_d = ℒ^d(B_1), we also have

    g_L^∞(Q_1/2)≤ C (g_L_t,x^∞ L_v^2(Q_1) + s_L^∞(Q_1)).

To replace g_L_t,x^∞ L_v^2(Q_1) with g_L_t,x^∞ L_v^1(Q_1), we use an interpolation argument. For 0<r≤ 1, define

    [ g_r(t,x,v)  :=  g(r^2t, r^3x,rv),    s_r(t,x,v) := s(r^2t,r^3x,rv),; A_r(t,x,v)  :=  A(r^2t,r^3x, rv),    B_r(t,x,v) := B(r^2t,r^3x,rv), ]

and note that g_r satisfies

    ∂_t g_r + v·∇_x g_r = ∇_v ·(A_r∇_v g_r) + rB_r ·∇_v g_r + r^2s_r

in Q_1. Since r≤ 1, we may apply (<ref>) to g_r, which gives

    g_L^∞(Q_r/2)≤ C(1/r^d/2g_L_t,x^∞ L_v^2(Q_r) + r^2s_L^∞(Q_r)),

for any r∈ (0,1]. Now, for θ,R∈ (0,1), apply (<ref>) in Q_(1-θ)R(z) for each z∈ Q_θ R to obtain

    g_L^∞(Q_θ R)   ≤ C(1/[(1-θ)R]^d/2g_L_t,x^∞ L_v^2(Q_R) + R^2s_L^∞(Q_R))
       ≤ C(1/[(1-θ)R]^d/2g_L_t,x^∞ L_v^2(Q_R) + s_L^∞(Q_1)).

By the Hölder and Young inequalities, we have

    g_L^∞(Q_θ R)   ≤ C(1/[(1-θ)R]^d/2g_L^∞(Q_R)^1/2g_L_t,x^∞ L_v^1(Q_R)^1/2 + s_L^∞(Q_1))
       ≤1/2g_L^∞(Q_R) + C(1/[(1-θ)R]^dg_L_t,x^∞ L_v^1(Q_R) + s_L^∞(Q_1)).

Define η(ρ) = g_L^∞(Q_ρ) for ρ∈ (0,1]. Then for any 0<r<R≤ 1, we have

    η(r) ≤1/2η(R) + C/(R-r)^dg_L_t,x^∞ L_v^1(Q_1) + Cs_L^∞(Q_1).

Applying Lemma <ref>, we obtain

    η(r) ≤C/(R-r)^dg_L_t,x^∞ L_v^1(Q_1) + Cs_L^∞(Q_1).

Let R→ 1- and set r=1/2 to conclude (<ref>).





Let g(t,x,v) solve (<ref>) weakly in Q_R(z_0) for some z_0∈^2d+1 and R>0, with

    0 < λ I ≤ A(t,x,v) ≤Λ I,        (t,x,v)∈ Q_R,
    
    |B(t,x,v)| ≤Λ/R,       (t,x,v)∈ Q_R,
    
    s∈ L^∞(Q_R).


Then the improved estimate

    g(t_0,x_0,v_0) ≤ C(g_L_t,x^∞ L_v^1(Q_R)^2/(d+2)s_L^∞(Q_R)^d/(d+2) + R^-dg_L_t,x^∞ L_v^1(Q_R))

holds, with C depending only on d, λ, and Λ.


By applying the change of variables

    (t,x,v) ↦( t-t_0/R^2, x-x_0 - (t-t_0)v_0/R^3, v-v_0/R)

to g and s, we may suppose (t_0,x_0,v_0) = (0,0,0) and R=1.

For r∈(0,1] to be determined, we make the transformation (<ref>) as in the proof of Proposition <ref> and get a function g_r satisfying (<ref>) in Q_1. Then Proposition <ref> implies

    g(0,0,0)    ≤ C(g_r_L^∞_t,xL^1_v(Q_1) + r^2s_r_L^∞(Q_1))
       = C(r^-dg_L^∞_t,xL^1_v(Q_r) + r^2s_L^∞(Q_r))
       ≤ C(r^-dg_L^∞_t,xL^1_v(Q_1) + r^2s_L^∞(Q_1)).

If g_L_t,x^∞ L_v^1(Q_1)≤s_L^∞(Q_1), then the choice r = (g_L^∞_t,xL_v^1(Q_1)/s_L^∞(Q_1))^1/(d+2) implies

    g(0,0,0) ≤ Cg_L_t,x^∞ L_v^1(Q_1)^2/(d+2)s_L^∞(Q_1)^d/(d+2).

On the other hand, if s_L^∞(Q_1)≤g_L_t,x^∞ L_v^1(Q_1), the choice r=1 implies g(0,0,0) ≤ Cg_L_t,x^∞ L_v^1(Q_1), so we have

    g(0,0,0) ≤ C(g_L_t,x^∞ L_v^1(Q_1)^2/(d+2)s_L^∞(Q_1)^d/(d+2) + g_L_t,x^∞ L_v^1(Q_1))

in both cases.




§ GLOBAL ESTIMATES


In this section, we prove global upper bounds for solutions f of (<ref>). Our bounds depend only on the estimates on the hydrodynamic quantities (<ref>), (<ref>) and (<ref>). Our bound does not depend on an upper bound of the initial data. We also get that the solution will have certain polynomial decay in v for t>0.

