Inverse modified scattering and polyhomogeneous expansions for the Vlasov--Poisson system
Volker Schlue Martin Taylor





* ]Volker Schlue

†]Martin Taylor

[†]Imperial College London,
Department of Mathematics,
South Kensington Campus, London SW7 2AZ, United Kingdom.2pc martin.taylor@imperial.ac.uk.2pc  

[*]University of Melbourne, School of Mathematics and Statistics, Parkville, VIC, 3010, Australia.2pc  volker.schlue@unimelb.edu.au.2pc   

[]Universität Münster,
Mathematisches Institut, Einsteinstrasse 62, 48149 Münster, Germany.2pc  volker.schlue@uni-muenster.de.2pc   





Inverse modified scattering and polyhomogeneous expansions for the Vlasov–Poisson system
    [
    April 24, 2024
========================================================================================




	We give a new proof of well posedness of the inverse modified scattering problem for the Vlasov–Poisson system: for every suitable scattering profile there exists a solution of Vlasov–Poisson which disperses and scatters, in a modified sense, to this profile.  Further, as a consequence of the proof, the solutions are shown to admit a polyhomogeneous expansion, to any finite but arbitrarily high order, with coefficients given explicitly in terms of the scattering profile.  The proof does not exploit the full ellipticity of the Poisson equation.







§ INTRODUCTION


The Vlasov–Poisson system describes the evolution of an ensemble of collisionless particles, interacting via a collectively generated gravitational or electrostatic potential force.  The system on ℝ^3, in the electrostatic case, takes the form

    _ϕ f = 0,


    Δ_ℝ^3ϕ (t,x) = ϱ(t,x),
    	    ϱ(t,x) = ∫_ℝ^3 f(t,x,p) dp,

where f:I ×ℝ^3×ℝ^3 → [0,∞), and ϕ, ϱ I ×ℝ^3→ℝ, for a suitable interval I ⊂ℝ, and the operator _ϕ is defined by,

    _ϕ = ∂_t + p^i ∂_x^i + ∂_x^iϕ (t,x) ∂_p^i.



This article concerns the inverse scattering problem for the system (<ref>)–(<ref>).
  The first main result is a new proof of the existence of future-global solutions which scatter in the modified sense to a given asymptotic profile.

 
	For any smooth compactly supported function f_∞ℝ^3 ×ℝ^3 → [0,∞), there exists T_0 > 0 and a smooth solution f  [T_0,∞) ×ℝ^3 ×ℝ^3 → [0,∞) of the Vlasov–Poisson system such that, for all x,p∈ℝ^3,
	
    lim_t→∞ f(t,x+tp - log t  ∇ϕ_∞ ( p), p )
    		=
    		f_∞(x,p),

	where ϕ_∞ is the unique solution of
	
    Δϕ_∞ (p) = ϱ_∞(p),
    		    ϕ_∞ (p) → 0,   as | p |→∞,
    		    ϱ_∞(p) = - ∫_ℝ^3 f_∞ (x,p) dx.



Theorem <ref> was first obtained by Flynn–Ouyang–Pausader–Widmayer <cit.>, using a proof based on a certain pseudo-conformal inversion of ℝ_t ×ℝ_x^3 ×ℝ_p^3.  Theorem <ref> has also recently been generalised to the Vlasov–Maxwell system independently, with a different method, by Bigorne <cit.>.



  For free transport (namely if f solves (<ref>) with ϕ≡0), 
  the quantity f(t,x+t p,p) is independent of t.  We would say that a solution f of Vlasov–Poisson scatters to free transport if this quantity  was to converge to a function independent of t as t →∞.
  We say that the solutions of Theorem <ref> scatter in a modified sense in view of the presence of the logarithmic correction involving ∇ϕ_∞ in (<ref>).  See Figure <ref> for a comparison of the trajectories t↦ x+tp - log t  ∇ϕ_∞ ( p) with the free trajectories t ↦ x+t p.





 



  Our proof of Theorem <ref> proceeds by constructing suitable approximate solutions to (<ref>)–(<ref>).
These are polyhomogeneous expansions of the form

    f_[K](t,x,p)
    	=
    	∑_k=0^K ∑_l=0^k(log t)^l/t^k f_k,l(x-tp + log t  ∇ϕ_∞ ( p), p ),


    ϕ_[K](t,x)
    	=
    	1/t∑_k=0^K ∑_l=0^k(log t)^l/t^kϕ_k,l( x/t),
    	  ϱ_[K](t,x)
    	=
    	1/t^3∑_k=0^K ∑_l=0^k(log t)^l/t^kϱ_k,l( x/t),

  which are determined explicitly from the scattering profile f_∞ alone.
  In fact, we show that these approximate solutions provide a detailed description the asymptotic behaviour of the solutions  in Theorem <ref>:




 
	For any smooth compactly supported function f_∞ℝ^3 ×ℝ^3 → [0,∞), there are sequences of smooth compactly supported functions
	
    f_k,lℝ^3 ×ℝ^3 →ℝ,
    		    ϕ_k,l, ϱ_k,lℝ^3 →ℝ,
    		    for 
    		k = 0,1,2,…,  and  l=0,… k,

	defined explicitly in terms of f_∞, with
	
    f_0,0 = f_∞,     ϱ_0,0 = ϱ_∞,     ϕ_0,0 = ϕ_∞,

	such that, for any K ∈ℕ, there exists T_0 > 0 such that the solution (f,ϱ,ϕ) of Theorem <ref> satisfies, for all (t,x,p) ∈ [T_0,∞) ×ℝ^3 ×ℝ^3,
	
    |
    		f(t,x,p)
    		- f_[K](t,x,p)
    		|≤
    		C_K ℱ_K ( log t/t)^K+1,

	where f_[K] is given by (<ref>), and with ϕ_[K] and ρ_[K] given by (<ref>),
	
    |∇ϕ(t,x)
    		-
    		∇ϕ_[K](t,x)
    		|   ≤
    		C_K ℱ_K (log t)^K+1/t^K+3,
    		
    |ϱ(t,x)
    		-
    		ϱ_[K](t,x)
    		|   ≤
    		C_K ℱ_K (log t)^K+1/t^K+4,

      	where C_K>1 is constant depending on K, and ℱ_K is defined in terms of the L^2 norm of f_∞, along with the L^2 norm of a large number (depending on K) of derivatives of f_∞.  Similar statements to (<ref>)–(<ref>) hold for, appropriately weighted, higher order derivatives.



Polyhomogeneous expansions of the form (<ref>)–(<ref>) have also very recently been obtained independently, in a closely related setting, by Bigorne–Velozo Ruiz <cit.>.  See Theorem <ref> below.


  While the explicit expressions for ϱ_k,l, ϕ_k,l and f_k,l are given in the proof of Theorem <ref> in Section <ref> below,
 we will discuss the constructions for K=0 and K=1  in detail in Section <ref>.
 The proof of Theorem <ref> is based on the existence of expansions of the form (<ref>)–(<ref>), and thus the proof of both Theorem <ref> and Theorem <ref> are established together.
 

  A similar approach, of using approximate solutions to finite order, has been used to obtain inverse scattering results for nonlinear wave equations in <cit.>. Homogeneous asymptotics as in (<ref>), and their implications for the scattering problem for wave equations have recently been considered in <cit.>.
  For another recent example in general relativity, of the broad strategy of using sequences of increasingly better approximate solutions to prove existence of solutions of an inverse scattering-type problem, see <cit.>.  




   
    A motivation to give a new proof of Theorem <ref> in this paper is to illustrate a method which can be applied to other collisionless kinetic equations, such as the Vlasov–Maxwell and Einstein–Vlasov systems.
In contrast to the  ellipticity of the Poisson equation,
the field equations in each of these two examples are, appropriately interpreted, hyperbolic.  As such, we have elected not to utilise the full ellipticity of the Poisson equation in the proof of Theorem <ref>.  In concrete terms, this means that the estimate (<ref>) below — which has appropriate analogues
for hyperbolic operators — is used, but not the estimate (<ref>).  We have also chosen, for this reason, to base the proof of Theorem <ref> entirely on L^2 estimates.




  The proof given in this paper simplifies considerably if the scattering profile f_∞ is further assumed to be suitably small.
In this case, an approximation to order K=0 suffices; see discussion in Section <ref> below.







The final main result concerns the uniqueness of the solutions of Theorem <ref>, which is shown to hold in the class of solutions agreeing with the polyhomogeneous expansions of Theorem <ref> to sufficiently high order.

 
	For any smooth compactly supported function f_∞ℝ^3 ×ℝ^3 → [0,∞), the solution (f,ϱ,ϕ) of Theorem <ref> is unique in the class of solutions which satisfy an expansion of the form (<ref>) to sufficiently high order.


The work <cit.> also provides a uniqueness statement as in Theorem <ref>, but without requiring the restriction to the class of solutions which a priori admit expansions of the form (<ref>).


For a more precise statement of Theorem <ref>, Theorem <ref>, and Theorem <ref>, see Section <ref> below.






 §.§ Scattering results and the small data regime




  It is well known that solutions of Vlasov–Poisson (<ref>)–(<ref>), for which f|_t=0 is appropriately small, scatter in modified sense, meaning that there exists a function f_∞ℝ^3 ×ℝ^3 → [0,∞) such that (<ref>) holds.
  This insight is due to Choi–Kwon <cit.>,   and an alternative proof was later given by Ionescu–Pausader–Wang–Widmayer <cit.>.

  
 
	For any function f_C ℝ_x ×ℝ_p → [0,∞) which is suitably regular and small, in an appropriate norm, there exists a solution (f,ϱ,ϕ) which exists globally in time, attains the Cauchy data, f(0,·,·) ≡ f_C, disperses as t →∞ and scatters in a modified sense.  More precisely, there exists a function f_∞ℝ_x^3 ×ℝ_p^3 → [0,∞) such that (<ref>) holds for all x,p∈ℝ^3.


The part of Theorem <ref> concerning global existence and dispersion is a classical result of Bardos–Degond <cit.>, for which there now exists a number of alternative proofs and improvements <cit.>, and has been generalised to certain large dispersive solutions <cit.>.  Theorem <ref> has also been generalised to higher dimensions by Pankavich <cit.> (see also Remark <ref> below), and to the Vlasov–Maxwell system by Bigorne <cit.> and Pankavich–Ben-Artzi <cit.>.  Furthermore the very recent work of Bigorne–Velozo Ruiz <cit.>, completed independently, provides a detailed description of the asymptotic behaviour of the solutions of Theorem <ref>.

 
	For any function f_C ℝ_x^3 ×ℝ_p^3 → [0,∞) which is suitably regular and small, in an appropriate norm, there exist functions f_k,lℝ^3 ×ℝ^3 →ℝ, ϕ_k,l, ϱ_k,lℝ^3 →ℝ for k = 0,1,2,…, and l=0,… k, such that the solutions of Theorem <ref> admit polyhomogeneous expansions of the form (<ref>)–(<ref>).




A scattering theory for the system (<ref>)–(<ref>) may be formulated in terms of the following questions:



	
  (i)
		Existence of the forward modified scattering operator: For a given time T_0 ∈ℝ, for which classes of Cauchy data f_C = f|_t=T_0 does there exist a corresponding scattering state f_∞?  What can be said about the forward scattering map 𝒮_F  f_C ↦ f_∞?
	
  (ii)
		Uniqueness of forward modified scattering states: Do solutions giving rise to the same scattering state coincide, i.e., when acting on appropriate spaces, is the forward scattering map 𝒮_F  f_C ↦ f_∞ injective?
	
  (iii)
		Asymptotic completeness of the forward modified scattering map:  For every f_∞, does there exist corresponding Cauchy data f_C, i.e., when acting on appropriate spaces, is the map 𝒮_F  f_C ↦ f_∞ surjective?



 Theorem <ref>  addresses question (iii),  and Theorem <ref> concerns question (ii).
    In the small data regime, Theorem <ref>    gives  a resolution of question (i).
    Despite global existence results for general classes of Cauchy data <cit.>, question (i) remains open in general. 



Due to the time reversibility of the system (<ref>)–(<ref>), a resolution of questions (i)–(iii) also provides a resolution of the corresponding questions for the backwards modified scattering operator 𝒮_B  f_C ↦ f_-∞.  Such an f_-∞ (-∞,-T_0] ×ℝ^3 ×ℝ^3 → [0,∞) satisfies, for all x,p∈ℝ^3,

    lim_t→ -∞ f(t,x+tp - log t  ∇ϕ_-∞ ( p), p )
    	=
    	f_-∞(x,p),

where f  (-∞,-T_0] ×ℝ^3 ×ℝ^3 → [0,∞) is the unique solution of the Vlasov–Poisson system attaining the Cauchy data f_C, and ϕ_-∞ is the unique solution of (<ref>), after replacing f_∞ with f_-∞.



    The time reversed analogue of Theorem <ref> states that for any smooth compactly supported function f_-∞ℝ^3 ×ℝ^3 → [0,∞), there exists T_0 > 0 and a solution f  (-∞,-T_0] ×ℝ^3 ×ℝ^3 → [0,∞) of the Vlasov–Poisson system such that, for all x,p∈ℝ^3, (<ref>) holds.
    Similarly, expansions of the form (<ref>)–(<ref>) hold for (t,x,p) ∈ (-∞,-T_0] ×ℝ^3 ×ℝ^3.
	






One can also ask analogous questions of the full modified scattering operator 𝒮 f_-∞↦ f_∞:

	
  (iv)
		Existence of the full modified scattering operator: For which classes of past scattering states f_-∞ does there exist a corresponding future scattering state f_∞?  What can be said about the full scattering map 𝒮 f_-∞↦ f_∞?
	
  (v)
		Uniqueness of full modified scattering states: Is the full scattering map, when acting on appropriate spaces, 𝒮 f_-∞↦ f_∞ injective?
	
  (vi)
		Asymptotic completeness of the full modified scattering map:  Is the full scattering map, when acting on appropriate spaces, 𝒮 f_-∞↦ f_∞ surjective?




In the small data regime, affirmative answers of questions (iv)–(vi)  are due to Flynn–Ouyang–Pausader–Widmayer <cit.>.


Alternatively,
in view of the time reversibility, Theorem <ref>  can also be combined directly with Theorem <ref> to answer questions (iv)–(vi) in the affirmative, under the additional assumption that f_-∞ is small:
For any smooth compactly supported function f_-∞ℝ^3 ×ℝ^3 → [0,∞), which is small in an appropriate sense, there exists a smooth compactly supported function f_∞ℝ^3 ×ℝ^3 → [0,∞) and a smooth solution f ℝ×ℝ^3 ×ℝ^3 → [0,∞) of the Vlasov–Poisson system (<ref>)–(<ref>) such that both (<ref>) and (<ref>) hold.  Moreover, the function f_∞ is quantitatively controlled by f_-∞.




Finally, we remark that there have been previous “scattering constructions” for Vlasov–Poisson where the spatial domain is the torus <cit.>.




 §.§ Further remarks on the main results


  Before we turn to the proof of Theorem <ref>, Theorem <ref>, and Theorem <ref>, several further remarks are given: 




	The proof of Theorem <ref> makes no essential use of the repulsive sign of the nonlinearity in (<ref>)–(<ref>) and thus equally applies, with the obvious modifications to the statements, to the gravitational Vlasov–Poisson system, i.e. the equation
	
    ∂_t f + p^i ∂_x^i f - ∂_x^iϕ ∂_p^i f = 0,

	coupled to equation (<ref>).  Attention is restricted here to (<ref>)–(<ref>) in order to ease notation.




	The convergence (<ref>) is shown to in fact hold in a much stronger sense, with a quantitative rate. In particular, see already the estimate (<ref>) with K=0.  See also equation (<ref>) in Theorem <ref> for a stronger statement. 





	The function f_∞ in Theorem <ref> is assumed, for simplicity, to be smooth.  The proof also holds, however, for far rougher f_∞ (for example for f_∞ lying in a suitable Sobolev space).  No attempt has been made to optimise the regularity requirements here.






 
	The proof of Theorem <ref> can be adapted to the case of d+1 dimensions, for d ≥ 4, to obtain solutions of Vlasov–Poisson which scatter to free transport, i.e. which satisfy
	
    lim_t→∞ f(t,x+tp,p) = f_∞(x,p),

	for all x,p∈ℝ^3.  The proof, in fact, simplifies considerably, and the 3+1 dimensional case should be viewed as critical, in a certain sense.  The borderline t^-1 behaviour discussed in Section <ref> below is replaced by integrable t^2-d behaviour (meaning that, for f̌(t,x,p) = f(t,x,p) - f_∞(x-tp,p), inequality (<ref>) becomes ℰ̌(t) ≤ℱ∫_t^T_f s^-d+2ℰ̌(s) ds + ℱt^-d+3), obviating the need for the approximate solutions (<ref>)–(<ref>) in the proof.  Recall that the analogue of Theorem <ref> in higher dimensions, where the solutions satisfy (<ref>), is due to Pankavich <cit.>.
	
	Theorem <ref> also generalises to higher dimensions to give expansions of the form
	
    f(t,x,p) = ∑_k=0^K 1/t^k f_k(x-tp, p ) + 𝒪(1/t^1+K),

	
    ϱ(t,x)
    		=
    		1/t^d∑_k=0^K 1/t^kϱ_k( x/t)
    		+
    		𝒪(1/t^d+K+1),
    		    ∇ϕ(t,x)
    		=
    		1/t^d-1∑_k=0^K 1/t^k (∇ϕ_k) ( x/t)
    		+
    		𝒪(1/t^d+K).







	As will be seen in the proof, in the context of Theorem <ref>, p and x/t are comparable in the support of f and thus, in the argument of the elements of the expansion (<ref>), x-tp + log t  ∇ϕ_∞ ( p) can be replaced with x-tp + log t  ∇ϕ_∞ (x/t), at the expense of further terms (involving further log t corrections) in the summation (<ref>).
	