From Lemma <ref>, we see that the bounds on a̅_ij(t,x,v) degenerate as |v|→∞. In the first lemma, we show how to change variables to obtain an equation with uniform ellipticity constants independent of |v|.

Let z_0 =(t_0,x_0,v_0)∈_+×^2d be such that |v_0|≥ 2, and let T be the linear transformation such that

    T e =    |v_0|^1+γ/2 e ,     e · v_0 = 0
    
    |v_0|^γ/2e,     e · v_0 = |v_0|.


Let T̃(t,x,v) = (t,Tx,Tv), and define

    𝒯_z_0(t,x,v)    := 𝒮_z_0∘T̃ (t,x,v)
        = (t_0+t,x_0+T x + t v_0 ,v_0 + T v).



Then,


  (a) There exists a constant C>0 independent of v_0∈^d∖ B_2 such that for all v∈ B_1,

    C^-1 |v_0| ≤ |v_0 + Tv| ≤ C |v_0|.


  (b) If f_T(t,x,v) := f(𝒯_z_0(t,x,v)), then f_T satisfies

    ∂_t f_T + v ·∇_x f_T = ∇_v [ A(z)∇_v f_T] +  B(z)·∇_v f_T +  C(z) f_T

in Q_R for any 0<R<min{√(t_0),c_1 |v_0|^-1-γ/2}, where c_1 is a universal constant, and

    λ I    ≤ A(z) ≤Λ I,
    
    |B(z)|    ≲ |v_0|^1+γ/2(1+f(t,x,·)_L^∞(B_ρ(v)))^-(γ+1)/d,    γ∈ [-2,-1),
    
    |v_0|^1+γ/2,    γ∈ [-1,0],
    
    |C(v)|    ≲ |v_0|^γ(1+f(t,x,·)_L^∞(B_ρ(v)))^-γ/d,    -2dd+2≤γ < 0,
    
    |v_0|^-2-2γ/d(1+f(t,x,·)_L^∞(B_ρ(v)))^-γ/d,    -2 < γ < -2dd+2,

with λ and Λ universal, and ρ≲ 1+ |v_0|^-2/d. 






Since |v|≤ 1 and |v_0| > 2,

    |v_0| - |v_0|^1+γ/2≤ |v_0| - |Tv| ≤ |v_0 + Tv| ≤ |v_0| + |Tv| ≤ |v_0| + |v_0|^1+γ/2.

 Thus, (a) follows since γ∈ (-2,0).

For (b), by direct computation, f_T satisfies (<ref>) with

    A(z) = T^-1a̅(𝒯_z_0(z)) T^-1,    B(z) = T^-1b̅(𝒯_z_0(z)),    C(z) = c̅(𝒯_z_0(z)).

In order to keep the proof clean, let us write a̅_ij and A_ij instead of a̅_ij(𝒯_z_0(z)) and A_ij(z) for the rest of the proof.

Fix z = (t,x,v)∈ Q_R, and let ṽ = v_0+T v. From part (a), we know that |ṽ| ≈ |v_0|. Applying Lemma <ref>, we have that for any unit vector e,

    a̅_ij e_ie_j ≲(1+|v_0|)^γ,    e = ṽ / |ṽ|,
    
    (1+|v_0|)^γ+2,     e ∈ S^d-1.

and,

    a̅_ij e_ie_j ≳(1+|v_0|)^γ,    e ∈ S^d-1,
    
    (1+|v_0|)^γ+2,     e ·ṽ = 0.


Our first step is to verify that we can switch ṽ for v_0 in (<ref>) and (<ref>).

Let us start with (<ref>). This is where the assumption |v| < R ≤ C_1 |v_0|^-1-γ/2 plays a role. We can choose c_1 so as to ensure that |Tv| ≤ 1. Since v_0 = ṽ - Tv and using the fact that a̅_ij is positive definite,

    a̅_ij (v_0)_i (v_0)_j ≤ 2 a̅_ijṽ_i ṽ_j + 2 a̅_ij(Tv)_i (Tv)_j ≤ C |v_0|^2+γ.

Let e_0 = v_0 / |v_0|. The computation above tells us that a̅_ij (e_0)_i (e_0)_j ≲ |v_0|^γ.

Let us now turn to (<ref>). We will show that

    a̅_ij w_iw_j ≳ (1+|v_0|)^γ+2 |w|^2     if  w · v_0 = 0.

Note that (1+|v_0|)^2+γ and (1+|v_0|)^γ are comparable when |v_0| is small, so we only need to verify (<ref>) for w · v_0 = 0 and |v_0| arbitrarily large. For such vector w, we write w= ηṽ + w' with w'·ṽ = 0. Since |ṽ - v_0| = |Tv| ≤ 1, we have |η| = |w·ṽ| / |ṽ|^2 = |w·(ṽ - v_0)|/|ṽ|^2 ≤ |w| |ṽ|^-2. Moreover, |w'| ≈ |w|.

Since a̅_ij is positive definite,

    a̅_ij (√(2)ηṽ - w'/√(2))_i (√(2)ηṽ - w'/ √(2))_j ≥ 0,

then we have

    a̅_ij w_i w_j    ≥1/2a̅_ij w'_iw'_j - η^2a̅_ijṽ_i ṽ_j
       ≥( c(1+|v_0|)^γ+2 - (1+|v_0|)^γ)|w|^2 ≳ (1+|v_0|)^γ+2  |w|^2,

as desired.



Let w ∈^d be arbitrary. We will estimate A_ij w_i w_j from above. Writing w = μ e_0 + w̃, with w̃· e = 0.

    A_ij w_i w_j    = |v_0|^-γ( μ^2 a̅_ij (e_0)_i (e_0)_j + 2 μ |v_0|^-1a̅_ij (e_0)_i w̃_j + |v_0|^-2a̅_ijw̃_i w̃_j ),
    and using that a̅_ij is positive definite,
    A_ijw_iw_j    ≤ 2|v_0|^-γ( μ^2 a̅_ij (e_0)_i (e_0)_j + |v_0|^-2a̅_ijw̃_i w̃_j ),
       ≤ C ( μ^2 + |w̃|^2 ) =: Λ |w|^2.

This establishes upper bound {A_ij}≤Λ I for some Λ>0.

Now we will prove the lower bound for A_ij. Again, we write w = μ e_0 + w̃ with e_0 ·w̃ = 0. We need to analyze the quadratic form associated with the coefficients a̅_ij more closely. From (<ref>), we have that for some universal constant c>0,

    c |v_0|^γ (μ^2 + |w̃|^2) ≤a̅_ij w_i w_j = μ^2 a̅_ij (e_0)_i (e_0)_j + 2 μa̅_ij (e_0)_i w̃_j + a̅_ijw̃_i w̃_j.

Moreover, (<ref>) implies that there is a universal constant δ > 0 so that

    c |v_0|^γ (μ^2 + |w̃|^2) ≥δμ^2 a̅_ij (e_0)_i (e_0)_j + δ |v_0|^-2a̅_ijw̃_i w̃_j.

Subtracting the two inequalities above,

    (1-δ) μ^2 a̅_ij (e_0)_i (e_0)_j + 2 μa̅_ij (e_0)_i w̃_j + (1-δ |v_0|^-2) a̅_ijw̃_i w̃_j ≥ 0.

The same inequality holds if we replace w = μ e_0 + w̃ with w = (1-δ/2)^-1/2μ e_0 + (1-δ/2)^1/2|v_0|^-1w̃, therefore

    1-δ/1-δ/2μ^2 a̅_ij (e_0)_i (e_0)_j + 2 μ |v_0|^-1a̅_ij (e_0)_i w̃_j + (1-δ/2)(1-δ |v_0|^-2) |v_0|^-2a̅_ijw̃_i w̃_j ≥ 0.

Recalling the formula above for A_ij w_i w_j, and replacing it in the left hand side, we get

    A_ij w_i w_j - ( 1 - 1-δ/1-δ/2) |v_0|^-γμ^2 a̅_ij (e_0)_i (e_0)_j - ( 1 - (1-δ/2)(1-δ |v_0|^-2) ) |v_0|^-2-γa̅_ijw̃_i w̃_j ≥ 0.

Therefore, using (<ref>) and (<ref>),

    A_ij w_i w_j    ≥( 1 - 1-δ/1-δ/2) |v_0|^-γμ^2 a̅_ij (e_0)_i (e_0)_j + ( 1 - (1-δ/2)(1-δ |v_0|^-2) ) |v_0|^-2-γa̅_ijw̃_i w̃_j, 
       ≥λ (μ^2 + |w̃|^2),

for some universal constant λ > 0. This establishes the lower bound {A_ij}≥λ I.




We now turn to the lower bound for

Next, (<ref>) and (<ref>) imply that there is a unit eigenvector e̅ of a̅_ij with eigenvalue λ̅≈ (1+|v_0|)^γ. For |v_0| sufficiently large, the angle between e_0 and e̅ must be small. Indeed, writing e_0 = (e_0·e̅)e̅ + e^⊥, we have a̅_ij(e_0)_i(e_0)_j = (e_0·e̅)^2 λ̅+ a̅_ij e^⊥_i e^⊥_j, so that (<ref>) and (<ref>) imply |e^⊥|^2 ≲ |v_0|^-2, and therefore, |e_0 - e̅|≲ |v_0|^-1. Similarly, for w̃ perpendicular to e_0, we can write w̃ = (w̃·e̅)e̅ + w̅ with |w̃ - w̅| ≲ |w̃| |v_0|^-1. Now, with w∈^d and w = μ e_0 + w̃ as above, we have

    a̅_ij(e_0)_i w̃_j    = a̅_ije̅_i (w̃ - w̅)_j + a̅_ij (e_0 - e̅)_i w̃_j
       ≥  - λ̅|w̃ - w̅| - |a̅ (e_0-e̅)| |w̃|
       ≥ -c(1+|v_0|)^γ |v_0|^-1 |w̃|,

for some constant c. Then, if |v_0|≥ρ_0 large enough, (<ref>) implies

    A_ij w_i w_j    ≥ |v_0|^-γ( μ^2 |v_0|^γ - 2cμ |v_0|^-2+γ|w̃| + |v_0|^γ|w̃|^2)
       ≥μ^2 + |w̃|^2 = |w|^2.