	Indeed, one sees that further terms are necessary in such an expansion in view of the following fact.  Since f_∞, and hence ϕ_∞, is smooth and, as will be seen in the proof of Theorem <ref>, x-tp + log t  ∇ϕ_∞ ( p) is uniformly bounded in the support of the solution, it follows that
	
    f_∞(x-tp + log t  ∇ϕ_∞( x/t), p )
    		=
    		f_∞(x-tp + log t  ∇ϕ_∞ ( p), p )
    		
    
    		+
    		(log t)^2/t∂_i ϕ_∞(p) ∂_i ∂_j ϕ_∞(p) 
    		(∂_x^j f_∞)(x-tp + log t  ∇ϕ_∞ ( p), p )
    		+
    		𝒪( log t/t).

	Thus, if x-tp + log t  ∇ϕ_∞ ( p) is replaced with x-tp + log t  ∇ϕ_∞ (x/t) in the argument of the elements of the expansion, it is necessary to include a (log t)^2/t term in the expansion (which is not present in (<ref>)).






	Under alternative assumptions on f_∞, such as analyticity, one may hope to show that the infinite series
	
    ∑_k=0^∞∑_l=0^k(log t)^l/t^k f_k,l(x-tp + log t  ∇ϕ_∞ ( p), p ),

	converges to the solution of Vlasov–Poisson of Theorem <ref>.  Such convergence is not considered here.




 §.§ Overview of the proof



In this section an overview of the proof of Theorem <ref> is given.  As noted above, the proof of Theorem <ref> is established at the same time.  The approximate solutions of Theorem <ref> are discussed in Section <ref>, and in Section <ref> it is outlined how the approximate solutions are used to prove the existence of a solution of attaining the scattering data f_∞, which moreover agrees with the approximate solutions to increasing order.



  §.§.§ The proof of the main results — the approximate solutions
 
In the logic of the proof of Theorem <ref>, the first step is to give explicit definitions of the functions f_k,l, ϕ_k,l, ϱ_k,l in terms of f_∞.  Each ϕ_k,l is defined in terms of the corresponding ϱ_k,l as the unique solution of the Poisson equation

    Δ_ℝ^3ϕ_k,l = ϱ_k,l,
    	    ϕ_k,l(w) → 0,   as | w |→∞,

for which there is a well known representation formula for solutions, and so it remains to define f_k,l and ϱ_k,l.

Given such functions define f_[K], ϱ_[K], and ϕ_[K], for each K ≥ 0, by (<ref>)–(<ref>).
The functions (f_[K], ϱ_[K], ϕ_[K]) are said to be an approximate solution of order K if, for each x,p∈ℝ^3,

    _ϕ_[K] f_[K] (t,x,p)
    	=
    	𝒪((log t)^1+K/t^2+K),
    	    ∫_ℝ^3 f_[K] (t,x,p) dp - ϱ_[K] (t,x) 
    	=
    	𝒪( (log t)^K/t^4+K).




The functions f_k,l, and ϱ_k,l are defined simply by inserting the expressions (<ref>)–(<ref>) into the Vlasov–Poisson system (<ref>)–(<ref>) as an ansatz and solving order by order so that (f_[K], ϱ_[K], ϕ_[K]) is an approximate solution of order K.  Remarkably, after inserting these expressions, though ϱ_k,l appears in the expression for f_k,l, the quantity f_k,l does not appear in the expression for ϱ_k,l and so each quantity can be explicitly defined.  In order to illustrate the procedure, the cases K=0 and K=1 are presented here explicitly.  The full details are given in Section <ref>.  See, in particular, Theorem <ref>.

The fact that each approximate solution (f_[K], ϕ_[K], ϱ_[K]) agrees with the true solution to order K is only later shown, as part of the proof of the existence of the true solution (see the discussion in Section <ref> below).

In order to simplify expressions, the notation

    y = y(t,x,p) = x-tp + log t  ∇ϕ_∞ ( p),

is used.  Since f_∞ is compactly supported, the function y is uniformly bounded by a constant M>0 in the support of f and in the support of f_[K] for all K ≥ 0.  In particular, it follows that, if T_0 is suitably large,

    | y(t,x,p) |≤ M,
    	    | x |≤ 3M t,
    	    | p |≤ 2M,
    	    in (f)  and (f_[K]).

For fixed p and y, the trajectory t↦ y + tp - log t ∇ϕ_∞ is as depicted in Figure <ref>.  Fixing y and p can be viewed as identifying a particle trajectory in phase space.


It follows from (<ref>) that, for each t,x, the p-supports, (f(t,x,·)) and (f_[K](t,x,·)), are contained inside a ball centred at x/t + log t/tϕ_∞(x/t) with a radius of order t^-1 (cf. (<ref>) below) and so

    ∫_ℝ^31_(f) (t,x,p)
    	+
    	∫_ℝ^31_(f_[K]) (t,x,p) dp ≲1/t^3.

It also follows that the right hand sides in (<ref>) also satisfy the support properties (<ref>).

For functions hℝ^3 ×ℝ^3 →ℝ, such as the functions f_k,l, for i=1,2,3, the notation ∂_x^i h is always used to denote the derivative with respect to the i-th component, and the notation ∂_p^i h is used to denote the derivative with respect to the i+3-th component.




  §.§.§ The case K=0

It is convenient to first consider the case K=0.  Defining

    f_0,0(y,p)
    	=
    	f_∞(y, p ),
    	    ϱ_0,0(w)
    	=
    	ϱ_∞( w ),

so that

    f_[0](t,x,p)
    	=
    	f_∞(y(t,x,p), p ),
    	    ϕ_[0](t,x)
    	=
    	1/tϕ_∞( x/t),
    	    ϱ_[0](t,x)
    	=
    	1/t^3ϱ_∞( x/t),

with y(t,x,p) defined by (<ref>), we check that (<ref>) holds for K=0.  One computes

    _ϕ_[0] f_[0](t,x,p)
    	=
    	   1/t( ∂_i ϕ_∞(p) - ∂_i ϕ_∞(x/t) ) (∂_x^i f_∞)(y,p)
    	
       
    	+
    	log t/t^2∂_j ϕ_∞(x/t) ∂_i ∂_j ϕ_∞(p) (∂_x^i f_∞)(y,p)
    	+
    	1/t^2∂_i ϕ_∞(x/t) (∂_p^i f_∞)(y,p).

For y in the support of f_∞, so that (<ref>) holds,

    sup_| y |≤ M|∂_i ϕ_∞(x/t) - ∂_i ϕ_∞(p) |≤‖∇^2 ϕ_∞‖_L^∞( M/t + log t ‖∇ϕ_∞‖_L^∞/t),

which ensures that there is a cancellation in the first line of (<ref>) and one indeed has

    _ϕ_[0] f_[0](t,x,p)
    	=
    	𝒪(log t/t^2).


For t suitably large, the expression (<ref>) can be inverted to give p as an implicit function of t,x,y, and so one moreover checks

    ∫ f_[0](t,x,p) dp
    	-
    	1/t^3ϱ_∞( x/t)
    	=
    	∫ f_∞(y,p) ∂ p/∂ ydy + 1/t^3∫ f_∞( y, x/t) dy
    	=
    	𝒪( log t/t^4),

where the uniform boundedness of y, and fact that ∂ p/∂ y is equal to -t^-3 to leading order (see Proposition <ref> below), has been used.




  §.§.§ The case K=1

Consider now the case K=1.  Recall f_[1] and ϕ_[1] defined by (<ref>)–(<ref>), so that

    f_[1](t,x,p) = f_∞(y,p) + log t/t f_1,1(y,p) + 1/t f_1,0(y,p),
      ϕ_[1](t,x)
    	=
    	1/tϕ_∞( x/t) +log t/t^2ϕ_1,1( x/t) + 1/t^2ϕ_1,0( x/t),

for some smooth functions f_1,1, f_1,0, ϕ_1,1 and ϕ_1,0.  Revisiting (<ref>) one sees that, in order to ensure that _ϕ_[1] f_[1](t,x,p) = 𝒪((log t)^2/t^3), it is necessary to not only exploit the cancellation (<ref>), but to moreover take into account the higher order terms in an expansion (whose coefficients are functions of y and p) for ∇ϕ_∞ (x/t), in the region | y |≤ M.  Replacing x/t using the expression (<ref>), and Taylor expanding around p, one computes, for | y |≤ M,

    ∇ϕ_[1](t,x)
    	=
    	1/t^2∇ϕ_∞ (p)
    	+
    	log t/t^3( ∇ϕ_1,1(p) - ∇ϕ_∞(p) ·∇^2ϕ_∞ (p) )
    	+
    	1/t^3( ∇ϕ_1,0(p) + y ·∇^2ϕ_∞ (p) )
    	+
    	𝒪((log t)^2/t^4).

It thus follows that

    _ϕ_[1] f_[1](t,x,p)
    	=
    	𝒪( (log t)^2/t^3)
    	+
    	log t/t^2[
    	- f_1,1(y,p)
    	-
    	( ∂_i ϕ_1,1(p)
    	-
    	2∂_jϕ_∞(p) ∂_i ∂_j ϕ_∞(p) 
    	) (∂_x^i f_∞)(y,p)
    	]
    	
         
    	+
    	1/t^2[
    	f_1,1(y,p)
    	-
    	f_1,0(y,p)
    	-
    	(
    	∂_i ϕ_1,0(p) + y^j ∂_i ∂_j ϕ_∞(p)
    	)
    	(∂_x^i f_∞)(y,p)
    	+
    	∂_iϕ_∞(p) (∂_p^i f_∞)(y,p)
    	]
    	,

and one sees that the former of (<ref>) holds for K=1, provided the functions f_1,1, f_1,0, ϕ_1,1 and ϕ_1,0 satisfy

    f_1,1(y,p)
    	   
    	=
    	( 2 ∂_j ϕ_∞(p) ∂_i ∂_j ϕ_∞(p) - ∂_i ϕ_1,1(p) ) (∂_x^i f_∞)(y,p),
    	
    
    	f_1,0(y,p)
    	   
    	=
    	f_1,1(y,p)
    	-
    	( ∂_i ϕ_1,0(p) + y^j ∂_i ∂_j ϕ_∞(p) ) (∂_x^i f_∞)(y,p)
    	+
    	∂_i ϕ_∞(p) (∂_p^i f_∞)(y,p).

Once ϱ_1,1 and ϱ_1,0 — and hence ϕ_1,1 and ϕ_1,0 — have been defined, f_1,1 and f_1,0 will indeed be defined by (<ref>) and (<ref>) respectively.  Observe that, regardless of how ϕ_1,1 and ϕ_1,0 are defined, the terms in (<ref>)–(<ref>) involving the functions ϕ_1,1 and ϕ_1,0 vanish upon integration in y:

    ∫_ℝ^3 f_1,1(y,p) dy = 0,
    	    ∫_ℝ^3 f_1,0(y,p) dy
    	=
    	Δϕ_∞(p)
    	∫_ℝ^3 f_∞(y,p) dy
    	+
    	∂_i ϕ_∞(p) ∫_ℝ^3 (∂_p^i f_∞)(y,p) dy
    	.

This observation is returned to below.


Turning now to the latter of (<ref>), one similarly sees, upon revisiting (<ref>), that it is necessary to not only exploit the fact that ∂ p/∂ y is equal to -t^-3 to leading order, but to moreover take into account higher order terms in an expansion for ∂ p/∂ y.  One computes

    ∂ p/∂ y(t,x,y)
    	=
    	- 1/t^3 - log t/t^4Δϕ_∞( x/t) + 𝒪( (log t)^2/t^5),

for | y | + | x/t |≤ M.  Similarly one has (recalling that (<ref>) can be inverted to give p as an implicit function of t,x,y), for | y | + | x/t |≤ M,

    f_∞(y,p(t,x,y))
    	=
    	f_∞( y, x/t)
    	+
    	log t/t∂_i ϕ_∞( x/t) (∂_p^i f_∞) ( y, x/t)
    	-
    	1/t
    	y^i (∂_p^i f_∞) ( y, x/t)
    	+
    	𝒪( (log t)^2/t^2).

Recall now ϱ_[1] defined by (<ref>), so that

    ϱ_[1](t,x)
    	=
    	1/t^3ϱ_∞( x/t)
    	+
    	log t/t^4ϱ_1,1( x/t)
    	+
    	1/t^4ϱ_1,0( x/t),

for some smooth functions ϱ_1,1 and ϱ_1,0.  It follows that

    ∫_ℝ^3 f_[1] (t,x,p) dp - ϱ_[1] (t,x)
    	
       
    	=
    	- log t/t^4[
    	ϱ_1,1( x/t)
    	+
    	∫_ℝ^3 f_1,1( y, x/t) dy
    	+
    	Δϕ_∞( x/t) ∫_ℝ^3 f_∞( y, x/t) dy
    	-
    	∂_i ϕ_∞( x/t) ∫_ℝ^3 (∂_p^i f_∞) ( y, x/t) dy
    	]
    	
         
    	- 1/t^4[
    	ϱ_1,0( x/t)
    	+
    	∫_ℝ^3 f_1,0( y, x/t) dy
    	-
    	∫_ℝ^3 y^i (∂_x^i f_∞) ( y, x/t) dy
    	]
    	+
    	𝒪( (log t)^2/t^5).

The latter of (<ref>) then holds if 

    ϱ_1,1 (w)
    	   
    	=
    	-
    	∫_ℝ^3 f_1,1( y, w) dy
    	-
    	Δϕ_∞(w) ∫_ℝ^3 f_∞ ( y, w) dy
    	+
    	∂_i ϕ_∞(w) ∫_ℝ^3 (∂_p^i f_∞) ( y, w) dy,
    	
    ϱ_1,0 (w)
    	   
    	=
    	-
    	∫_ℝ^3 f_1,0 ( y, w) dy
    	+
    	∫_ℝ^3 y^i (∂_x^i f_∞) ( y,w) dy.


Thus, (f_[1], ϱ_[1], ϕ_[1]) is an approximate solution of order 1 if f_1,1, f_1,0, ϱ_1,1, and ϱ_1,0, satisfy both (<ref>)–(<ref>), and (<ref>)–(<ref>).  Remarkably, as noted above, the terms involving ϕ_1,1 and ϕ_1,0 in (<ref>)–(<ref>) vanish upon integration in y and thus, when the integrated expressions (<ref>) are inserted into (<ref>), one obtains explicit expressions for ϱ_1,1, and ϱ_1,0:

    ϱ_1,1 (w)
    	   
    	=
    	-
    	Δϕ_∞(w) ∫_ℝ^3 f_∞ ( y, w) dy
    	-
    	∂_i ϕ_∞(w) ∫_ℝ^3 (∂_p^i f_∞) ( y, w) dy.
    	
    ϱ_1,0 (w)
    	   
    	=
    	-
    	Δϕ_∞(w)
    	∫_ℝ^3 f_∞(y,w) dy
    	-
    	∂_i ϕ_∞(w) ∫_ℝ^3 (∂_p^i f_∞) ( y, w) dy
    	+
    	∫_ℝ^3 y^i (∂_x^i f_∞) ( y,w) dy.

One therefore defines ϱ_1,1 and ϱ_1,0 by (<ref>) and (<ref>) respectively, and then defines f_1,1 and f_1,0 by (<ref>) and (<ref>) respectively (recalling that ϕ_1,1 and ϕ_1,0 are defined explicitly in terms of ϱ_1,1 and ϱ_1,0 as the unique solutions of the Poisson equation (<ref>)).




  §.§.§ The case K≥ 2

For K≥ 2, one proceeds inductively by similarly inserting the expressions (<ref>)–(<ref>) into the Vlasov–Poisson system (<ref>)–(<ref>) as an ansatz and solving for f_k,l, and ϱ_k,l order by order so that (f_[K], ϱ_[K], ϕ_[K]) is an approximate solution of order K.  Higher order analogues of the steps discussed in the K=1 case above are required.  These steps form the content of Section <ref>.


In Proposition <ref>, a higher order analogue of the first order expansion (<ref>) for ∇ϕ_[K] is given.  The coefficients of the expansion for ∇ϕ_[K] are denoted Ψ_k,l.  

In Section <ref> (see (<ref>)–(<ref>)) higher order analogues of the explicit expressions (<ref>)–(<ref>) for f_k,l are given in terms of Ψ_k,l (which are in turn given explicitly in term of ϕ_k,l in (<ref>)–(<ref>)) and in Proposition <ref> it is shown that, if these expressions are satisfied for all 0≤ k ≤ K, 0 ≤ l ≤ k, then the former of (<ref>) holds.  Moreover, an analogue of the observation (<ref>) — namely that, upon integrating the expression for f_k,l with respect to its first argument, the terms involving ϕ_k,l all vanish and only terms involving ϕ_k',l', for k'≤ k-1, remain — persists to higher orders.  See Remark <ref>.

Turing again to the latter of (<ref>), one sees that a higher order analogue of the first order expansion (<ref>) for (∂ p / ∂ y) is required.  Such an expansion is given in Proposition <ref>, where the coefficients of this expansion are denoted J_k,l.  Higher order analogues of the expressions (<ref>)–(<ref>) for ϱ_k,l are then given in terms of f_k,l and J_k,l.  See (<ref>)–(<ref>).  The analogue of the observation (<ref>), discussed in Remark <ref>, means that these expressions, together with the expressions (<ref>)–(<ref>) for f_k,l, define each ϱ_k,l and f_k,l explicitly.  In Proposition <ref> it is then shown that, with these definitions, the latter of (<ref>) holds.



The reader is referred to Section <ref> for more details.