To derive the bound on B(z), Lemma <ref> and conclusion (a) imply

    |B(z)|    ≲T^-1 |b̅(𝒯_z_0(z))|
    
    
       ≲ (1+|v_0|)^γ/2+1(1+f_L^∞(B_ρ'(ṽ)))^-(γ+1)/d,    γ∈ (-2,-1),
    
    (1+|v_0|)^γ/2+1,    γ∈ [-1,0],

where ρ' = |ṽ|^-2/d. From the triangle inequality, we have that B_ρ'(ṽ) ⊂ B_ρ(v_0), with ρ≲ (1+|v_0|)^-2/d+R(1+|v_0|)^(γ+2)/2≤ 1+(1+|v_0|)^-2/d.
The bound on C(z) follows in a similar manner, using Lemma <ref>.


The key lemma in the proof of Theorem <ref> is the following pointwise estimate on f:

Let γ∈ (-2,0], T_0>0, and let f:[0,T_0]×^2d→_+ solve the Landau equation (<ref>) weakly. If

    f(t,x,v) ≤ K (1+t^-d/2)(1+|v|)^-α

in [0,T_0]×^2d for some α∈ [0,1] and K≥ 1, then

    f(t,x,v) ≤ C((K (1+t^-d/2))^(d-γ)/(d+2) (1+|v|)^P(d,α,γ) + K^Q(γ)(1+ t^-d/2) (1+|v|)^-1),

for some C universal and

    P(d,α,γ) =  -1 - d(1+α)/(d+2),    γ∈[-2dd+2, 0],
    
    -[d(4+γ)+2 +2γ + α d]/(d+2),    γ∈(-2, -2dd+2),


    Q(γ) =  0,    γ∈ [-1,0]
    
    -(1+γ),    γ∈ (-2,-1).






Case 1: γ∈ [-1,0]. 
Let z_0 = (t_0,x_0,v_0) be such that such that |v_0|≥ 2. Define r_0 = min{1,√(t_0)}, and note that r_0^-d≈ (1+t_0^-d/2). Letting f_T be as in Lemma <ref>, we will estimate f_T(t,x,v) in Q_R, where

    R := c_1(r_0/2)(1+|v_0|)^-(2+γ)/2,

with c_1 as in Lemma <ref>(b). We have that f_T solves (<ref>) in Q_R, and by Lemma <ref>(a) and our assumption on f,

    f_T(t,x,v) ≲ K r_0^-d (1+|v_0|)^-α

in Q_R. Feeding (<ref>) into Lemma <ref>(b), we have

    0<   λ I ≤  A(z) ≤Λ I,
    
    |B(z)|   ≲ (1+|v_0|)^(2+γ)/2,
    
    |C(z)|    ≲(Kr_0^-d)^-γ/d(1+|v_0|)^γ,

in Q_R.











Let Q_T,R be the image of Q_R under z ↦𝒯_z_0(z), and note that

    f_T _L_t,x^∞ L_v^1(Q_R)   = (T^-1)f_L_t,x^∞ L_v^1(Q_T,R)
        = (1+|v_0|)^-[(d-1)(2+γ)/2 + γ/2]f_L_t,x^∞ L_v^1(Q_T,R)
       ≤ (1+|v_0|)^-(1+d(2+γ)/2)E_0,

where the last inequality comes from the energy bound (<ref>) and Lemma <ref>(a).

By (<ref>) and our choice of R, we can apply Lemma <ref> in Q_R with g = f_T and s = C(z) f_T to obtain

    f(t_0,x_0,v_0)    ≤ C(f_T_L_t,x^∞ L_v^1(Q_R)^2/(d+2) C(z) f_T_L^∞(Q_R)^d/(d+2)+ r_0^-d(1+|v_0|)^d(2+γ)/2f_T_L_t,x^∞ L_v^1(Q_R))
       ≤ C((Kr_0^-d)^(d-γ)/(d+2) (1+|v_0|)^-1-d(1+α)/(d+2) + r_0^-d(1+|v_0|)^-1),

using (<ref>), (<ref>), and (<ref>). Note that we derived (<ref>) assuming that |v_0|≥ 2. When |v_0|≤ 2, the matrix a̅_ij(z) is uniformly elliptic and we can apply Lemma <ref> directly to f to obtain (<ref>) in this case as well.

Case 2: γ∈ (-2, -1]. The argument is the same as in Case 1, but the estimates are quantitatively different as a result of the different bounds on B(z) and C(z) in Lemma <ref>. The changes are as follows: the radius R of the cylinder Q_R is chosen to be

    R := K^(1+γ)/d(r_0/2)(1+|v_0|)^-(2+γ)/2,

the bound on B(z) becomes

    |B(z)|≲  K^-(1+γ)/d r_0^1+γ(1+|v_0|)^(2+γ)/2≤Λ/R,    z∈ Q_R,

and for C(z) we have

    |C(z)| ≲(Kr_0^-d)^-γ/d(1+|v_0|)^γ,    γ∈[-2dd+2, -1],
    (Kr_0^-d)^-γ/d(1+|v_0|)^-2-2γ/d,    γ∈(-2, -2dd+2),

for z∈ Q_R. After applying Lemma <ref> and (<ref>), we obtain

    f(t_0,x_0,v_0) ≤ C((K r_0^-d)^(d-γ)/(d+2) (1+|v_0|)^P(d,α,γ) + K^-(1+γ) r_0^-d (1+|v_0|)^-1),

as desired, with P(d,α,γ) as in the statement of the lemma.