  §.§.§ The proof of the main results — existence of the solution


The existence part of the proof of Theorem <ref> proceeds by considering some arbitrarily large time T_f, and considering an associated “finite problem” for each such time.  One defines “final data” at time t=T_f to coincide with the approximate solution f_[K](T_f,·,·) (discussed in Section <ref>):

    f |_t=T_f = f_[K](T_f,·,·),

for some sufficiently large K, to be determined later.  One then shows that the corresponding solution of Vlasov–Poisson exists on the interval [T_0,T_f], for some T_0 which is independent of T_f.  The main step in the proof is in establishing uniform estimates for the solution on this interval, with constants independent of T_f.  Once such estimates have been established, the proof of the existence of a solution on [T_0,∞) follows from considering, on their common domain, the differences f^(T_f) - f^(T_f') of such solutions corresponding to different final times T_f<T_f', and showing that such differences vanish in the limit T_f→∞ (see Section <ref>).  The prescription (<ref>), for each T_f, ensures that the limiting solution on [T_0,∞) attains the scattering data f_∞ in the sense (<ref>).


The solution is not estimated directly but, for some fixed appropriate K ≥ 0, the quantity

    f̌_[K](t,x,p)
    	:=
    	f(t,x,p)
    	-
    	f_[K](t,x,p),

is considered, where f_[K] is the approximate solution discussed in Section <ref> (see equation (<ref>)).  This quantity satisfies an inhomogeneous equation of the form

    _ϕf̌_[K] = F_[K],

where

    F_[K] := - _ϕ f_[K]
    	=
    	- ∂_x^i(ϕ - ϕ_[K]) ∂_p^i f_[K] + 𝒪((log t)^1+K/t^2+K),

by virtue of the fact that (f_[K], ϱ_[K],ϕ_[K]) is an approximate solution of order K (see (<ref>)).  The definition (<ref>) of the “final data” ensures that

    f̌_[K](T_f,x,p) = 0,

for all x,p ∈ℝ^3.  For fixed N≥ 6, L^2 based energies of the form

    ℰ̌_[K](t)
    	: =
    	∑_| I | +| J | = 0^N ‖ L^I (t^-1∂_p)^J f̌_[K](t,·,·) ‖_L^2_x L^2_p,

are considered, where I, J are multi-indices and L is an appropriate collection of vector fields (see Section <ref> below).

In the remainder of this section, the estimates for f̌_[K] via the energy (<ref>), on the interval [T_0,T_f] are outlined.  After describing the procedure for obtaining L^2 based estimates, the case that f_∞ is assumed to be small, in an appropriate sense, is discussed.  When f_∞ is assumed to be appropriately small, it is only necessary to consider the energy (<ref>) for K=0 (and the thus the approximate solutions (<ref>)–(<ref>) are only considered up to order zero).  It is helpful to present first this simpler case.  The case for general f_∞ is then presented, which involves considering the energy (<ref>) for K suitably large, depending on an appropriate norm of f_∞.


For simplicity, only the estimate for the energy (<ref>) with N=0 is discussed in the present overview.  In the proof of Theorem <ref> it is important, however, to estimate derivatives of f and, in order to control the nonlinear terms arising from commuting the equations, derivatives up to order N ≥ 6 are estimated.  As noted above, a collection of vector fields {L_1, L_2, L_3}, are introduced.  These vector fields are defined in terms of the scattering profile ϕ_∞ so as to have good commutation properties with the Vlasov equation.  See Section <ref>.



  §.§.§ L^2 estimates

We have chosen, in this work, to make all estimates L^2 based (see Remark <ref>, and note further the simplicity with which (<ref>) and (<ref>) are derived).  For any smooth, suitably decaying function f, and any smooth ϕ, one has

    ∂_t f^2 + p^i ∂_x^i f^2 + ∂_x^iϕ ∂_p^i f^2 = 2 f _ϕ f,

and, since the second two terms on the left vanish after integrating in x and p,

    ‖ f (t,·, ·) ‖_L^2_xL^2_p≲‖ f (T_f,·, ·) ‖_L^2_xL^2_p
    	+
    	∫_t^T_f‖_ϕ f (s,·, ·) ‖_L^2_xL^2_p ds.

Moreover, for any smooth function ϕ, multiplying Δϕ by ϕ and integrating by parts gives,

    ∫_ℝ^3|∇ϕ|^2 dx
    	=
    	-
    	∫_ℝ^3ϕΔϕ dx
    	≤‖ r^-1ϕ‖_L^2‖ r Δϕ‖_L^2≲‖∇ϕ‖_L^2‖ r Δϕ‖_L^2
    	,

where r(x) = | x |, and a Hardy inequality (see Proposition <ref> below) is used in the last step.
Thus, if Δϕ is supported in the region | x |≲ t, one has

    ‖∇ϕ (t,·) ‖_L^2_x≲ t ‖Δϕ (t,·) ‖_L^2_x.




  §.§.§ Small f_∞ and the case K=0


One already sees the utility of the quantity f̌_[K] for K=0, in that f̌_[0] leads to an estimate for f, independent of T_f, provided f_∞ is small in an appropriate norm.  Indeed, define f̌ = f̌_[0], and consider the equation (<ref>)–(<ref>).  The approximate solution f_[0] has the property that, for all t ∈ [T_0,T_f],

    sup_x ∈ℝ^3( ∫_ℝ^3|∂_p f_[0](t,x,p) |^2 dp )^1/2≤
    	t^-1/2ℱ,

where ℱ is a constant depending on an appropriate norm of the scattering data f_∞.  The final term on the right hand side of (<ref>) is supported in (<ref>) (and so in particular satisfies (<ref>)).  Hence

    ‖𝒪(log t/t^2) ‖_L^2_xL^2_p≤ℱlog t/t^2.

Applying the inequality (<ref>) with f̌ and setting K=0 in the final condition (<ref>), it follows that the energy ℰ̌ = ℰ̌_[0], defined by (<ref>) with N=0, satisfies, for all T_0 ≤ t ≤ T_f,

    ℰ̌(t)
    	≲∫_t^T_f‖ F_[0] (s,·, ·) ‖_L^2_xL^2_p ds
    	≲ℱ∫_t^T_f s^-1/2‖∇_x ϕ̌ (s,·) ‖_L^2_x ds
    	+
    	ℱlog t/t≲ℱ∫_t^T_f s^1/2‖ϱ̌ (s,·) ‖_L^2_x ds
    	+
    	ℱlog t/t
    	,

where the last step uses (<ref>).  Using the Cauchy–Schwarz inequality and the fact (<ref>), it follows that the energy ℰ̌(t) satisfies, for all T_0 ≤ t ≤ T_f,

    ℰ̌(t)
    	≤ℱ∫_t^T_fℰ̌(s)/s ds + ℱ/t^1-,

where ℱ is a constant depending on an appropriate norm of the scattering data f_∞, and the - in t^1- indicates a logarithmic loss.  The Grönwall inequality (see Proposition <ref>) then implies that

    ℰ̌(t)
    	≤ℱ/t^1-
    	+
    	ℱ^2/t^ℱ∫_t^T_f1/s^-ℱ+2 - ds.

If ℱ is sufficiently small — which can be achieved by making the relevant norm of the scattering profile f_∞ suitably small — then this inequality gives

    ℰ̌(t)
    	≤ℱ +ℱ^2/t^1-,

as desired.  For larger ℱ, however, this inequality fails to provide an estimate for ℰ̌(t) which is uniform in T_f.





  §.§.§ General f_∞ and the case K≥ 1



In order to obtain uniform estimates of the solution without assuming any smallness on the scattering data f_∞, one considers the energy ℰ̌_[K](t), for general K ≥ 0.  Consider the equation (<ref>)–(<ref>).  

The approximate solution f_[K] retains the property (<ref>), i.e. for all t ∈ [T_0,T_f],

    sup_x ∈ℝ^3( ∫_ℝ^3|∂_p f_[K](t,x,p) |^2 dp )^1/2≤
    	t^-1/2ℱ,

and, further, the final term on the right hand side of (<ref>) satisfies (<ref>) with (log t)^1+K/t^2+K in place of log t/t^2.
Revisiting the steps above one therefore sees that the energy ℰ̌_[K](t) satisfies an inequality which, in the case N=0, takes the form

    ℰ̌_[K](t)
    	≤ℱ∫_t^T_fℰ̌_[K](s)/s ds + ℱ/t^K+1-,

(see the proof of Proposition <ref>, in particular equation (<ref>), for the inequality satisfied by ℰ̌_[K](t) for general N ≥ 6).  Note the improvement, compared to (<ref>).  The Grönwall inequality (see Proposition <ref>) then gives

    ℰ̌_[K](t)
    	≤ℱ/t^K+1-
    	+
    	ℱ^2/t^ℱ∫_t^T_f1/s^-ℱ+K+2- ds.

Now, if K is chosen to be suitably large, with respect to ℱ (and hence to the size of f_∞), it follows that

    ℰ̌_[K](t)
    	≤ℱ +ℱ^2/t^K+1-,

as desired.




 §.§ Outline of the paper


Section <ref> concerns certain preliminaries, including various inequalities which are used throughout, along with a collection of vector fields used in the proof of Theorem <ref> and a discussion of their basic properties.  In Section <ref>, more precise versions of Theorem <ref>, Theorem <ref> and Theorem <ref> are stated.  Section <ref> concerns the the functions f_k,l, ϱ_k,l, ϕ_k,l, which define approximate solutions of the Vlasov–Poisson system (<ref>)–(<ref>), as outlined in Section <ref> above.  In Section <ref> a sequence of “finite problems” is introduced, and the approximate solutions of Section <ref> are used to obtain estimates on the solutions of these finite problems, as outlined in Section <ref> above.  Finally, in Section <ref>, the proof of the precise versions of Theorem <ref>, Theorem <ref> and Theorem <ref>, stated in Section <ref>, are given using the solutions of these finite problems and the estimates obtained in Section <ref>.





 §.§ Acknowledgements


We would like to thank MATRIX, the mathematical research institute in Creswick, Australia, for their hospitality and  the collaborative environment during the workshop on “Hyperbolic PDEs and Nonlinear Evolution Problems”, 18–29th September 2023, where part of this research was conducted.
     M.T. acknowledges support through Royal Society Tata University Research Fellowship URF\R1\191409.





§ PRELIMINARIES



This section contains certain preliminaries which will be used throughout the remainder of the article.  In Section <ref> some functional inequalities, such as Sobolev and Hardy inequalities, are collected.  Section <ref> contains basic L^2 based estimates for the Vlasov equation and the Poisson equation.  In Section <ref> a collection of vector fields is introduced and their commutation properties with the Vlasov equation are discussed.  Further, a weighted Sobolev inequality, which features these vector fields, is shown.  Finally, in Section <ref>, conventions for the constants appearing in this work are stated.






 §.§ Functional inequalities



Define the norms, for any smooth function hℝ^3 →ℝ,

    ‖ h ‖_L^∞(ℝ^3) = sup_x∈ℝ^3| h(x) |,
    	    ‖ h ‖_L^2(ℝ^3)^2 = ∫_ℝ^3| h(x) |^2 dx,


    ‖ h ‖_H^k(ℝ^3)^2
    	=
    	∑_i_1,i_2,i_3=0^k ∫_ℝ^3|∂_x^1^i_1∂_x^2^i_2∂_x^3^i_3 h(x) |^2 dx,
    	    ‖ h ‖_H^k(ℝ^3)^2
    	=
    	∑_i_1,i_2,i_3=k∫_ℝ^3|∂_x^1^i_1∂_x^2^i_2∂_x^3^i_3 h(x) |^2 dx.

Functions hℝ^6 →ℝ of both x and p will also often be considered and so, for emphasis, we also write

    ‖ h ‖_L^2_xL^2_p^2 := ‖ h ‖_L^2(ℝ^6)^2 = ∫_ℝ^3_x∫_ℝ^3_p| h(x,p) |^2 dp dx.



The first result is the following L^∞—L^2 Sobolev inequality.

 
	There exists a constant C such that, for any function h ∈ H^2(ℝ^3),
	
    ‖ h ‖_L^∞(ℝ^3)≤
    		C
    		‖ h ‖_L^2(ℝ^3)^1/4‖ h ‖_H^2(ℝ^3)^3/4.




	First note that
	
    ‖ h ‖_L^∞(ℝ^3)≤
    		C(
    		‖ h ‖_L^2(ℝ^3) + ‖ h ‖_H^2(ℝ^3)).

	Indeed, since h ∈ H^2(ℝ^3) the Fourier inversion formula holds and so, for any x∈ℝ^3,
	
    | h (x) |≤∫_ℝ^3|f̂ (ξ) | d ξ≤( ∫_|ξ|≤ 1 1   dξ)^1/2( ∫|f̂ (ξ) |^2 d ξ)^1/2
    		+
    		( ∫_|ξ| > 1|ξ|^-4 dξ)^1/2( ∫|ξ|^4 |f̂ (ξ) |^2 d ξ)^1/2.

	The proof of (<ref>) then follows from the Plancherel formula and the fact that
	
    ∫_|ξ| > 1|ξ|^-4 dξ < ∞,

	in 3 dimensions.
	
	Now, for any λ>0, define
	
    h_λ(x) = h(λ x).

	It follows that
	
    ‖ h_λ‖_L^2(ℝ^3) = λ^-3/2‖ h ‖_L^2(ℝ^3),
    		    ‖ h_λ‖_H^1(ℝ^3) = λ^-1/2‖ h ‖_H^1(ℝ^3),
    		    ‖ h_λ‖_H^2(ℝ^3) = λ^1/2‖ h ‖_H^2(ℝ^3),

	and thus, applying (<ref>) with h_λ in place of h,
	
    ‖ h ‖_L^∞(ℝ^3)
    		=
    		‖ h_λ‖_L^∞(ℝ^3)≤
    		C(
    		λ^-3/2‖ h ‖_L^2(ℝ^3) + λ^1/2‖ h ‖_H^2(ℝ^3)).

	The proof follows from setting λ = ‖ h ‖_L^2(ℝ^3)^1/2‖ h ‖_H^2(ℝ^3)^-1/2 (and noting that the inequality is trivial if ‖ h ‖_H^2(ℝ^3) = 0).


 
	In the proof of the main results of this article, weighted operators of the form t∇_x will often be considered, and so the Sobolev inequality of Proposition <ref> will typically be used in the form, for any appropriate function h [T_0,∞) ×ℝ^3 →ℝ,
	
    ‖ h(t,·) ‖_L^∞(ℝ^3)≤
    		C ‖ h(t,·) ‖_L^2(ℝ^3)^1/4‖ h(t,·) ‖_H^2(ℝ^3)^3/4   ≤C/t^3/2‖ h(t,·) ‖_L^2(ℝ^3)^1/4∑_i,j=1^3 ‖ (t∂_x^i) (t∂_x^j) h(t,·) ‖_L^2(ℝ^3)^3/4
       ≤C/t^3/2(
    		‖ h(t,·) ‖_L^2(ℝ^3)
    		+
    		∑_i,j=1^3 ‖ (t∂_x^i) (t∂_x^j) h(t,·) ‖_L^2(ℝ^3)),

	where the final step follows from Young's Inequality.
	See, in particular, the use of this fact in the proof of Proposition <ref>.


Similarly, one has the following Sobolev inequality which can be applied to functions a ℝ^3_x ×ℝ^3_p →ℝ.

 
	There exists a constant C such that, for any function a ∈ H^4(ℝ^6),
	
    ‖ a ‖_L^∞(ℝ^6)≤
    		C
    		‖ a ‖_L^2(ℝ^6)^1/4‖ a ‖_H^4(ℝ^6)^3/4≤
    		C
    		‖ a ‖_H^4(ℝ^6).



Consider also the following Hardy inequality.

 
	For any function h ∈ H^1(ℝ^3),
	
    ∫_ℝ^3| x |^-2 h^2 dx
    		≤
    		4 ∫_ℝ^3|∇ h |^2 dx.




	Consider polar coordinates (r,θ^1,θ^2) on ℝ^3.  For fixed (θ^1,θ^2), since h ∈ H^1(ℝ^3), there is a sequence r_n such that lim_n →∞ (rh^2)(r_n, θ^1,θ^2) = 0.  Integrating ∂_r (r h^2) between 0 and r_n, and taking n →∞, it follows that
	
    0
    		=
    		∫_0^∞∂_r (r h^2) dr
    		=
    		∫_0^∞ h^2 + 2 r h ∂_r h dr,

	and so, by the Cauchy–Schwarz inequality,
	
    ∫_0^∞ h^2 dr
    		≤
    		2 ( ∫_0^∞ h^2 dr)^1/2(∫_0^∞ (∂_r h)^2 r^2 dr)^1/2.

	The result follows after dividing by ( ∫_0^∞ h^2 dr)^1/2, squaring, and integrating over S^2.


The following form of Taylor's Theorem will be used.

 
	For any smooth function h ℝ^n →ℝ, any x_0 ∈ℝ^n, and any K ≥ 0,
	
    h(x)
    		=
    		   ∑_k=0^K
    		1/k! (x-x_0)^i_1… (x-x_0)^i_k (∂_i_1…∂_i_k h)(x_0)
    		
       
    		+
    		1/K! (x-x_0)^i_1… (x-x_0)^i_K+1∫_0^1 (1-s)^K (∂_i_1…∂_i_K+1 h)(sx) ds,

	where (x-x_0)^i ∈ℝ denotes the i-th component of x-x_0 = ((x-x_0)^1,…,(x-x_0)^n) ∈ℝ^n.


Finally, the following Grönwall type inequality will also be used.

 
	Suppose 0<T<T_f and v  [T,T_f]→ℝ satisfies
	
    v(t) ≤ b(t) + ∫_t^T_f a(s) v(s) ds,
 
	for all T ≤ t ≤ T_f, for some a,b  [T,T_f]→ [0,∞).  Then
	
    v(t) ≤ b(t) + ∫_t^T_f a(s) b(s) e^∫_t^s a(s')ds'ds,

	for all T_0 ≤ t ≤ T.



	First note that
	
    d/dt(  ∫_t^T_f a(s) v(s) ds   e^-∫^T_f_t a(s')ds')
    		=
    		- a(t)   e^∫_t^T_f a(s')ds'( v(s) - ∫_t^T_f a(s) v(s) ds )
    		≥
    		- a(t) b(t)   e^-∫_t^T_f a(s')ds'.

	Integrating between T_f and t gives
	
    ∫_t^T_f a(s) v(s) ds   e^-∫_t^T_f a(s')ds'≤∫_t^T_f a(s) b(s)   e^-∫_s^T_f a(s')ds' ds,

	and the result follows.




 §.§ Elliptic and transport estimates



The following gradient estimate for the Poisson equation will be used.

 
	For any function h∈ H^2(ℝ^3),
	
    ‖∇ h ‖_L^2≲‖ r Δ h ‖_L^2.




	Integrating by parts,
	
    ∫_ℝ^3|∇ h |^2 dx
    		=
    		-
    		∫_ℝ^3 h Δ h dx
    		≤‖ r^-1 h ‖_L^2‖ r Δ h ‖_L^2,

	where r(x) = | x |.  The proof then follows from the Hardy inequality, Proposition <ref>.



 
	In addition to Proposition <ref>, one also has the full elliptic estimate
	
    ∑_i,j=1^3‖∂_i ∂_j h ‖_L^2≲‖Δ h ‖_L^2,

	which holds for all functions h∈ H^2(ℝ^3).  In order to illustrate an approach which can be applied to other equations, we have elected not to use the estimate (<ref>) in this work (see Remark <ref>).  In particular, the estimate (<ref>) has appropriate analogues — in terms of number of derivatives appearing on each side of the inequality — for hyperbolic operators, in contrast to the full elliptic estimate (<ref>).


The next result is an L^2 estimate for the Vlasov equation.

 
	Let a [T,T_f] ×ℝ^3_x ×ℝ^3_p →ℝ be a suitably decaying smooth function satisfying
	
    ∂_t a + p^i ∂_x^i a + ∂_x^iϕ ∂_p^i a = H,

	for some smooth H [T,T_f] ×ℝ^3_x ×ℝ^3_p →ℝ and some smooth ϕ [T,T_f] ×ℝ^3_x →ℝ.  Then a satisfies the L^2 estimate,
	
    ‖ a(t,·,·) ‖_L^2_x L^2_p≲‖ a(T_f,·,·) ‖_L^2_x L^2_p
    		+
    		∫^T_f_t ‖ H(s,·,·) ‖_L^2_x L^2_p ds,

	for all t ∈ [T,T_f], where
	
    ‖ a(t,·,·) ‖_L^2_x L^2_p^2
    		=
    		∫_ℝ^3∫_ℝ^3| a(t,x,p) |^2 dp dx.




	First note that a^2 satisfies
	
    ∂_t a^2 + p^i ∂_x^i a^2 + ∂_x^iϕ ∂_p^i a^2 = 2 a H,

	and so
	
    ∂_t ‖ a(t,·,·) ‖_L^2_x L^2_p^2
    		=
    		∫_ℝ^3∫_ℝ^3
    		-
    		p ·∇_x (| a(t,x,p) |^2)
    		-
    		∇_x ϕ·∇_p (| a(t,x,p) |^2)
    		+
    		2 a(t,x,p) H(t,x,p)
    		dp dx.

	Since a decays, it follows that the first two terms on the right hand side vanish and so, by Cauchy–Schwarz, for any t ≤ t_0 ≤ T_f,
	
    ∂_t ‖ a(t,·,·) ‖_L^2_x L^2_p
    		=
    		∂_t ‖ a(t,·,·) ‖_L^2_x L^2_p^2/2 ‖ a(t,·,·) ‖_L^2_x L^2_p≤‖ H(t,·,·) ‖_L^2_x L^2_p.

	The result follows after integrating from t to T_f.





 §.§ Vector fields and multi-index notation



Suppose a smooth function ϕ_∞ℝ^3 →ℝ is given.  For k=1,2,3, define vector fields

    L_k
    	=
    	t ∂_x^k + ∂_p^k
    	+
    	log t/t∂_k ∂_i ϕ_∞(p) ∂_p^i.




Given a multi-index I=(I_1,I_2,I_3), with I_1,I_2,I_3 ≥ 0, define the operators

    L^I = (L_1)^I_1 (L_2)^I_2 (L_3)^I_3,
    	  
    	(t^-1∂_p)^I = (t^-1∂_p^1)^I_1 (t^-1∂_p^2)^I_2 (t^-1∂_p^3)^I_3,
    	  
    	(t ∂_x)^I = (t ∂_x^1)^I_1 (t ∂_x^2)^I_2 (t ∂_x^3)^I_3.



The main reason for introducing the vector fields L_k is due to the form of the commutator [ 𝕏,L_k], given in the following proposition.  See the discussion in Remark <ref> below. The vector fields L_k also behave well when applied to functions of x-tp + log t ∇ϕ_∞( p ) and p.  See Remark <ref> below.



The vector fields L_i and t^-1∂_p^i have the following commutation properties with the Vlasov equation.

 
	For i=1,2,3, the vector fields L_i satisfy,
	
    [
    		_ϕ
    		,
    		L_i
    		]
    		=
    		   (
    		t^-2∂_i ∂_kϕ_∞(p)
    		-
    		t ∂_x^i∂_x^kϕ(t,x)
    		)
    		∂_p^k
    		-
    		log t/t^2∂_i ∂_jϕ_∞(p)  ( t ∂_x^j + ∂_p^j)
    		
       
    		+
    		log t/t∂_x^kϕ(t,x)  ∂_i ∂_k ∂_lϕ_∞(p)  ∂_p^l
    		,

	and t^-1∂_p^i satisfy,
	
    [
    		_ϕ
    		,
    		t^-1∂_p^i]
    		=
    		-
    		1/t^2( t ∂_x^i + ∂_p^i)
    		.




	The proof is a direct computation.





	Note that the vector fields obtained by setting ϕ_∞≡ 0 in (<ref>) commute with the free transport operator (i.e. the operator ∂_t + p^i ∂_x^i).  The logarithmic corrections in (<ref>) are crucial in ensuring the good commutation properties of Proposition <ref> (see Remark <ref>), and the boundedness property of the right hand side of (<ref>) below.  Corrections to vector fields with good commutation properties of the linearised part of the equation, of this form, played a central role in the proof <cit.>, and the works <cit.> on the stability of Minkowski space for the Einstein–Vlasov system.  In fact, it is exactly because the solutions only scatter in a modified sense in 3+1 dimensions that corrections to these vector fields are required in these works.  Note that these corrections are, in a sense, simpler in the present setting since they are defined directly with respect to the scattering profile ϕ_∞.


 
	In order to understand the improvement that the corrected vector fields L_i have over the non-corrected vector fields t∂_x^i + ∂_p^i, it is a helpful exercise to assume the conclusions of the main results, in particular that solutions satisfy the pointwise estimates
	
    | L_j f | + | t^-1∂_p f|≤ℱ,
    		    |∇^2 ϕ|≤ℱ/t^3,
    		    |∇^2 ϕ̌_[0]|≤ℱlog t/t^4,

	where ϕ̌_[0](t,x) = ϕ(t,x) - t^-1ϕ_∞(x/t), and check the behaviour of the respective commutators when applied to a solution.
	Indeed, assuming (<ref>), the commutators (<ref>), when applied to a solution f, have the property that
	
    |[
    		_ϕ
    		,
    		L_i
    		] f |≲
    		t^2 |∇^2 ϕ̌_[0]| | t^-1∂_p f |
    		+
    		log t/t^2|∇^2 ϕ_∞| | L_j f |
    
    		+
    		(log t)^2/t^2|∇^2 ϕ_∞|^2 | t^-1∂_p f |
    		+
    		log t/t^2|∇^3 ϕ_∞||∇ϕ|| t^-1∂_p f |≲log t/t^2ℱ
    		.

	Contrast with the borderline behaviour exhibited by the commutator of _ϕ with the non-corrected vector fields (obtained by setting ϕ_∞≡ 0 in (<ref>)):
	
    |[
    		_ϕ
    		,
    		(t∂_x^i + ∂_p^i)
    		] f |≲
    		t^2 |∇^2 ϕ| | t^-1∂_p f |≲ℱ/t.

	This borderline behaviour would lead to a logarithmic divergence upon integrating globally in time.



 
	Many of the objects considered in the following are functions of x - tp + log t ∇ϕ_∞(p) and p, and the vector fields L_i also have good properties when applied to such functions.  Indeed, for any smooth h ℝ^3 ×ℝ^3 →ℝ,
	
    L_i ( h(x - tp + log t ∇ϕ_∞(p),p) )
    		=
    		(log t)^2/t∂_k ∂_i ϕ_∞(p) (∂_x^kh)
    		+
    		( δ_ik + log t/t∂_k ∂_i ϕ_∞(p) ) (∂_p^kh),

	and the coefficients of the derivatives of h are not growing in t.  Contrast with the non-corrected vector fields t∂_x^i + ∂_p^i.  Note also that
	
    (t^-1∂_p^i) ( h(x - tp + log t ∇ϕ_∞(p),p) )
    		=
    		( - δ_ik + log t/t∂_k ∂_i ϕ_∞(p) ) (∂_x^kh)(y,p)
    		+
    		t^-1 (∂_p^ih)(y,p)
    		.

	These facts are in particular used in the proofs of Proposition <ref> and Proposition <ref> below.


One also has a weighted version of the Sobolev inequality in which the derivatives are taken with respect to the above vector fields.

 
	There exists a constant C such that, for any function h ∈ H^2(ℝ^3_x ×ℝ^3_p) and any t ≥ 1,
	
    sup_x ∈ℝ^3( ∫_ℝ^3| h(x,p) |^2 dp )^1/2≤C/t^3/2∑_| I | =1^3 ∑_n = 0^2 ( log t/t‖∂^I Δϕ_∞‖_L^∞)^n
    		∑_| J | =0^3 ‖ L^J h ‖_L^2_x L^2_p
    		.




	First note that, for any smooth compactly supported function h ℝ^3_x ×ℝ^3_p →ℝ,
	
    sup_x ∈ℝ^3|∫_ℝ^3  h(x,p) dp |≤C/t^3∑_| I | =1^3 ∑_n = 0^3 ( log t/t‖∂^I Δϕ_∞‖_L^∞)^n
    		∑_| J | =0^3
    		∫_ℝ^3∫_ℝ^3| L^J h(x,p) | dp dx.

	Indeed, for any x=(x^1,x^2,x^3) ∈ℝ^3,
	
    ∫_ℝ^3  h(x,p) dp
    		   
    		=
    		1/t∫_-∞^x^3 t ∂_x^3( ∫_ℝ^3  h(x^1,x^2,x̃^3,p) dp ) dx̃^3
    		
       
    		=
    		1/t^3∫_-∞^x^1∫_-∞^x^2∫_-∞^x^3
    		(t ∂_x^1) (t ∂_x^2) (t ∂_x^3) (∫_ℝ^3  h(x̃,p) dp )
    		dx̃.

	Now,
	
    t ∂_x^i(∫_ℝ^3  h(x,p) dp )
    		=
    		∫_ℝ^3 L_i h dp
    		+
    		log t/t∫_ℝ^3∂_i Δϕ_∞(p) h(x,p) dp,

	where
	
    ∫_ℝ^3∂_p^i h(x,p) dp = 0,
    		    ∫_ℝ^3∂_p^k( ∂_i ∂_k ϕ_∞(p) h(x,p) ) dp = 0,

	is used.  The estimate (<ref>) follows from repeatedly applying to (<ref>).
	
	Replacing h with h^2 in (<ref>), and using the fact that
	
    ∑_| I | =0^3 | L^I  ( h(x,p)^2 ) |≤
    		C
    		∑_| I | =0^3 | L^I h(x,p) |^2,

	the proof follows.





 §.§ Constants
 
In what follows a constant B>0 will be fixed and the assumption that

    (f_∞) ⊂{ (x,p) ∈ℝ^3 ×ℝ^3 || x | + | p |≤ B},

will be made.  An integer N≥ 6, corresponding to the number of derivatives of the solution which are estimated, will also be assumed fixed.  Recall the approximate solution, (f_[K], ϱ_[K], ϕ_[K]), introduced in (<ref>)–(<ref>).  The notation K ∈ℕ will be used to denote the order of this approximate solution (also denoted k ∈ℕ in Section <ref>).

The notation

    a ≲ b,

will be used when there is a constant C>0, which may depend on B and N, such that

    a ≤ C b.

This constant C implicit in (<ref>) is not allowed to depend on K.  Any constants which depend on K will be denoted C_K.




§ PRECISE VERSIONS OF THE MAIN RESULTS




In this section, more precise versions of Theorem <ref>, Theorem <ref> and Theorem <ref> are stated.

Define first, for any given sequence {𝔫(n) }_n=0^∞,
for any smooth compactly supported function f_∞ℝ^3 ×ℝ^3 → [0,∞), and n ≥ 0,

    ℱ_∞^n
    	=
    	ℱ_∞^n[f_∞,{𝔫(n) }]
    	=
    	∑_| I | + | J |≤ n(
    	‖∂_x^I ∂_p^J f_∞‖_L^2_x L^2_p
    	+
    	‖∂_x^I ∂_p^J f_∞‖_L^2_x L^2_p^𝔫(n)).

The sequences considered will always be increasing and have the property that 𝔫(n)∈ℕ and 𝔫(n) > n for all n.
In order to ease notation, the dependence of ℱ_∞^n on f_∞, and the sequence {𝔫(n) }, is typically suppressed.  In expressions to follow, the sequence {𝔫(n) } may change from line to line.

Recall the function ϕ_∞ℝ^3 →ℝ, defined to be the unique solution of

    Δϕ_∞ (p) = ϱ_∞(p),
    	    ϕ_∞ (p) → 0,   as | p |→∞,
    	    ϱ_∞(p) = - ∫_ℝ^3 f_∞ (y,p) dy.

Note that, assuming that f_∞ satisfies (<ref>) for some B>0, for any n ≥ 2, the Sobolev inequality (see Proposition <ref>) and the gradient estimate (<ref>) imply that

    ∑_| I | =0^n-2‖∂^I∇ϕ_∞‖_L^∞≲∑_| I | =0^n‖∂^I∇ϕ_∞‖_L^2≲∑_| I | =0^n‖∂^Iϱ_∞‖_L^2≲ℱ_∞^n.


Given functions f_k,lℝ^3 ×ℝ^3 →ℝ and ϱ_k,lℝ^3 →ℝ for k = 0,1,2,… and l=0,… k, define ϕ_k,lℝ^3 →ℝ to be the unique solutions of the Poisson equation sourced by ϱ_k,l,

    Δ_ℝ^3ϕ_k,l = ϱ_k,l,
    	    ϕ_k,l(w) → 0,   as | w |→∞,

and define functions, for K ≥ 0,

    f_[K](t,x,p)
    	:=
    	∑_k=0^K ∑_l=0^k(log t)^l/t^k f_k,l(x-tp + log t  ∇ϕ_∞ ( p), p )
    	,


    ϱ_[K](t,x)
    	:= 
    	1/t^3∑_k=0^K ∑_l=0^k(log t)^l/t^kϱ_k,l( x/t)
    	,
    	  ϕ_[K](t,x)
    	:=
    	1/t∑_k=0^K ∑_l=0^k(log t)^l/t^kϕ_k,l( x/t)
    	.




The following theorem is a precise statement of Theorem <ref>.

 
	Consider a smooth compactly supported function f_∞ℝ^3 ×ℝ^3 → [0,∞), satisfying (<ref>) for some B>0, and some N ≥ 6.  There exists k_* = k_*(ℱ_∞^N+1) large and T_0 = T_0(ℱ_∞^N+2k_*+4) ∈ℝ large such that:
	
		
  * Existence of solution:
			There exists a solution (f,ϱ,ϕ) of the Vlasov–Poisson system (<ref>)–(<ref>) on [T_0,∞)×ℝ^3_x ×ℝ^3_p which attains the scattering data f_∞ in the sense that, for all t ≥ T_0,
	
    ∑_| I | + | J |≤ N(
    		∫∫|∂_x^I ∂_p^J
    		(
    		f(t,x+tp - log t  ∇ϕ_∞(p), p )
    		-
    		f_∞(x,p)
    		)
    		|^2
    		dp dx
    		)^1/2≲ℱ_∞^N+4 (log t)^2/t,

	and
	
    sup_x,p∈ℝ^3∑_| I | + | J |≤ N-4|∂_x^I ∂_p^J
    		(
    		f(t,x+tp - log t  ∇ϕ_∞(p), p )
    		-
    		f_∞(x,p)
    		)
    		|≲ℱ_∞^N+4 (log t)^2/t.

	For any t ≥ T_0, f satisfies the support property
	
    (f(t,·,·))
    		⊂{
    		(x,p) ⊂ℝ^3 ×ℝ^3
    		|| x - t p + log t ∇ϕ_∞(p) |≤ 2ℱ_∞^3 + B, | p |≤ 2 B
    		},

	where B>0 is as in (<ref>).  
	



The proof of Theorem <ref> is based on a sequence of approximate solutions of the Vlasov–Poisson system, defined explicitly in terms of the scattering profile f_∞.  The following theorem concerns the existence of these approximate solutions, along with their key properties.