We are now in a position to prove our main theorem.





Define

    K:= sup_(0,T_0]×^2dmin{t^d/2,1}f(t,x,v).

First, we will show that K≤ K_*, where K_* is universal. We can assume K > 1. For each γ∈ (-2, 0], define p_γ: (1,∞)→ by

    p_γ(K) =  C(K^(d-γ)/(d+2) + 1),    γ∈ (-1, 0], 
    
    C(K^(d-γ)/(d+2) + (K)^-(1+γ)),    γ∈(-2, -1],

where C is the appropriate constant from Lemma <ref> for each γ.  Then since -(1+γ)<1 and d-γ/d+2<1 for γ>-2, there is a K_*>1 such that

    K_*    = p_γ(K_*), 
    K   >p_γ(K),    K>K_*.

Let >0. By the definition of K, there exists some (t_0,x_0,v_0)∈ (0,T]×^2d such that f(t_0,x_0,v_0)> (K-)max{t_0^-d/2, 1}.  Therefore, Lemma <ref> implies that

    K-≤ p_γ(K).

Since this is true for all >0, we have that K≤ K_*.

If γ∈[-2d/d+2,0], we apply Lemma <ref> with α=0 to conclude (<ref>) with K_0 =C K_*. If γ∈(-2,-2d/d+2), Lemma <ref> with α=0 implies

    f(t,x,v) ≤ C K (1+t_0^-d/2) (1+|v_0|)^-[d(4+γ) + 2 + 2γ]/(d+2),

so we can apply Lemma <ref> again with α = [d(4+γ) + 2 + 2γ]/(d+2). We iterate this step, and since for any α∈ (0,1], we have α≤ 1 < d(4+γ)/2 + 1 + γ, the gain of decay at each step, -P(d,α,γ) - α, is bounded away from 0. Therefore, after finitely many steps (with the number of steps depending only on d and γ), we obtain (<ref>) for some K_0.



The next result shows that the generating decay in Theorem <ref> cannot be improved to polynomial decay with power greater than d+2, or to exponential decay. Note that since b̅_i = -∂_j a̅_ij, for smooth solutions (<ref>) may be written equivalently in non-divergence form as

    ∂_t f + v·∇_x f= a̅(t,x,v)D_v^2 f + c̅(t,x,v)f.


Let γ∈ [-2,0] and p > d+2. Assume f solves (<ref>) in [0,T_0]×^2d with

    f_in(x,v) ≥ c_0 (1+|v|)^-p

for v, x∈^d, for some c_0>0. Then there exist c_1>0 and β>0 such that

    f(t,x,v) ≥ c_1 e^-β t (1+|v| ) ^-p

for all |v|≥ 1, x∈^d, and t∈ [0,T_0].



Let η:_+→_+ be a smooth, decreasing function such that η(r) ≡ 2 when r∈[0,1/2] and η(r) = r^-p when r∈ [1,∞). Note η(r) ≈ (1+r)^-p. Let us define ψ(t,x,v) = e^-β tη(|v|) with β to be chosen later. Choose an arbitrary R_0 >1, and recall from Lemma <ref> that a̅_ij∂_ijψ≥ - C(1+|v|)^γ+2|D^2ψ|. (Throughout this proof, a̅_ij and c̅ are defined in terms of f.) From our choice of η, it is clear that |D^2ψ|/ψ is uniformly bounded from above in _+×^d×{v: |v|≤ R_0 + 1}, so for β≥β_1 sufficiently large, we have

    -∂_t ψ + a̅_ij∂_ijψ + c̅ψ≥βψ - C(1+|v|)^γ+2|D^2ψ| ≥ 0,    |v| ≤ R_0+1.


For |v|≥ R_0, we estimate a̅_ij∂_ijψ more carefully. Since |v|≥ 1, we have

    ∂_ijψ = ∂_rrψ/|v|^2   v_i v_j + ∂_rψ/|v|( δ_ij - v_iv_j/|v|^2) = [p(p+1)|v|^-4 v_i v_j -  p|v|^-2( δ_ij - v_iv_j/|v|^2)] e^-β t|v|^-p,

and Lemma <ref> implies

    -∂_t ψ + a̅_ij∂_ijψ   ≥βψ + [p(p+1) C_1|v|^-2+γ  -  pC_2|v|^γ] ψ≥(β - C |v|^γ) ψ.

For β≥β_2 sufficiently large, the right-hand side is positive for all |v|≥ R_0. Since c̅(t,x,v)≥ 0, this implies ψ(t,x,v) = e^-β tη(|v|) with β = max(β_1,β_2) is a subsolution of (-∂_t + a̅_ij∂_ij + c̅)g = 0 in the entire domain _+×^2d. By (<ref>), there is some c_1≥ c_0 so that f_in(x,v) ≥ c_1ψ(0,x,v) in ^2d. Now we can apply the maximum principle (see Appendix <ref>) to c_1ψ - f to conclude (<ref>).