 
	Under the assumptions of Theorem <ref>:
	

		
  * Approximate solutions:
			There are sequences of smooth functions
			
    f_k,lℝ^3 ×ℝ^3 →ℝ,
    				    ϱ_k,lℝ^3 →ℝ,

			for k = 0,1,2,… and l=0,… k — defined explicitly in terms of f_∞ — 
			such that (f_[K], ϱ_[K], ϕ_[K]), defined by (<ref>)–(<ref>), is an approximate solution of order K, in the sense that, for all t ≥ T_0,
			
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J ( _ϕ_[K] f_[K]) (t,·,·) ‖_L^2_xL^2_p   ≤
    				C_K (ℱ_∞^N+2K+3)^2
    				(log t)^1+K/t^2+K,
    			
    ∑_| I |≤ N‖ (t∂_x)^I ( ∫_ℝ^3 f_[K] (t,·,p) dp - ϱ_[K] (t,·) ) ‖_L^2_x   ≤
    				C_Kℱ_∞^N+2K+3(log t)^K+1/t^5/2+K,

			and the scattering data f_∞ is attained in the sense (<ref>).
			
			The functions f_k,l and ϱ_k,l satisfy the support property
			
    (f_k,l)
    				⊂{ (x,p) ∈ℝ^3 ×ℝ^3 || x |≤ B, | p |≤ B },
    				    (ϱ_k,l)
    				⊂{ x ∈ℝ^3 || x |≤ B },

			and the estimates
			
    ∑_| I | + | J |≤ N‖∂_x^I ∂_p^J f_k,l‖_L^2_x L^2_p≤
    				C_kℱ_∞^N+k,
    				    ∑_| I | + | J |≤ N‖∂_x^I ϱ_k,l‖_L^2_x
    				+
    				‖∂_x^I ∇ϕ_k,l‖_L^2_x≤
    				C_kℱ_∞^N+k.

		
	


The functions f_k,l and ϱ_k,l are defined in Section <ref>, where the proof of Theorem <ref> is given.  See in particular Theorem <ref>, which also contains further properties of the approximate solutions.

The next theorem is a precise statement of Theorem <ref>, on the polyhomogeneous expansions satisfied by the solutions of Theorem <ref>.

 
	Under the assumptions of Theorem <ref>:
	
			
		
  * Polyhomogeneous expansion in L^2:
			The approximate solution (f_[K], ϱ_[K], ϕ_[K]) of Theorem <ref> agrees with true solution, of Theorem <ref>, to order K in the sense that, for any K ∈ℕ and t≥ T_0, the difference satisfies the L^2 estimates
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J (f - f_[K]) (t,·,·) ‖_L^2_x L^2_p   ≤
    		C_K
    		(ℱ_∞^N+2K+4)^2
    		 (log t)^1+K/t^1+ K,
    	
    ∑_| I |≤ N‖ (t∂_x)^I ( ϱ - ϱ_[K]) (t,·) ‖_L^2_x   ≤
    		C_Kℱ_∞^N+2K+4(log t)^1+K/t^5/2 + K,
    	
    ∑_| I |≤ N‖ (t∂_x)^I ∇ (ϕ - ϕ_[K]) (t,·) ‖_L^2   ≤
    		C_Kℱ_∞^N+2K+4(log t)^1+K/t^3/2 + K.

			
	
		
  * Polyhomogeneous expansion in L^∞:
			For any K ∈ℕ and t≥ T_0, the difference moreover satisfies the L^∞ estimates
			
    ∑_| I | + | J |≤ N-4sup_x,p∈ℝ^3| L^I (t^-1∂_p)^J (f - f_[K]) (t,x,p) |   ≤
    				C_K
    				(ℱ_∞^N+2K+4)^2
    				 (log t)^1+K/t^1+ K,
    				
    ∑_| I |≤ N-2sup_x∈ℝ^3| (t∂_x)^I ( ϱ - ϱ_[K]) (t,x) |   ≤
    				C_Kℱ_∞^N+2K+4(log t)^1+K/t^4 + K,
    				
    ∑_| I |≤ N-2sup_x∈ℝ^3| (t∂_x)^I ∇ (ϕ - ϕ_[K]) (t,x) |   ≤
    				C_Kℱ_∞^N+2K+4(log t)^1+K/t^3 + K.

	
	


Finally, the next theorem is a more precise statement of Theorem <ref>, on the uniqueness of the solutions of Theorem <ref>.

 

	Under the assumptions of Theorem <ref>:
	
		
  * Uniqueness of solution:
			The solution (f,ϱ,ϕ) of Theorem <ref> is unique in the class of solutions which agree with the approximate solution (f_[K], ϱ_[K], ϕ_[K]) of Theorem <ref> for sufficiently large K.  More precisely, if (f',ϱ',ϕ') is another solution such that there is a constant C satisfying
	
    ‖ (f' - f_[K]) (t,·,·) ‖_L^2_x L^2_p≤
    		C
    		 (log t)^1+K/t^1+ K,

	for some K=K(ℱ_∞^N+1) and for all t sufficiently large, and moreover f' satisfies the support property (<ref>), then (f',ϱ',ϕ') = (f,ϱ,ϕ).
	



The proofs of Theorem <ref> and Theorem <ref> are based on a sequence of “finite problems”, which are discussed in Section <ref>.  The proofs of Theorem <ref> and Theorem <ref>, using the results of Section <ref>, are then given in Section <ref>, where the proof of Theorem <ref> is also given.

The reader interested only in seeing how the approximate solutions are used to establish Theorem <ref>, Theorem <ref> and Theorem <ref>, who is willing to accept the statement of Theorem <ref> (or rather the version with additional statements, in Theorem <ref> below) without seeing the details of the construction, may skip ahead to Section <ref>.







§ APPROXIMATE SOLUTIONS



In this section explicit definitions of f_k,lℝ^3 ×ℝ^3 →ℝ and ϱ_k,lℝ^3 →ℝ are given, for k = 0,1,2,… and l=0,… k, and the corresponding (f_[K],ϱ_[K], ϕ_[K]), given by (<ref>)–(<ref>), are shown to be approximate solutions to the Vlasov–Poisson system (<ref>)–(<ref>).

Recall the definition (<ref>) of ℱ_∞^n[f_∞,{𝔫(n) }] and the convention that the dependence on the sequence {𝔫(n)} is omitted, and moreover that this omitted sequence can change from line to line.  This convention is used throughout this section.

The main result of this section is the following.

 
	Let N≥ 6 be fixed and consider a smooth compactly supported function f_∞ℝ^3 ×ℝ^3 → [0,∞) satisfying (<ref>) for some B>0.  There exists T_0>0 and smooth functions f_k,lℝ^3 ×ℝ^3 →ℝ and ϕ_k,l, ϱ_k,lℝ^3 →ℝ for k = 0,1,2,… and l=0,… k, defined explicitly in terms of f_∞ with
	
    f_0,0 = f_∞,     ϱ_0,0 = ϱ_∞,     ϕ_0,0 = ϕ_∞,

	such that:
	
		
  * The functions f_[K], ϱ_[K] and ϕ_[K], defined by (<ref>)–(<ref>), are an approximate solution of the Vlasov–Poisson system (<ref>)–(<ref>) of order K, in the sense that, for any K ≥ 1 and for all t ≥ T_0,
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J ( _ϕ_[K] f_[K]) (t,·,·) ‖_L^2_xL^2_p≤
    		C_K (ℱ_∞^N+2K+3)^2
    		(log t)^1+K/t^2+K,

	
    ∑_| I |≤ N‖ (t∂_x)^I ( ∫_ℝ^3 f_[K] (t,·,p) dp - ϱ_[K] (t,·) ) ‖_L^2_x≤
    		C_Kℱ_∞^N+2K+3(log t)^K+1/t^5/2+K.


		
		
		
  * The functions f_[K] moreover satisfy the esimtates, for all t≥ T_0,
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J f_[K] (t,·,·) ‖_L^2_xL^2_p   ≤ℱ_∞^N
    		+
    		C_Kℱ_∞^N+Klog t/t,
    		
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J (f_[K] - f_[0]) (t,·,·) ‖_L^2_xL^2_p   ≤C_Kℱ_∞^N+Klog t/t.

		
	
		
  * Finally, each f_k,l and each ϱ_k,l satisfies
	
    (f_k,l)
    		⊂{ (x,p) ∈ℝ^3 ×ℝ^3 || x |≤ B, | p |≤ B },
    		    (ϱ_k,l)
    		⊂{ x ∈ℝ^3 || x |≤ B },

	along with the estimate, for k ≥ 1,
	
    ∑_| I | + | J |≤ N‖∂_x^I ∂_p^J f_k,l‖_L^2_x L^2_p≤
    		C_k (ℱ_∞^N+k)^2,
    		    ∑_| I | + | J |≤ N‖∂_x^I ϱ_k,l‖_L^2_x
    				+
    				‖∂_x^I ∇ϕ_k,l‖_L^2_x≤
    				C_kℱ_∞^N+k.

	
		
	




Throughout this section, it is supposed that a smooth compactly supported function f_∞ℝ^3 ×ℝ^3 → [0,∞) is given, and B>0 is such that (<ref>) holds.  It is also assumed that N≥ 6 is fixed.


In Section <ref> notation is introduced which is used throughout.

In Section <ref>, for the purpose of an inductive argument, the definitions of f_k,l are given for k ≤ K, under the assumption that smooth functions ϕ_k,l are given for all k ≤ K.  The properties (<ref>), (<ref>)–(<ref>) and the former of (<ref>) and (<ref>) are established, assuming that ϕ_k,l satisfy the latter of (<ref>).  These properties hold independently of how ϕ_k,l are defined.

In Section <ref>, it is assumed, for the purpose of the inductive argument, that K≥ 1 is such that functions f_k,l are given for k ≤ K-1 and satisfy the former of (<ref>) and (<ref>).  Functions f_K,l are defined, along with functions ϱ_k,l for all k ≤ K.  These functions are shown to satisfy (<ref>), along with the latter of (<ref>) and (<ref>).  Again, these properties hold independently of how f_k,l, for k ≤ K-1, are defined.

Finally the proof of Theorem <ref> is given in Section <ref> using the results of Section <ref> and Section <ref> together with an induction argument.

The reader is referred to the discussion in Section <ref> for explicit treatments of the K=0 and K=1 cases of Theorem <ref>, and a guide to the K ≥ 2 case.




 §.§ Notation



The following notation will be used throughout this section.
First, y [T_0,∞) ×ℝ^3 ×ℝ^3 →ℝ^3 will denote the function

    y(t,x,p) = x - tp + log t ∇ϕ_∞(p).

Next, for any X∈ℝ^3, define

    ⊗_k X := X ⊗…⊗ X_k  times,

and for X_1, …, X_k ∈ℝ^3, with X_1 = (X_1^1, X_1^2, X_1^3) etc., and any function Φℝ^3 →ℝ^3, define

    X_1 ⊗…⊗ X_k ·∇^k Φ
    	:=
    	∑_i_1,…, i_k=1^3 X_1^i_1… X_k^i_k∂_i_1…∂_i_kΦ∈ℝ^3.

For a given subset A⊂ℕ^d, let ι_A be the indicator function of this set

    ι_A(k)
    	=
    	
    		1,    if k∈ A,
    
    		0,    otherwise.





 §.§ The approximate solution f_[K]




For the purpose of an inductive argument, given later in Section <ref>, suppose that K≥ 1 is such that some smooth functions ϕ_k,l are given, for k=0,…,K, l=0,…,k, with ϕ_0,0=ϕ_∞, which satisfy, for some constants C_n,k, for all n ≥ 6,

    ∑_| I |≤ n‖∂^I ∇ϕ_k,l‖_L^2≤
    	C_n,kℱ_∞^n+k.

Let ϕ_[K] be defined in terms of these functions by (<ref>).
In this section the functions f_k,l are defined in terms of ϕ_k,l and it is shown that, under the assumption that ϕ_k,l satisfy (<ref>), the Vlasov equation is satisfied by f_[K] and ϕ_[K] to order K, in the sense that the estimate (<ref>) holds.



First it is shown that, for p∈ℝ^3, if the function y(t,x,p) is bounded then, in the coefficients of ∇ϕ_[K](t,x) — i.e. in ∇ϕ_k,l(x/t) — the argument can be replaced using (<ref>) and Taylor expanded to give a series whose coefficients are functions of y(t,x,p) and p.  See (<ref>) for the example of the K=1 case.


Define smooth functions Ψ_k,lℝ^3 ×ℝ^3 →ℝ^3, for k = 0,…, K, l= 0, …, k, by

    Ψ_0,0(y,p)
    	=
    	∇ϕ_∞ (p),
    	  Ψ_1,1(y,p)
    	=
    	∇ϕ_1,1(p) - ∇ϕ_∞(p) ·∇^2ϕ_∞ (p),
    	  Ψ_1,0(y,p)
    	=
    	∇ϕ_1,0(p) + y ·∇^2ϕ_∞ (p),

and, for k ≥ 2,

    Ψ_k,l(y,p)
    	=
    	∑_ k_1+k_2=k 
     l_1+l_2 = lk_2l_2(-1)^l_2/k_2!⊗_k_2-l_2 y ⊗_l_2∇ϕ_∞(p) ·∇^k_2+1ϕ_k_1,l_1 (p).

where, in all summations in this section, unless explicitly stated otherwise, 0≤ l_1 ≤ k_1 and 0≤ l_2 ≤ k_2.  It is helpful to also rewrite the expression (<ref>) with the k_1=k term isolated,

    Ψ_k,l(y,p)
    	=
    	∇ϕ_k,l(p)
    	+
    	∑_ k_1+k_2=k 
     l_1+l_2 = l 
     k_1 ≤ k-1k_2l_2(-1)^l_2/k_2!⊗_k_2-l_2 y ⊗_l_2∇ϕ_∞(p) ·∇^k_2+1ϕ_k_1,l_1 (p).



 
	Consider some K≥ 0 and smooth functions ϕ_k,lℝ^3 →ℝ satisfying (<ref>) for all k=0,…,K, l=0,…,k.
	The explicit functions Ψ_k,lℝ^3 →ℝ^3, defined by (<ref>)–(<ref>), for k = 0,1,2,…, l= 0, …, k, satisfy, 
	for any p ∈ℝ^3, any B > 0, and any t ≥ T_0,
	
    sup_| y(t,x,p) |≤ B∑_| I | + | J |≤ N|
    		L^I (t^-1∂_p)^J
    		(
    		∇_xϕ_[K](t,x)
    		-
    		1/t^2∑_k=0^K ∑_l = 0^k (log t)^l/t^kΨ_k,l(y(t,x,p),p)
    		)
    		|
    ≤
    		C_Kℱ_∞^N+2K+3(log t)^K+1/t^K+3
    		.

	Moreover, for all n ≥ 2,
	
    ∑_| I |≤ n‖∂^I Ψ_k,l‖_L^2_x L^2_p≤
    		C_n,kℱ_∞^n+k.






	First note that
	
    ∇ϕ_[K](t,x)
    		=
    		1/t^2[
    		∇ϕ_∞( x/t)
    		+
    		∑_k=1^K ∑_l=0^k(log t)^l/t^k∇ϕ_k,l( x/t)
    		].

	Note also that the assumption (<ref>), along with the Sobolev inequality of Proposition <ref>, implies that, for any n≥ 2,
	
    ∑_| I |≤ n-2‖∂^I ∇ϕ_k,l‖_L^∞≤
    		C_n,kℱ_∞^n+k.

	Now,
	
    x/t =  p + y/t - log t/t∇ϕ_∞(p),

	and so, for any function Φℝ^3 →ℝ^3, by Taylor's Theorem (see Proposition <ref>) and the Binomial Theorem,
	
    sup_| y(t,x,p) |≤ B|Φ( x/t)
    		-
    		[
    		Φ(p)
    		+
    		∑_k=1^K ∑_l=0^k (log t)^l/t^kkl(-1)^l/k!⊗_k-l y ⊗_l ∇ϕ_∞(p) ·∇^k Φ (p)
    		]
    		|
    ≲
    		(B + ‖∇ϕ_∞‖_L^∞)^K+1‖∇^K+1Φ‖_L^∞( log t/t)^K+1.

	Using this expression with Φ = ∇ϕ_∞ and Φ = ∇ϕ_k,l, for k =1,…,K, l=0,…,k, after inserting into (<ref>) it follows that there is a sequence {𝔫(n)} such that
	
    |∇ϕ_[K](t,x)
    		-
    		1/t^2∑_k=0^K ∑_l=0^k(log t)^l/t^k∑_ k_1+k_1=k 
     l_2+l_2 = lk_2l_2(-1)^l_2/k_2!⊗_k_2-l_2 y ⊗_l_2∇ϕ_∞(p) ·∇^k+1ϕ_k_1,l_1 (p)
    		|
    ≲ℱ_∞^2K+3(log t)^K+1/t^K+3,

	and the zeroth order part of the proof of (<ref>) follows.  The estimates for the higher order derivatives follow similarly, using Remark <ref> and the fact that L_i(x^j/t) = δ_i^j for all i,j=1,2,3.
	
	
	Finally, the proof of (<ref>) follows from the assumption (<ref>) and the Sobolev inequality of Proposition <ref>, using the inductive definitions (<ref>)–(<ref>).



We will now give explicit definitions of f_k,l, in terms of Ψ_k,l, to arrange that _ϕ_[K] f_[K] vanishes to sufficiently high order, in the sense that (<ref>) holds (see also (<ref>)–(<ref>) and the surrounding discussion for the k=1 case).






 
	Consider some K≥ 0 and smooth functions ϕ_k,lℝ^3 →ℝ satisfying (<ref>) for all k=0,…,K, l=0,…,k.
	The functions f_k,lℝ^3 ×ℝ^3 →ℝ defined by (<ref>)–(<ref>) below (with the functions Ψ_k,l defined explicitly in terms of ϕ_k,l by (<ref>)–(<ref>)), satisfy,
	
    (f_k,l)
    		⊂{ (x,p) ∈ℝ^3 ×ℝ^3 || x |≤ B, | p |≤ B },

	and, for all k ≥ 1 and any n ≥ 6,
	
    ∑_| I | + | J |≤ n‖∂_x^I ∂_p^J f_k,l‖_L^2_x L^2_p≤
    		C_n,k (ℱ_∞^n+k)^2.

	For any t ≥ T_0, the function f_[K] [T_0,∞) ×ℝ^3_x ×ℝ^3_p →ℝ, defined by (<ref>), satisfies
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J f_[K] (t,·,·) ‖_L^2_xL^2_p   ≤ℱ_∞^N
    		+
    		C_Kℱ_∞^N+Klog t/t,
    		
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J (f_[K] - f_[0]) (t,·,·) ‖_L^2_xL^2_p   ≤C_Kℱ_∞^N+Klog t/t.

	and
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J ( _ϕ_[K] f_[K]) (t,·,·) ‖_L^2_xL^2_p≤
    		C_K (ℱ_∞^N+2K+3)^2
    		(log t)^1+K/t^2+K.




The functions f_k,lℝ^3 ×ℝ^3 →ℝ, for k = 0,…, K, l= 0, …, k, are defined by

    f_0,0(y,p) = f_∞(y,p),

and, inductively, for 1 ≤ k ≤ K,

    f_k,k(y,p)
    	=
    	   
    	-
    	1/k∑_
    	k_1+k_2=k 
     l_1+l_2=k 
     k_1 ≤ k-1
    	
    	(∂_x^i f_k_1,l_1)(y,p) Ψ^i_k_2,l_2(y,p)
    	+
    	1/k∑_
    	k_1+k_2=k-1 
     l_1+l_2=k -1
    	
    	(∂_x^i f_k_1,l_1)(y,p) Ψ^j_k_2,l_2(y,p) ∂_i ∂_j ϕ_∞(p)
    	,

and, for l=k-1,…,0,

    f_k,l(y,p)
    	=
    	l+1/k f_k,l+1(y,p)
    	-
    	1/k∑_
    	k_1+k_2=k 
     l_1+l_2=l 
     k_1 ≤ k-1
    	
    	(∂_x^i f_k_1,l_1)(y,p) Ψ^i_k_2,l_2(y,p)
    	
       
    	+
    	ι_l≥ 1/k∑_
    	k_1+k_2=k-1 
     l_1+l_2=l -1
    	
    	(∂_x^i f_k_1,l_1)(y,p) Ψ^j_k_2,l_2(y,p) ∂_i ∂_j ϕ_∞(p)
    	+
    	1/k∑_
    	k_1+k_2=k-1 
     l_1+l_2=l
    	
    	(∂_p^i f_k_1,l_1)(y,p) Ψ^j_k_2,l_2(y,p) ∂_i ∂_j ϕ_∞(p)
    	,

where ι is defined by (<ref>).
Note that k_1=k is excluded from each of the summations in (<ref>) and (<ref>), and so the definitions are indeed inductive.


The relevant aspect of the form of these expressions — apart from the fact that they achieve exact cancellations in the following proof — is explained in Remark <ref> below.




	The proof of (<ref>) and (<ref>) follow from the definitions (<ref>)–(<ref>) by a straightforward induction argument, using (<ref>) and the Sobolev inequality of Proposition <ref> for (<ref>).

	The estimates (<ref>) and (<ref>) follow from the properties (<ref>)–(<ref>) of the vector fields L_i and t^-1∂_p^i acting on functions of y and p, and the Sobolev inequality of Proposition <ref>, using also (<ref>) and (<ref>).
	
	
	For (<ref>), note that, with y(t,x,p) defined by (<ref>),
	
    _ϕ_[K] (y(t,x,p)^j)
    		=
    		t ( 1/t^2∂_i ϕ_∞(p) - ∂_x^iϕ_[K](t,x) ) + log t  ∂_x^iϕ_[K](t,x) ∂_i ∂_j ϕ_∞ (p),

	and thus,
	
    _ϕ_[K] f_[K]
    		=
    		∑_k=0^K ∑_l=0^k[
    		l(log t)^l-1 - k (log t)^l/t^k+1
    		f_k,l(y,p)
    		+
    		(log t)^l/t^k∂_x^iϕ_[K](t,x) (∂_p^i f_k,l)(y,p)
    		
                   
    		+
    		(log t)^l/t^k[
    		t ( 1/t^2∂_i ϕ_∞(p) - ∂_x^iϕ_[K](t,x) ) + log t  ∂_x^iϕ_[K](t,x) ∂_i ∂_j ϕ_∞ (p)
    		]
    		(∂_x^i f_k,l)(y,p)
    		]
    		
       
    		=
    		∑_k=0^2K+1∑_l=0^k(log t)^l/t^k+1[
    		( - k f_k,l(y,p) + (l+1) f_k,l+1(y,p) ) ι_k ≤ K
       		-
    		∑_
    		k_1+k_2=k 
     l_1+l_2=l 
     k_2 ≥ 1
    		ι_k_1,k_2 ≤ K
    		(∂_x^i f_k_1,l_1)(y,p) Ψ^i_k_2,l_2(y,p)
    		+
    		ι_l ≥ 1∑_
    		k_1+k_2=k-1 
     l_1+l_2=l -1
    		ι_k_1,k_2 ≤ K
    		(∂_x^i f_k_1,l_1)(y,p) Ψ^j_k_2,l_2(y,p) ∂_i ∂_j ϕ_∞(p)
    		
           
    		+
    		ι_l ≠ k∑_
    		k_1+k_2=k-1 
     l_1+l_2=l
    		ι_k_1,k_2 ≤ K
    		(∂_p^i f_k_1,l_1)(y,p) Ψ^j_k_2,l_2(y,p) ∂_i ∂_j ϕ_∞(p)
    		]
    		
           
    		+
    		(∂_x^iϕ_[K](t,x)
    		-
    		1/t^2∑_k=0^K ∑_l = 0^k (log t)^l/t^kΨ_k,l^i(y,p)
    		)
    		∂_p^i f_[K](t,x,p).

	After inserting the definitions (<ref>)–(<ref>), the terms in the first summation corresponding to k=0,…,K all vanish.
	Recall now (<ref>) and (<ref>).  It follows from the properties (<ref>)–(<ref>) of the vector fields L_i and t^-1∂_p^i acting on functions of y and p, and the Sobolev inequality of Proposition <ref>, that, for any k≥ K+1 and l ≤ k and k_1,k_2 ≤ K, l_1,l_2 such that k_1+k_2 = k, l_1+l_2=l, and any N ≥ 6, 
	
    ∑_| I | + | J |≤ N‖
    		L^I (t^-1∂_p)^J ( (log t)^l/t^k+1∂_x^i f_k_1,l_1(y,p) Ψ^j_k_2,l_2(y,p) )
    		‖_L^2_x L^2_p≤
    		C_K
    		(ℱ_∞^N+k+1)^2
    		(log t)^K+1/t^K+2.

	Similarly for all but the final term on the right hand side of (<ref>).  The final term is estimated using Proposition <ref> and (<ref>),
	
    ∑_| I | + | J |≤ N‖
    		L^I (t^-1∂_p)^J
    		(
    		(∂_x^iϕ_[K](t,x)
    		-
    		1/t^2∑_k=0^K ∑_l = 0^k (log t)^l/t^kΨ_k,l^i(y,p)
    		)
    		∂_p^i f_[K](t,x,p)
    		)
    		‖_L^2_x L^2_p
    ≤
    		C_K (ℱ_∞^N+2K+3)^2
    		(log t)^1+K/t^2+K.

	The proof of (<ref>) then follows from Proposition <ref>.




 
	For fixed 1 ≤ k≤ K and 0≤ l ≤ k, the expressions (<ref>)–(<ref>) for f_k,l involve ϕ_k',l' for all 0 ≤ k'≤ k and 0≤ l' ≤ k'.  Note however that, upon integration in y, the terms involving ϕ_k,l' vanish, for all l' = 0,…,k — regardless of how ϕ_k,l are defined — and only the terms involving ϕ_k',l' with k'<k and 0≤ l'≤ k' remain.  See (<ref>), and the surrounding discussion, for this observation in the special case k=1.
	
	Indeed, considering the expression (<ref>), it follows that the integral of (<ref>) takes the form
	
    ∫_ℝ^3 f_k,k(y,p) dy
    		=
    		1/k∑_
    		k_1+k_2=k-1 
     l_1+l_2=k -1
    		∫_ℝ^3 (∂_x^i f_k_1,l_1)(y,p) Ψ^j_k_2,l_2(y,p) ∂_i ∂_j ϕ_∞(p) dy
    		
               
    		-
    		1/k∑_
    		k_1+k_2=k 
     l_1+l_2=k 
     k_2 ≥ 1
    		∑_ k_3+k_4=k_2 
     l_3+l_4 = l_2 
     k_4 ≥ 1k_4l_4(-1)^l_4/k_4!∫_ℝ^3⊗_k_4-l_4 y ⊗_l_4∇ϕ_∞(p) ·∇^k_4+1ϕ_k_3,l_3 (p)
    		(∂_x^i f_k_1,l_1)(y,p) 
    		dy
    		,

	and the integral of (<ref>) takes the form, for l=k-1,…,0,
	
    ∫_ℝ^3 f_k,l(y,p) dy
    		=
    		l+1/k∫_ℝ^3 f_k,l+1(y,p) dy
    		
               
    		+
    		ι_l≥ 1/k∑_
    		k_1+k_2=k-1 
     l_1+l_2=l -1
    		∫_ℝ^3
    		(∂_x^i f_k_1,l_1)(y,p) Ψ^j_k_2,l_2(y,p) ∂_i ∂_j ϕ_∞(p)
    		dy
    		
               
    		+
    		1/k∑_
    		k_1+k_2=k-1 
     l_1+l_2=l
    		∫_ℝ^3
    		(∂_p^i f_k_1,l_1)(y,p) Ψ^j_k_2,l_2(y,p) ∂_i ∂_j ϕ_∞(p)
    		dy
    		
               
    		-
    		1/k∑_
    		k_1+k_2=k 
     l_1+l_2=l 
     k_2 ≥ 1
    		∑_ k_3+k_4=k_2 
     l_3+l_4 = l_2 
     k_4 ≥ 1k_4l_4(-1)^l_4/k_4!∫_ℝ^3⊗_k_4-l_4 y ⊗_l_4∇ϕ_∞(p) ·∇^k_4+1ϕ_k_3,l_3 (p)
    		(∂_x^i f_k_1,l_1)(y,p)
    		dy
    		.

	Note again that terms of the form ϕ_k,l' are precluded from the final summations of (<ref>) and (<ref>) by the condition that k_4 ≥ 1.
	
	This fact that the top order ϕ_k,l terms vanish upon integration in y is used in the proof of Theorem <ref> in Section <ref> below.  See also Section <ref> below.






 §.§ The approximate solution ϱ_[K]






Throughout this section suppose, for the purpose of the inductive argument given in Section <ref>, that K≥ 1 is such that some smooth functions f_k,l are given, for k=0,…,K-1, l=0,…,k, with f_0,0=f_∞, which satisfy the support property (<ref>) and, for some constants C_n,k, for all n ≥ 6,

    ∑_| I | + | J |≤ n‖∂_x^I ∂_p^J f_k,l‖_L^2_x L^2_p≤
    	C_n,kℱ_∞^n+k.

In this section there are two main aims:

	
  * The first aim is to give explicit definitions of ϱ_k,l in terms of f_k,l, for 0≤ k ≤ K-1 and 0≤ l ≤ k, so that ϱ_[K-1] defined by (<ref>) is approximately equal to the spatial density of the function f_[K-1] defined by (<ref>), i.e. so that
		
    ∑_k=0^K-1∑_l=0^k (log t)^l/t^k( ∫ f_k,l(y(t,x,p),p) dp - 1/t^3ϱ_k,l( x/t) )
    			=
    			𝒪( (log t)^K-1/t^4+(K-1)),

		or, more precisely, so that the estimate (<ref>) holds for K-1.  This property is equivalent to the requirement that the relations (<ref>) below hold, and it is indeed the relations (<ref>) which are used to define the functions ϱ_k,l.
	
  * The second aim is to give explicit definitions of functions f_K,l and ϱ_K,l, for l=0,…,K, in terms of the functions f_k,l, for 0≤ k ≤ K-1, l=0,…,k.	
		Using the observation of Remark <ref>, these functions can be defined in such a way that the relation (<ref>) also holds for k=K, and thus that the property (<ref>) moreover holds for K in place of K-1.  See Remark <ref> below.

Both of these aims are achieved together in Proposition <ref> below.




  §.§.§ Expansions for p(t,x,y) and (∂ y/∂ p)



Before giving the definitions of ϱ_k,l, certain properties of the expression (<ref>) for y(t,x,p) are established.  A computation gives
	
    ∂ y/∂ p
    		=
    		-
    		t^3
    		+
    		t^2 log t J_1 (p)
    		+
    		t (log t)^2 J_2 (p)
    		+
    		(log t)^3 J_3 (p)
    		,

	where
	
    J_1
    		=
    		Δϕ_∞,
    	    J_2
    		=
    		(∂_1 ∂_2 ϕ_∞)^2
    		+
    		(∂_1 ∂_3 ϕ_∞)^2
    		+
    		(∂_2 ∂_3 ϕ_∞)^2
    		-
    		∂_1^2 ϕ_∞∂_2^2 ϕ_∞
    		-
    		∂_1^2 ϕ_∞∂_3^2 ϕ_∞
    		-
    		∂_2^2 ϕ_∞∂_3^2 ϕ_∞,

	and
	
    J_3
    		=
    		∂_1^2 ϕ_∞∂_2^2 ϕ_∞∂_3^2 ϕ_∞
    		+
    		2 ∂_1 ∂_2 ϕ_∞∂_1 ∂_3 ϕ_∞∂_2 ∂_3 ϕ_∞
    		-
    		∂_1^2 ϕ_∞ (∂_2 ∂_3 ϕ_∞)^2
    		-
    		∂_2^2 ϕ_∞ (∂_1 ∂_3 ϕ_∞)^2
    		-
    		∂_3^2 ϕ_∞ (∂_1 ∂_2 ϕ_∞)^2
    		.

	It therefore follows from the Implicit Function Theorem that, for bounded p, if T_0 is sufficiently large, the expression (<ref>) can be inverted to give p(t,x,y) as a function of t, x and y.  
	
	
In order to define the expansion for ϱ it is helpful to first provide an expansion for this function.



 
	There are explicit functions p_k,lℝ^3 ×ℝ^3 →ℝ^3, for k = 1,2,…, l= 0, …, k, such that, for any K ≥ 1, the function p(t,x,y) satisfies, for any B ≥ 1 and for all t ≥ T_0,
	
    sup_| x/t | + | y |≤ B|
    		p(t,x,y)
    		-
    		[
    		x/t
    		+
    		∑_k=1^K ∑_l=0^k
    		(log t)^l/t^k p_k,l( x/t,y )
    		]
    		|≲( log t/t)^K+1
    		.






	The expression (<ref>) can be rewritten
	
    p = x/t + y/t + log t/t∇ϕ_∞(p).

	Rather than give an explicit expression for the (k,l)-th element of the expression, the general procedure for obtaining such an expression is described.  Expression (<ref>) moreover gives
	
    p
    		=
    		x/t + y/t
    		+
    		log t/t∇ϕ_∞( x/t + y/t + log t/t∇ϕ_∞(p) ),

	and, for any K ≥ 1, by Taylor's Theorem,
	
    sup_| x/t | + | y |≤ B|
    		p(t,x,y)
    		-
    		[
    		x/t + y/t
    		+
    		log t/t[
    		ϕ_∞( x/t)
    		+
    		∑_k=1^K
    		⊗_k ( y/t + log t/t∇ϕ_∞(p) )
    		·∇^k+1ϕ_∞( x/t)
    		]
    		]
    		|≲( log t/t)^K+1.

	The explicit expressions are obtained by iteratively inserting the expression (<ref>) for p.
	
	The first elements of the sequence {p_k,l} take the form
	
    p_1,1(w,y) = ∇ϕ_∞(w),
    		    
    		p_1,0(w,y) = y,

	
    p_2,2(w,y) = ∇ϕ_∞(w) ·∇^2 ϕ_∞(w),
    		    
    		p_2,1(w,y) = y ·∇^2 ϕ_∞(w),
    		    
    		p_2,0(w,y) = 0.



One similarly has an expansion for the Jacobian of the change p ↦ y(t,x,p).  See (<ref>) for a discussion of the first terms in this expansion.

 
	There are explicit functions J_k,lℝ^3 ×ℝ^3 →ℝ, for k = 0,1,2,…, l= 0, …, k, such that, for any K ≥ 1 and any B ≥ 1, the determinant of ∂ p/∂ y satisfies, for all t ≥ T_0,
	
    sup_| x/t | + | y |≤ B|∂ p/∂ y(t,x,y)
    		-
    		1/t^3∑_k=1^K ∑_l=0^k
    		(log t)^l/t^k J_k,l( x/t,y )
    		|≲(log t)^K+1/t^K+4
    		.




	Recall (<ref>)–(<ref>).  It follows that, for any K ≥ 1,
	
    sup_| x/t | + | y |≤ B|∂ p/∂ y(t,x,y)
    		-
    		[
    		-
    		1/t^3
    		-
    		1/t^3∑_k=1^K
    		(
    		log t/tJ_1 (p)
    		+
    		(log t)^2/t^2J_2 (p)
    		+
    		(log t)^3/t^3J_3 (p)
    		)^k
    		]
    		|≲(log t)^K+1/t^K+4
    		.

	The proof then follows from using the expansions for p(t,x,y) from Proposition <ref>.  The first elements of the sequence {J_k,l} take the explicit form
	
    J_0,0(w,y) = - 1,
    		    
    		J_1,1(w,y) = - J_1(w),
    		    
    		J_1,0(w,y) = 0,

	
    J_2,2(w,y) = - J_2 (w) - (J_1 (w))^2 - ∇ϕ_∞(w) ·∇J_1 (w),
    		    
    		J_2,1(w,y) = - y ·∇J_1 (w),
    		    
    		J_2,0(w,y) = 0.