The bound on the energy ∫_^d |v|^2 f(t,x,v) v ≤ E_0 < ∞ implies that f_in(x,v) cannot be bounded below by  c_0|v|^-p with p≤ d+2 as |v|→∞.



In particular, Theorem  tells us that there is no generation of moments when γ∈ [-2,0].




§ GAUSSIAN BOUNDS


We show the propagation of Gaussian upper bounds. The first lemma says that a sufficiently slowly decaying Gaussian is a supersolution of the linear Landau equation for large velocities. As above, the coefficients a̅_ij and c̅ in (<ref>) are defined in terms of f.

Let γ∈(-2,0]. Let f be a bounded function satisfying (<ref>), (<ref>), and (<ref>). Let a̅ and c̅ be given by (<ref>) and (<ref>) respectively. If α >0 is sufficiently small, then there exists R_0>0 and C>0, depending on d, γ, M_0, m_0, E_0, H_0 and f_L^∞, such that

    ϕ(v) := e^-α |v|^2

satisfies


    a̅_ij∂_ijϕ + c̅ϕ≤ -C|v|^γ+2ϕ,

for |v|≥ R_0.






Since ϕ is radial, we have

    ∂_ijϕ = ∂_rrϕ/|v|^2   v_i v_j + ∂_rϕ/|v|( δ_ij - v_iv_j/|v|^2) = [4α^2|v|^2 - 2α/|v|^2   v_i v_j -  2α( δ_ij - v_iv_j/|v|^2)] e^-α |v|^2,

and the bounds (<ref>) and (<ref>) imply

    a̅_ij∂_ijϕ   ≤[(4α^2|v|^2 - 2α)C_1 |v|^γ  -  2α C_2 |v|^γ+2] e^-α |v|^2
       = ((4α^2 C_1 - 2α C_2) |v|^γ+2 - 2α C_1 |v|^γ) e^-α|v|^2
       ≤ -C |v|^γ+2ϕ(v),

for |v| sufficiently large, provided α < C_2 / (2C_1). With Lemma <ref> (this is the point where f_L^∞ plays a role), this implies

    a̅_ij∂_ijϕ  + c̅ϕ≤[-C|v|^γ+2 + C|v|^-2-2γ/d]ϕ(v).

For -2<γ≤ 0, the first term on the right-hand side will dominate for large |v|, since γ+2 > 0 > -2-2γ/d.









Theorem <ref> gives us an upper bound for a solution f to the Landau equation which is useful away from t=0. If the initial data f(0,x,v) is a bounded function, we can improve our upper bound for small values of t using the upper bound for f(0,x,v). That is the purpose of the next lemma.

Let f: [0,T_0]×^2d→ be a solution of the Landau equation (<ref>) for some γ∈ (-2, 0], and suppose that g: [0,T_0]×^2d→ is bounded from above and a subsolution to the equation

    ∂_t g(t,x,v) + v·∇_x g(t,x,v) ≤a̅_ij(t,x,v) ∂_ij g(t,x,v) + c̅(t,x,v) g(t,x,v),

where a̅_ij and c̅ are defined in terms of f as in (<ref>) and (<ref>). Let κ(t) be defined by

    κ(t) = {[  β/1+γ/2 t^1+γ/2,          0≤ t ≤ 1; β/1+γ/2 + β(t-1),              t≥ 1 ]. ,

where β>0 depends only on d, γ, m_0, M_0, E_0, and H_0.
Then

    sup_[0,T_0]×^d e^-κ(t)g_+(t,x,v) = sup_^2d g_+(0,x,v).





By Theorem <ref>, we have that f(t,x,v)≤ K_0t^-d/2 for 0<t<1.  Hence by Lemma <ref>, we have that c̅(t,x,v) ≲ t^γ/2. Since γ > -2, for some universal β>0, κ(t) satisfies c̅(t,x,v)≤κ'(t) for all t>0.  Thus g̃(t,x,v) = e^-κ(t)g(t,x,v) satisfies

    ∂_t g̃(t,x,v) + v·∇_xg̃(t,x,v)    ≤a̅_ij(t,x,v)∂_ijg̃(t,x,v) + (c̅(t,x,v)-κ'(t))g̃(t,x,v)
       ≤a̅_ij(t,x,v)∂_ijg̃(t,x,v).

We apply Lemma <ref> from the Appendix to g̃(t,x,v) - sup_^2d g(0,x,v) to conclude (<ref>).



















Applying Lemma <ref> with g=f and t ∈ [0,1] and Theorem <ref> for t>1, we have that there is some constant C_2 (depending on C_0, d, γ, M_0, m_0, E_0, and H_0) so that f(t,x,v) ≤ C_2 for all t ≥ 0, x ∈^d and v ∈^d.

Let ϕ(v) := e^-α|v|^2. From Lemma <ref>, we have that there is a C, depending on C_2, d, γ, M_0, m_0, E_0, and H_0, such that

    sup_(0,T_0]×^d ×^da̅_ij∂_ijϕ + c̅ϕ≤ C ϕ.

Thus C_0 e^Ctϕ(v) is a supersolution of the equation and f(t,x,v) ≤ C_0 e^Ctϕ(v) for all t >0, x ∈^d and v ∈^d.