  §.§.§ The definitions of ϱ_k,l for k ≤ K-1


The definitions of ϱ_k,l, for k ≤ K-1, are now given.
For smooth functions h ℝ^3 ×ℝ^3 →ℝ, define,

    H_0,0[h](w,y) := h(y,w)

and, for k ≥ 1 and 0 ≤ l ≤ k,

    H_k,l[h](w,y)
    	=
    	∑_m=1^k
    	1/m!∑_k_1+…+k_m=k 
     l_1+…+l_m=l 
     1 ≤ k_i ≤ k,  0 ≤ l_i ≤ k_i
    	p_k_1,l_1 (w,y) ⊗…⊗ p_k_m,l_m (w,y)
    	·
    	(∇_p^m h) (y,w),

where ∇_p h denotes the gradient of h with respect to the second argument, and the functions p_k,l are as in Proposition <ref>.

Define now,

    ϱ_0,0(w)
    	   
    	=
    	- ∫ f_∞(y,w)dy,

and, for 1 ≤ k ≤ K-1, l=0,…,k,

    ϱ_k,l(w)
    	=
    	∑_k_1+k_2+k_3=k 
     l_1+l_2+l_3=l 
     0 ≤ k_i ≤ k,  0 ≤ l_i ≤ k_i∫_ℝ^3
    	H_k_2,l_2[f_k_1,l_1] (w,y) J_k_3,l_3(w,y)
    	dy.

Property (<ref>) is the relation required to achieve cancellations of the form (<ref>).  See the proof of Proposition <ref> below, where these cancellations are exploited to show estimate (<ref>).




  §.§.§ The definitions of f_K,l and ϱ_K,l



Functions f_K,l and ϱ_K,l, for 0≤ l ≤ K, are now defined — explicitly in terms of the functions f_k,l, for 0≤ k ≤ K-1, 0≤ l ≤ k — so that the property (<ref>) moreover holds for k=K.  This fact ensures that cancellations of the form (<ref>) are in fact achieved with K in place of K-1.

First define smooth functions ϕ_k,lℝ^3 →ℝ, for k=0,…, K-1, l=0,…,k as the unique solution of the Poisson equation (<ref>) sourced by ϱ_k,l as defined above.  The functions ϕ_k,l are expressed explicitly in terms of the functions ϱ_k,l via the well known representation formula for the Poisson equation

    ϕ_k,l(w) = 1/4π∫_ℝ^3ϱ_k,l(y)/| y - w | dy.




Define a smooth function 𝒫_K,Kℝ^3 →ℝ by setting 𝒫_K,K(p) to be equal to the right hand side of (<ref>) with k=K.  Next define 𝒫_K,lℝ^3 →ℝ inductively, for l=K-1,K-2,…,0, by setting 𝒫_K,l(p) to be equal to the right hand side of (<ref>) with k=K.  (Recall the discussion in Remark <ref> — these definitions of 𝒫_K,l indeed only involve f_K,l and ϕ_K,l up to order K-1, and so 𝒫_K,l are well defined.)

Define now, for l=0,…,K,

    ϱ_K,l(w)
    	=
    	-
    	𝒫_K,l(w)
    	+
    	∑_k_1+k_2+k_3=K 
     l_1+l_2+l_3=l 
     0 ≤ k_i ≤ K,  0 ≤ l_i ≤ k_i, 
     k_1≤ K-1,  l_1 ≤ l-1∫_ℝ^3
    	H_k_2,l_2[f_k_1,l_1] (w,y) J_k_3,l_3(w,y)
    	dy.


Finally, define a smooth function f_K,Kℝ^3 ×ℝ^3 →ℝ by equation (<ref>), and define f_K,lℝ^3 ×ℝ^3 →ℝ inductively, for l=K-1,K-2,…,0, by equation (<ref>), with ϕ_K,l defined by (<ref>) sourced by ϱ_K,l.



 
	Note that, with these definitions, f_K,l satisfy

    ∫_ℝ^3 f_K,l(y,w) dy = 𝒫_K,l(w),

and thus (<ref>) also holds for k=K.  This important property is used in the proof of Proposition <ref> below.





  §.§.§ The main properties of ϱ_k,l



The main result of this section is the following proposition, which concerns properties of the functions ϱ_k,l as defined above.

 
	Consider some K≥ 0 and smooth functions f_k,lℝ^3 ×ℝ^3 →ℝ satisfying (<ref>) for all k=0,…,K-1, l=0,…,k.
	
	The functions ϱ_k,lℝ^3 →ℝ defined by (<ref>)–(<ref>) and (<ref>) satisfy, for all k=0,…,K, l=0,…,k and any n ≥ 6,
	
    ∑_| I |≤ n‖∂_x^I ϱ_k,l‖_L^2_x≤
    		C_n,kℱ_∞^n+k,
    		    (ϱ_k,l)
    		⊂{ x∈ℝ^3 || x |≤ B }.

	The functions ϕ_k,lℝ^3 →ℝ, as defined above, satisfy, for all k=0,…,K, l=0,…,k and any n ≥ 6,
	
    ∑_| I |≤ n‖∂^I ∇ϕ_k,l‖_L^2≤
    		C_n,kℱ_∞^n+k.

	For any t ≥ T_0, the function ϱ_[K] [T_0,∞) ×ℝ^3 →ℝ, defined by (<ref>), satisfies
	
    ∑_| I |≤ N‖ (t∂_x)^I ( ∫_ℝ^3 f_[K] (t,·,p) dp - ϱ_[K] (t,·) ) ‖_L^2_x≤
    		C_Kℱ_∞^N+2K+3(log t)^K+1/t^5/2+K,

	where f_[K] is defined by (<ref>), with f_K,l as defined above.



	Consider first the properties (<ref>).  For k =0,…,K-1, the properties (<ref>) follow from the fact that each f_k,l, for k=0,…,K-1, l=0,…,k, satisfy the support property (<ref>) the estimates (<ref>).
	
	The property (<ref>), for k =0,…,K-1, then follows from (<ref>) and the gradient estimate of Proposition <ref> applied to the Poisson equation (<ref>).
	
	For k = K, the fact that (<ref>), the support property (<ref>) and the estimates (<ref>) hold for k =0,…,K-1 imply that 𝒫_K,l satisfy, for all n ≥ 6,
	
    (𝒫_K,l) ⊂{ x ∈ℝ^3 || x |≤ B },
    		    ∑_| I |≤ n‖∂_x^I 𝒫_K,l‖_L^2_x≤
    		C_n,K (ℱ_∞^n+K)^2.

	It also follows, from Proposition <ref>, that f_K,l satisfies the support property (<ref>) and the estimate (<ref>).  These facts imply that the properties (<ref>) also hold for k=K.  The property (<ref>) for k=K then follows again from the gradient estimate of Proposition <ref>.
	
	Consider now the estimate (<ref>).  First note that, for any smooth h ℝ^3 ×ℝ^3 →ℝ, satisfying
	
    (h) ⊂{ (x,p) ∈ℝ^3 ×ℝ^3 || x |≤ B, | p |≤ B },

	by Taylor's Theorem (see Proposition <ref>), using the notation of Section <ref>,
	
    h(y,p)
    		=
    		   
    		h(y,x/t)
    		+
    		∑_n=1^K
    		1/n!⊗_n (p - x/t) · (∇^n_p h)(y,x/t)
    		+
    		1/K!⊗_K+1(p - x/t) ·∫_0^1 (1-s)^K (∇^K+1_p h)(y,sp) ds.
	
	Thus Proposition <ref> implies that
	
    h(y,p)
    		=
    		h ( y, x/t)
    		
             
    		+
    		∑_n=1^K 1/n!∑_k_1=1^K ∑_l_1=0^k_1…∑_k_n=1^K ∑_l_n=0^k_n(log t)^l_1+… + l_n/t^k_1+ … + k_n
    		p_k_1,l_1( x/t,y ) ⊗…⊗ p_k_n,l_n( x/t,y )
    		·∇_p^n h (y,x/t)
    		+
    		E_h^K+1
         
    		=
    		h(y, x/t)
    		+
    		∑_k=1^K ∑_l=0^k
    		(log t)^l/t^k∑_m=1^k
    		1/m!∑_k_1+…+k_m=k 
     l_1+…+l_m=l
    		p_k_1,l_1(x/t,y) ⊗…⊗ p_k_m,l_m(x/t,y)
    		·∇_p^m h (y,x/t)
    		+
    		E_h^K+1
         
    		=
    		h(y, x/t)
    		+
    		∑_k=1^K ∑_l=0^k
    		(log t)^l/t^k
    		H_k,l[h](x/t,y)
    		+
    		E_h^K+1,

	where H_k,l[h] is defined by (<ref>), and E_h^K+1(y(t,x,p),p) satisfies
	
    ∫_ℝ^3| E_h^K+1(y(t,x,p),p) | dp
    		≲(log t)^K+1/t^K+1∑_| I |≤ K+1∫|∂_p^I h (y,p(t,x,y)) | |∂ p/∂ y| dy
    		
    ≲(log t)^K+1/t^K+4∑_| I |≤ K+1sup_p ∈ℝ^3( ∫|∂_p^I h (y,p) |^2 dy )^1/2≲(log t)^K+1/t^K+4∑_| I |≤ K+3‖∂_p^I h ‖_L^2_x L^2_p,

	where the support property (<ref>) is used, along with the Sobolev inequality of Proposition <ref>.  Moreover,
	
    (E_h^K+1(y(t,·,p),p)) ⊂{ x∈ℝ^3 || x |≤ (2B+ℱ_∞^3) t },

	which follows from the fact that | x |≤ B + t B + log t ℱ_∞^3 if | y |≤ B and | p |≤ B.
	Thus
	
    ∫ h(y(t,x,p),p) dp
    		   
    		=
    		∫ h(y,p(t,x,y)) ∂ p/∂ y(t,x,y) dy
    		
       
    		=
    		1/t^3∑_k_1=0^K ∑_l_1=0^k_1∑_k_2=0^K ∑_l_2=0^k_2(log t)^l_1+l_2/t^k_1 + k_2∫_ℝ^3
    		H_k_1,l_1[h](x/t,y)
    		J_k_2,l_2(x/t,y)
    		dy
    		+
    		∫ E_h^K+1 dp
    		
       
    		=
    		1/t^3∑_k=0^K ∑_l=0^k(log t)^l/t^k∑_m_1 + m_2=k 
     n_1+n_2=l∫_ℝ^3
    		H_m_1,n_1[h](x/t,y)
    		J_m_2,n_2(x/t,y)
    		dy
    		+
    		∫ E_h^K+1 dp,

	where E_h^K+1 varies from line to line, but always satisfies (<ref>)–(<ref>).
	Applying with h=f_k,l, the definitions (<ref>)–(<ref>) of ϱ_k,l for 0 ≤ k ≤ K-1 and the fact that ϱ_K,l also satisfies (<ref>) (see the definition (<ref>) and the important property (<ref>), discussed in Remark <ref>) imply that
	
    ∑_k=0^K ∑_l=0^k(log t)^l/t^k∫ f_k,l(y(t,x,p),p) dp
    		   
    		=
    		1/t^3∑_k=0^K ∑_l=0^k(log t)^l/t^kϱ_k,l( x/t)
    		+
    		∑_k=0^K ∑_l=0^k
    		E_f_k,l^K+1(y(t,x,p),p).

	The properties (<ref>) and (<ref>) imply that, for each k,l,
	
    ‖∫_ℝ^3
    		E_f_k,l^K+1(y(t,·,p),p)
    		dp
    		‖_L^2≲(log t)^K+1/t^K+5/2∑_| I |≤ K+3‖∂_p^I f_k,l‖_L^2_x L^2_p≲(log t)^K+1/t^K+5/2ℱ_∞^k+K+3,

	where the final inequality follows from (<ref>).  The proof of the zeroth order part of (<ref>) then follows.
	
	For the part of (<ref>) involving higher order derivatives, it suffices to show that, for each | I |≤ N, there exists ϱ_[K]^I such that
	
    ‖ (t∂_x)^I ( ∫_ℝ^3 f_[K] (t,·,p) dp)
    		-
    		∑_k=0^K ∑_l=0^k (log t)^l/t^kϱ_k,l^I (·/t) ‖_L^2_x≤
    		C_Kℱ_∞^N+2K+3(log t)^K+1/t^5/2+K.

	It then follows that ϱ_k,l^I = ∂^Iϱ_k,l, and thus that (<ref>) holds.
	
	To check that there are functions satisfying (<ref>), consider first the case | I | =1, so that there is i∈{1,2,3} such that ∂_x^I = ∂_x^i. Note first that, for any smooth h ℝ^3 ×ℝ^3 →ℝ,
	
    t ∂_x^i( ∫_ℝ^3 h(y(t,x,p),p) dp )
    		=
    		∫_ℝ^3 (t ∂_x^ih)(y(t,x,p),p) dp.

	Moreover,
	
    0
    		   
    		=
    		∫∂_p^i(
    		h(y(t,x,p),p)
    		)
    		dp
    		
       
    		=
    		- t
    		∫
    		(∂_x^ih)(y,p)
    		dp
    		+
    		log t 
    		∫∂_i ∂_k ϕ_∞(p)
    		(∂_x^kh)(y,p)
    		dp
    		+
    		∫
    		(∂_p^ih)(y,p)
    		dp,

	and so, adding to (<ref>) gives,
	
    t ∂_x^i( ∫_ℝ^3 h(y(t,x,p),p) dp )
    		=
    		log t 
    		∫∂_i ∂_k ϕ_∞(p)
    		(∂_x^kh)(y,p)
    		dp
    		+
    		∫
    		(∂_p^ih)(y(t,x,p),p)
    		dp.

	One can continue:
	
    0
    		   
    		=
    		log t/t∫∂_p^k(
    		∂_i ∂_k ϕ_∞(p)
    		h(y(t,x,p),p)
    		)
    		dp
    		
       
    		=
    		- log t
    		∫∂_i ∂_k ϕ_∞(p)
    		(∂_x^kh)(y,p)
    		dp
    		+
    		(log t)^2/t∫∂_i ∂_k ϕ_∞(p) ∂_k ∂_k_2ϕ_∞(p) (∂_x^k_2h)(y,p)
    		dp
    		
         
    		+
    		log t/t∫∂_i Δϕ_∞(p)
    		· h(y,p)
    		+
    		∂_i ∂_k ϕ_∞(p) (∂_p^kh)(y,p)
    		dp,

	and more generally, for any l ≥ 0,
	
    0
    		   
    		=
    		(log t)^l+1/t^l+1∫∂_p^k_l+1(
    		∂_i ∂_k_1ϕ_∞(p) ∏_j=1^l∂_k_j∂_k_j+1ϕ_∞(p)  h(y(t,x,p),p)
    		)
    		dp
    		
       
    		=
    		- (log t)^l+1/t^l∫∂_i ∂_k_1ϕ_∞(p) ∏_j=1^l∂_k_j∂_k_j+1ϕ_∞(p)
    		(∂_x^k_l+1h)(y,p)
    		dp
    		
         
    		+
    		(log t)^l+2/t^l+1∫∂_i ∂_k_1ϕ_∞(p) ∏_j=1^l+1∂_k_j∂_k_j+1ϕ_∞(p) 
    		(∂_x^k_l+2h)(y,p)
    		dp
    		
         
    		+
    		(log t)^l/t^l∫[
    		∂_p^k_l+1( ∂_i ∂_k_1ϕ_∞(p) ∏_j=1^l∂_k_j∂_k_j+1ϕ_∞(p) )
    		· h(y,p)
    		
             
    		+
    		∂_i ∂_k_1ϕ_∞(p) ∏_j=1^l∂_k_j∂_k_j+1ϕ_∞(p) (∂_p^k_l+1 h)(y,p)
    		]
    		dp,

	where Π_j=1^0 ∂_k_j∂_k_j+1ϕ_∞(p) : = 1.
	Summing (<ref>), (<ref>), along with (<ref>) for l=0,…, K+1, gives
	
    t ∂_x^i( ∫_ℝ^3 h(y(t,x,p),p) dp )
    		=
    		∑_l=0^K+1(log t)^l/t^l∫[
    		∂_p^k_l+1( ∂_i ∂_k_1ϕ_∞(p) ∏_j=1^l∂_k_j∂_k_j+1ϕ_∞(p) )
    		· h(y,p)
    		
             
    		+
    		∂_i ∂_k_1ϕ_∞(p) ∏_j=1^l∂_k_j∂_k_j+1ϕ_∞(p) (∂_p^k_l+1 h)(y,p)
    		]
    		dp
    		+
    		∫
    		(∂_p^ih)(y,p)
    		dp
    		
             
    		+
    		(log t)^K+3/t^K+2∫∂_i ∂_k_1ϕ_∞(p) ∏_j=1^K+2∂_k_j∂_k_j+1ϕ_∞(p) 
    		(∂_x^k_K+3h)(y,p)
    		dp
    		.

	Thus, if one defines
	
    G_0[h](y,p)
    		=
    		∂_p^k( ∂_i ∂_kϕ_∞(p) ) · h(y,p)
    		+
    		∂_i ∂_kϕ_∞(p) (∂_p^k h)(y,p)
    		+
    		(∂_p^ih)(y,p),

	and, for l ≥ 1,
	
    G_l[h](y,p)
    		=
    		∂_p^k_l+1( ∂_i ∂_k_1ϕ_∞(p) ∏_j=1^l∂_k_j∂_k_j+1ϕ_∞(p) )
    		· h(y,p)
    		+
    		∂_i ∂_k_1ϕ_∞(p) ∏_j=1^l∂_k_j∂_k_j+1ϕ_∞(p) (∂_p^k_l+1 h)(y,p),

	it follows that
	
    |
    		t ∂_x^i( ∫_ℝ^3 h(y(t,x,p),p) dp )
    		-
    		∑_l=0^K(log t)^l/t^l∫ G_l[h](y(t,x,p),p) dp
    		|
    ≲(log t)^K+1/t^K+1|∫ G_l[h](y(t,x,p),p) dp |
    		+
    		(log t)^K+3/t^K+2ℱ^4_∞ t^-3/2‖∇_p h ‖_L^2_x L^∞_p≲(log t)^K+1/t^K+5/2ℱ^4_∞‖∇_p h ‖_L^2_x L^∞_p,

	where the latter holds for t ≥ T_0 if T_0 is sufficiently large.  Hence
	
    t ∂_x^i( ∫_ℝ^3 f_[K](y(t,x,p),p) dp )
    		=
    		∑_k=0^K ∑_l=0^k (log t)^l/t^k t ∂_x^i( ∫_ℝ^3 f_k,l(y(t,x,p),p) dp ),

	satisfies
	
    |
    		t ∂_x^i( ∫_ℝ^3 f_[K](y(t,x,p),p) dp )
    		-
    		∑_k=0^K ∑_l=0^k (log t)^l/t^k∫f̃_k,l(y(t,x,p),p) dp
    		|≲(log t)^K+1/t^K+5/2
    		(ℱ^4_∞+ℱ^k+3_∞),

	where
	
    f̃_k,l(y,p)
    		:=
    		∑_m+k'=k 
     m≤ l∫ G_m[f_k',l-m](y,p).

	Revisiting the zeroth order case, with f̃_k,l in place of f_k,l, it follows that (<ref>) holds where ϱ_k,l^I is defined by replacing f_k,l with f̃_k,l in (<ref>).
	
	The cases involving higher order derivatives follows similarly.












 §.§ Approximate solutions: the proof of Theorem <ref>



The proof of Theorem <ref> can now be given.



	The first step in the proof of Theorem <ref> is to give explicit definitions of the functions
	
    f_k,lℝ^3 ×ℝ^3 →ℝ,
    		     and     ϕ_k,l, ϱ_k,lℝ^3 →ℝ,

	for k = 0,1,2,… and l=0,… k.  Define first
	
    f_0,0 = f_∞,     ϱ_0,0 = ϱ_∞,     ϕ_0,0 = ϕ_∞.

	Next suppose, inductively, that K≥ 1 is such that (<ref>) have been defined — and satisfy (<ref>) and (<ref>) — for all 0 ≤ k ≤ K-1 and 0 ≤ l ≤ k.  Define smooth functions
	
    𝒫_K,lℝ^3 →ℝ,
    		    
    		0 ≤ l ≤ K,

	by first setting 𝒫_K,K(p) to be equal to the right hand side of (<ref>) with k=K.  Then define 𝒫_K,lℝ^3 →ℝ inductively, for l=K-1,K-2,…,0, by setting 𝒫_K,l(p) to be equal to the right hand side of (<ref>) with k=K.  (Recall the discussion in Remark <ref> — these definitions of 𝒫_K,l indeed only involve (<ref>) up to order K-1 and so 𝒫_K,l are well defined.)

Define now, for l=0,…,K, functions ϱ_K,l, for 0≤ l ≤ K, by equation (<ref>).  The functions ϕ_K,l are then defined as the unique solutions of the Poisson equation (<ref>) sourced by ϱ_K,l.  (Recall that ϕ_K,l are expressed explicitly in terms of ϱ_K,l by the well known representation formula (<ref>) for the Poisson equation.)  Finally, define f_K,Kℝ^3 ×ℝ^3 →ℝ by equation (<ref>), and define f_K,lℝ^3 ×ℝ^3 →ℝ inductively, for l=K-1,K-2,…,0, by equation (<ref>) (where Ψ_k,l are defined in terms of ϕ_k,l by (<ref>)–(<ref>)).  Recall that these definitions imply that (<ref>) holds with k=K (see Remark <ref>) which is an important component of the estimate (<ref>) of Proposition <ref>.
	
	The proof of (<ref>) and the latter of (<ref>), and the estimate for ϱ_K,l in (<ref>), for k =K, follow from Proposition <ref>.  The gradient estimate of Proposition <ref> then implies that ϕ_K,l satisfy (<ref>) (or, equivalently, the estimate for ϕ_K,l in (<ref>)).
	The proof of properties (<ref>), (<ref>)–(<ref>) and the former of (<ref>) and (<ref>), for k =K, then follow from Proposition <ref>.






§ THE FINITE PROBLEMS



In this section solutions of Vlasov–Poisson will be considered on certain finite intervals of the form [T,T_f].  “Remainder quantities”, defined by subtracting the approximate solution of Section <ref>, will be considered, and the solutions are defined so as to have trivial “finite data” for the remainder at time t=T_f.  These problems are referred to as “finite problems” in view of the finite length of the interval [T,T_f].  The proof of the main results, in Section <ref>, will follow from considering these finite problems and letting T_f→∞.

Throughout this section, suppose a smooth compactly supported function f_∞ℝ^3 ×ℝ^3 → [0,∞), satisfying (<ref>) for some B>0, and large times T_f ≫ T_0 ≫ 1 are given, where T_0 is assumed to be suitably large so as to satisfy the hypothesis of Theorem <ref>.  Suppose also that N≥ 6 is fixed.  Recall the functions f_[k] [T_0,∞) ×ℝ^3 ×ℝ^3 →ℝ, ϱ_[k], ϕ_[k] [T_0,∞) ×ℝ^3 →ℝ — defined, for all k ≥ 1, explicitly in terms of f_∞ — of Theorem <ref>.

Given a solution (f,ϱ, ϕ) of the Vlasov–Poisson system (<ref>)–(<ref>), define, for each k≥ 1, functions f̌_[k] [T_0,∞) ×ℝ^3 ×ℝ^3 →ℝ, and ϱ̌_[k], ϕ̌_[k] [T_0,∞) ×ℝ^3 →ℝ, by

    f̌_[k](t,x,p) = f(t,x,p) - f_[k](t,x,p),
    	    ϱ̌_[k](t,x) = ϱ(t,x) - ϱ_[k](t,x),
    	    ϕ̌_[k](t,x) = ϕ(t,x) - ϕ_[k](t,x).

These functions are referred to as “remainders”.

The main result of this section is the following.

 
	There exists k_* = k_*(ℱ_∞^N+1) large such that the following holds. 
	
	For any fixed k ≥ k_*, suppose there exists a solution f [T,T_f] ×ℝ^3×ℝ^3 → [0,∞) of Vlasov–Poisson (<ref>)–(<ref>), for some T_0 ≤ T < T_f, which satisfies the final condition
	
    f (T_f,x,p)
    	=
    	f_[k](T_f,x,p),

	i.e. f̌_[k](T_f,·,·) ≡ 0, and suppose moreover that f̌_[k] satisfies the bootstrap assumption
	
    sup_t∈ [T,T_f]∑_| I | + | J|≤ N‖ L^I (t^-1∂_p)^Jf̌_[k] (t,·,·) ‖_L^2_x L^2_p≤
    		(ℱ_∞^N+2k+4)^2,

	and that ϕ satisfies the bootstrap assumption, for all T≤ t ≤ T_f,
	
    sup_x∈ℝ^3|∇ϕ (t,x) - t^-2∇ϕ_∞(x/t)
    		|≤ℱ_∞^N+2k+4/t^5/2.

	Then, if T_0 is sufficiently large (with respect to k and ℱ_∞^N+2k+4), the solution in fact satisfies 
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J f̌_[k] (t,·,·) ‖_L^2_x L^2_p   ≤
    		C_k
    		(ℱ_∞^N+2k+4)^2
    		 (log t)^1+k/t^1+ k,
    	
    ∑_| I |≤ N‖ (t∂_x)^I ϱ̌_[k] (t,·) ‖_L^2_x   ≤
    		C_k ℱ_∞^N+2k+4(log t)^1+k/t^5/2 + k,

	and
	
    ∑_| I |≤ N‖ (t∂_x)^I ∇ϕ̌_[k] (t,·) ‖_L^2≤
    		C_kℱ_∞^N+2k+4(log t)^1+k/t^3/2 + k.

	In particular, if T_0 is sufficiently large, then (<ref>) holds with the right hand side replaced by 1/2 (ℱ_∞^N+2k+4)^2 and (<ref>) holds with the right hand side replaced by ℱ_∞^N+2k+4/2t^5/2.  Finally,
	
    (f̌_[k](t,·,·))
    		⊂{
    		(x,p) ⊂ℝ^3 ×ℝ^3
    		|| x - t p + log t ∇ϕ_∞(p) |≤ 2ℱ_∞^3 + B, | p |≤ 2 B
    		},

	where B>0 is as in (<ref>).


An overview of the main aspects of the proof of of Theorem <ref> is given in Section <ref>.

Recall again the convention, discussed at the beginning of Section <ref>, for the suppressed sequence appearing in ℱ_∞^n.  We remark, in particular, that if {𝔫(n)} is the suppressed sequence appearing in (<ref>)–(<ref>), then the suppressed sequence in (<ref>)–(<ref>) will be a different sequence {𝔫'(n)}, where 𝔫'(n) ≥𝔫(n) for all n.

The results of this section will all be in the above setting.
In particular, N ≥ 6 will be assumed to be fixed throughout, and the dependence of constants on N is suppressed.
In Section <ref> the system of equations satisfied by the remainder quantities is obtained.  In Section <ref> these reminder quantities are estimated, and in Section <ref> the proof of Theorem <ref> is completed.



 §.§ System of equations for remainders




The proof of Theorem <ref> will proceed by considering the remainders (<ref>).  Recall that the final condition (<ref>) is defined so that the remainder f̌_[k] vanishes at time t=T_f:

    f̌_[k](T_f,x,p)
    	=
    	0,
    	    for all  x,p ∈ℝ^3.

The remainders satisfy the system of equations,

    _ϕf̌_[k]
    	=
    	F_[k],
    	    Δϕ̌_[k]
    	=
    	ϱ̌_[k],
    	    ϱ̌_[k](t,x) - ∫_ℝ^3f̌_[k](t,x,p) dp
    	=
    	P_[k],

where the functions F_[k] [T_0,∞) ×ℝ^3 ×ℝ^3 →ℝ and P_[k] [T_0,∞) ×ℝ^3 →ℝ are given by

    F_[k]
    	=
    	_ϕf̌_[k] = - _ϕ f_[k] = - ∂_x^iϕ̌_[k]∂_p^i f_[k] - _ϕ_[k] f_[k],

and

    P_[k](t,x)
    	=
    	ϱ̌_[k](t,x) - ∫_ℝ^3f̌_[k](t,x,p) dp
    	=
    	- ϱ_[k](t,x) + ∫_ℝ^3 f_[k](t,x,p) dp.

The relevant properties of F_[k] and P_[k] are given by the following proposition, which is a consequence of Theorem <ref>.

 
	For each k ≥ 0, if T_0 = T_0(ℱ_∞^N+k) is sufficiently large, the functions F_[k] and P_[k] satisfy, for all t ≥ T_0,
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J F_[k] (t,·, ·) ‖_L^2_x L^2_p≤C ℱ_∞^N+1/t^3/2∑_| I| =1^N+1‖ (t ∂_x)^I ϕ̌_[k](t,·) ‖_L^2
    		+
    		C_k
    		(ℱ_∞^N+2k+3)^2
    		(log t)^1+k/t^2+k
    		,

	where the constant C does not depend on k, and
	
    ∑_| I |≤ N‖ (t∂_x)^I P_[k] (t,·) ‖_L^2_x≤
    		C_kℱ_∞^N+2k+3(log t)^1+k/t^5/2 + k
    		.





	The estimate (<ref>) follows immediately from Theorem <ref> (see (<ref>)).
	
	For (<ref>), consider first the first term on the right hand side of (<ref>).  Note that, since N ≥ 3,
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J ( ∂_x^iϕ̌_[k]∂_p^i f_[k]) (t,·, ·) ‖_L^2_x L^2_p≲
                   ∑_| I | = 1^N-2sup_x∈ℝ^3| (t∂_x)^I ϕ̌_[k] (t,·) |∑_| I | + | J |≤ N+1
    | J |≥ 1‖ L^I (t^-1∂_p)^J f_[k] (t,·, ·) ‖_L^2_x L^2_p
                   
    		+
    		∑_| I | = 1^N+1‖ (t∂_x)^I ϕ̌_[k] (t,·) ‖_L^2_x∑_| I | + | J |≤ N-2
    | J |≥ 1sup_x∈ℝ^3‖ L^I (t^-1∂_p)^J f_[k] (t,x, ·) ‖_L^2_p,

	It follows from the Sobolev inequality — see Remark <ref> — that
	
    sup_x∈ℝ^3∑_| I| =1^N-2| (t ∂_x)^I ϕ̌_[k] (t,x) |≤
    		C
    		t^-3/2∑_| I| =1^N ‖ (t ∂_x)^I ϕ̌_[k] (t,·) ‖_L^2_x,

	and similarly, by Theorem <ref> (see (<ref>)) and the Sobolev inequality of Proposition <ref>,
	
    ∑_| I | + | J |≤ N-2sup_x∈ℝ^3
    		t^3/2‖ L^I (t^-1∂_p)^J f_[k] (t,x, ·) ‖_L^2_p
    		+
    		∑_| I | + | J |≤ N+1‖ L^I (t^-1∂_p)^J f_[k] (t,·, ·) ‖_L^2_xL^2_p
    ≤
    		Cℱ_∞^N+1
    		+
    		C_kℱ_∞^N+klog t/t≤
    		Cℱ_∞^N+1,

	where the constant C does not depend on k, and the latter inequality follows after taking T_0 suitably large.  The second term on the right hand side of (<ref>) is estimated in Theorem <ref> (see (<ref>)).  The proof of (<ref>) then follows.



In the following proposition the size of the support of f̌_[k] is estimated.

 
	If T_0 = T_0(ℱ_∞^N+2k+4) is sufficiently large, then
	
    (f̌_[k](t,·,·))
    		   ⊂{
    		(x,p) ⊂ℝ^3 ×ℝ^3
    		|| x - t p + log t ∇ϕ_∞(p) |≤ 2ℱ_∞^3 + B, | p |≤ 2 B
    		}
       ⊂{ (x,p) ⊂ℝ^3 ×ℝ^3 || x |≤ 2ℱ_∞^3 + 3 B t, | p |≤ 2 B }.




	Theorem <ref>, in particular the property (<ref>), implies that
	
    (f_[k])
    		⊂{
    		(x,p) ⊂ℝ^3 ×ℝ^3
    		|| x - t p + log t ∇ϕ_∞(p) |≤ B, | p |≤ B
    		}.

	Consider, therefore the support of f.
	Note that, for f_T_f(x,p) := f(T_f,x,p),
	
    | x-T_f p + log T_f  ∇ϕ_∞ (p) |≤
    		B,
    		    for all  (x,p) ∈(f_T_f),

	which follows from the definition (<ref>) of f_T_f and the fact that each f_k,l satisfies (<ref>).  Define the characteristic curves of , denoted X, P  [T,T_f] × [T,T_f] ×ℝ^3 ×ℝ^3 →ℝ^3, as solutions of
	
    dX(s,t,x,p)/ds = P(s,t,x,p),
    		    dP(s,t,x,p)/ds
    		=
    		(∇ϕ)(s,X(s,t,x,p)),

	along with the initial conditions
	
    X(t,t,x,p) = x,
    		    
    		P(t,t,x,p) = p.

	Clearly, for any x,p ∈ℝ^3, T ≤ s ≤ T_f,
	
    f(s,X(s,T_f,x,p), P(s,T_f,x,p)) = f(T_f,x,p),

	and so
	
    (f)
    		=
    		{ (s,X(s,T_f,x,p), P(s,T_f,x,p)) | (x,p) ∈(f_T_f), T ≤ s ≤ T_f }.

	We proceed by subtracting from X and P suitable approximate solutions of the characteristic equations and estimating the difference (cf. <cit.>, where similar approximate solutions of the characteristic equations are introduced in the context of the Einstein–Vlasov system).  Define
	
    X(s,t,x,p) = X(s,t,x,p) - ( x - (t-s)p - log( s/t) ∇ϕ_∞(p) ),
    		    P(s,t,x,p) = P(s,t,x,p) - ( p - 1/s∇ϕ_∞(p) ),

	so that
	
    dX(s,t,x,p)/ds = P(s,t,x,p),
    		    dP(s,t,x,p)/ds
    		=
    		(∇ϕ)(s,X(s,t,x,p)) - 1/s^2∇ϕ_∞(p),

	along with the initial conditions
	
    X(t,t,x,p) = 0,
    		    
    		P(t,t,x,p) = 1/t∇ϕ_∞(p).

	
	Consider some (x,p) ∈(f_T_f), T ≤ t ≤ s ≤ T_f.  Using the fact that
	
    sup_x∈ℝ^3|∇ϕ (t,x) - t^-2∇ϕ_∞(x/t)
    		|≤ℱ_∞^N+2k+4/t^5/2,
    		    sup_p∈ℝ^3( |∇ϕ_∞(p) | + |∇^2 ϕ_∞(p) |) ≲ℱ_∞^3,

	(see the bootstrap assumption (<ref>) and the fact (<ref>)) and,
	
    |X(s,T_f,x,p)/s - p |   
    		=
    		s^-1|X(s,T_f,x,p) - log s  ∇ϕ_∞(p) + x - T_f p + log T_f ∇ϕ_∞(p) |
       ≲|X(s,T_f,x,p) |/s + log s/sℱ_∞^3 + B/s
    		,

	it follows that
	
    |dP(s,T_f,x,p)/ds|   ≲| (∇ϕ)(s,X(s,T_f,x,p)) - 1/s^2 (∇ϕ_∞) (X(s,T_f,x,p)/s) |
         
    		+
    		1/s^2| (∇ϕ_∞) ( X(s,T_f,x,p)/s) - (