This upper bound is good for small values of t. We see that there is some time t_0 > 0 so that C_0 e^Ct_0ϕ(v) > C_2 for |v| < R_0. Here C_2 is the upper bound for f mentioned above and R_0 is the radius from Lemma <ref>. Thus, the function

    g(t,x,v) :=[ f(t_0+t,x,v) - C_0 e^Ct_0ϕ(v) ]_+

is a supersolution of

    g_t + v ·∇_x g ≤a̅_ij∂_ij g + c̅ g.

Applying the maximum principle (Lemma <ref>), we have that g ≤ 0 for all t>0, so f(t,x,v) ≤ C_0 e^t_0 Cϕ(v) for all t>t_0, and we conclude the proof.

















By combining Theorem <ref> with the local Hölder estimates proved in <cit.> or <cit.>, we derive a global Hölder estimate for solutions of (<ref>) under the assumption that f_in(x,v) ≤ C_0 e^-α |v|^2. The following local estimate is essentially the same as Theorem 2 of <cit.>:

Let f be a weak solution of

    ∂_t f + v·∇_x f = ∇_v ·(A∇_v f) + B ·∇_v f + s

in Q_1, with λ I ≤ A ≤Λ I, |B|≤Λ, and s∈ L^∞(Q_1). Then f is Hölder continuous with respect to (t,x,v) in Q_1/2, and

    |f(z_1) - f(z_1)|/|t_1-t_2|^β/2+|x_1-x_2|^β/3+|v_1+v_2|^β≤ C(f_L^2(Q_1) + s_L^∞(Q_1)),

for all z_1,z_2∈ Q_1/2, where β and C depend on d, λ, and Λ.




















To state our theorem as a global Hölder estimate, we will need an appropriate notion of distance in ×^d ×^d which is invariant by Galilean transformations. A natural choice is the following

    d_P(z_1,z_2) := min{ r : ∃ z ∈×^d ×^d : z_1 ∈ Q_r(z)  and  z_2 ∈ Q_r(z) }.


We can easily estimate the value of d_P(z_1,z_2) by the simpler formula

    d_P(z_1,z_2) ≈ |t_1-t_2|^1/2 + |x_1 - x_2 - (t_1-t_2)(v_1+v_2)/2|^1/3 + |v_1-v_2|.


It turns out that we need to deform this distance using the transformation 𝒯_z described in Lemma <ref>. We define

    d_L(z_1,z_2) := min{ |v|^1+γ/2 r: z ∈×^d ×^d : 𝒯_z^-1 z_1 ∈ Q_r and 𝒯_z^-1 z_2 ∈ Q_r}.


(Here, we make the convention that 𝒯_z = 𝒮_z when |v|< 2.) An explicit expression for d_L(z_1,z_2) is messy. It involves the affine transformation 𝒯 which is anisotropic and affects both the x and v variables. In the case that we compare two points with identical values of t and x, it is straightforward to check that when d_L((t,x,v_1),(t,x,v_2))<1, then d_L is equivalent to the metric introduced by Gressman-Strain <cit.> in their study of the Boltzmann equation.















Under the assumptions of Theorem , there exist C>0 and β∈ (0,1) depending on C_0, α, d, γ, m_0, M_0, E_0, and H_0, such that for any z_1,z_2∈ [0,T_0]×^2d, one has

    |f(z_1) - f(z_2)|    ≤ C(e^-α|v_1|^2 + e^-α|v_2|^2)min{ 1, (1+ t_1^-β/2+t_2^-β/2)
    d_L(z_1,z_2)^β}.





If |v_1|≤ 2 or |v_2| ≤ 2, the result follows by applying Theorem <ref> directly to f, noting that 1≲ e^-α|v_1|^2+e^-α|v_2|^2. So, we can assume that |v_1| > 2 and |v_2|>2.

Let z̅ = (t̅, x̅, v̅) be the point achieving the minimum in the definition of d_L(z_1,z_2). Thus z̃_1 := 𝒯_z̅^-1 z_1 ∈ Q_δ and z̃_1 := 𝒯_z̅^-1 z_1 ∈ Q_δ, where δ = |v̅|^-1-γ/2d_L(z_1,z_2).

Let r := min(t_1^1/2, t_2^1/2,(1+|v̅|)^-1-γ/2). If δ≥ r/2, then we simply estimate |f(z_1) - f(z_2)| ≤ C_1(e^-α|v_1|^2+e^-α|v_2|^2) from Theorem <ref>. We need to concentrate on the case δ < r/2.

Let us consider the function f_T as in Lemma <ref>, with base point z̅. By our choice of r, f_T satifies an equation of the form (<ref>) in Q_r, and since Theorem <ref> gives us a bound on f_T_L^∞, we have that A is uniformly elliptic (with constants independent of z̅), |B|≲ |v̅|^1+γ/2, and |s|=|C(z)f_T| ≲ |f_T|. Defining f̃_T(t,x,v) := f_T(r^2t,r^3x,rv), we see that f̃_T satisfies another equation of the form (<ref>) with the new |B| bounded independently of |v̅|. Moreover, the points (r^-2t̃_1, r^-3x̃_1, r^-1ṽ_1) and (r^-2t̃_2, r^-3x̃_2, r^-1ṽ_2) belong to Q_r^-1δ⊂ Q_1/2. Therefore, we can apply Theorem <ref> to f̃_T in Q_1 to obtain

    |f(z_1) - f(z_2)|/r^-β d_L(z_1,z_2)^β|v̅|^β(1+γ/2)   = |f_T(z̃_1) - f_T(z̃_2)|/r^-βδ^β
       = |f̃_T(r^-2t̃_1, r^-3x̃_1, r^-1ṽ_1) - f̃_T(r^-2t̃_2, r^-3x̃_2, r^-1ṽ_2)|/r^-βδ^β, 
       ≲f̃_T_L^1(Q_1) + f̃_T_L^∞(Q_1)≲sup_v∈ Q_r/2e^-α|v̅+Tv|^2≲ e^-α|v̅|^2.

We have used Theorem <ref> to estimate the L^∞ norm of f̃_T in Q_1. Rewriting this estimate, we obtain

    |f(z_1) - f(z_2)|    ≲ r^-β |v̅|^-β(1+γ/2) d_L(z_1,z_2)^β e^-α|v̅|^2
       ≲(1+t_1^-β/2+t_2^-β/2) d_L(z_1,z_2)^β(e^-α|v̅_1|^2 + e^-α|v̅_2|^2) .








§ MAXIMUM PRINCIPLE FOR WEAK SOLUTIONS TO KINETIC FOKKER-PLANCK EQUATIONS


In this appendix, we give a proof of the maximum principle in a form that is convenient for our purposes.

The following proposition is perhaps a classical result. We prove it here, since we could not find any easy reference and also for completeness. The result is for equations on a bounded domain with general coefficients (not necessarily defined by integrals as above).


Let Q = [0,T_0]×Ω, where Ω⊂^2d is a bounded domain, and assume that g is a subsolution of the equation

    ∂_t g + v·∇_x g ≤∇_v· [a(t,x,v) ∇_v g] + b(t,x,v)·∇_v g + c(t,x,v) g,

in the weak sense in Q, where a is uniformly elliptic in Q with constants λ and Λ, and b and c are uniformly bounded in Q.

If g≤ 0 on the parabolic boundary of Q, then g≤ 0 in Q.



Choosing the test function ϕ = g_+, the weak formulation of (<ref>) gives

    ∫_Q g_+ (∂_t g +  v·∇_x g)  x  v  t    ≤∫_Q (-a ∇_v g ∇_v g_+ - g_+b ·∇ g + c g_+^2) x  v  t,

or

    ∫_Q1/2d/dt (g_+)^2  x  v  t    ≤∫_Q (-λ |∇_v g_+|^2 - b g ∇ g_+ + c g_+^2) x  v  t
       ≤(b_L^∞/4λ + c_L^∞) ∫_Q g_+^2 x  v  t,

by Young's inequality. We apply Gronwall's Lemma to ∫_Ω (g_+)^2 x  v on [0, T_0] to conclude g_+ ≡ 0 in Q.


Next, we derive a maximum principle on the whole space for subsolutions of a Landau-type equation without a zeroth-order term:

Let g be a bounded function on [0,T_0]×^2d that satisfies

    ∂_t g + v·∇_x g ≤a̅(t,x,v) D^2_v g,

in the weak sense. Here, a̅(t,x,v) is defined as in (<ref>) in terms of a function f satisfying (<ref>), (<ref>), and (<ref>). If g(0,x,v) ≤ 0 in ^2d, then g(t,x,v) ≤ 0 in [0,T_0]×^2d.


By the bounds on a̅ given in Lemma <ref>, we have

    a̅_ij∂_ij (1+|v|) ≤ C_1(1+|v|)^1+γ,

for some constant C_1, and thus ϕ_1(t,v) := e^C_1t (1+|v|) satisfies
 
    ∂_t ϕ_1(t,v) ≥a̅_ij(t,x,v) ∂_ijϕ_1(t,v).

Let _1>0 be a small constant. Since g is bounded, there is R(_1)>0 such that g - _1 ϕ_1<0 whenever |v|≥ R(_1). Let R_1> R(_1), and choosing C_2>0 large enough depending on R_1, we can define ϕ_2(t,x) := (1+|x|) e^C_2 t, and we have

    ∂_t ϕ_2 + v·∇_x ϕ_2 ≥ 0,

whenever |v|<R_1. Finally, for _2>0 arbitrary, we define

    g̃(t,x,v) := [g(t,x,v) - _1 ϕ_1(t,v) - _2 ϕ_2(t,x)]_+.

It is clear that g̃ is a subsolution as in (<ref>) with c≡ 0, whenever |v|<R_1. For R(_2) sufficiently large, we have that g - _1ϕ_1 - _2ϕ_2 <0 for |x|≥ R(_2) or |v|≥ R(_1). Then for any R_2>R(_2), we have that g̃ = 0 on the parabolic boundary of [0,T_0]× B_R_2× B_R_1, so Proposition <ref> applied to g̃ gives

    g - _1ϕ_1 - _2 ϕ_2 ≤ 0,    |v|<R_1,  |x|< R_2.

Take R_2→∞ and _2→ 0 to conclude

    g - _1ϕ_1 ≤ 0,    |v|<R_1.

Take R_1→∞ and _1→ 0, and the proof is complete.

abbrv


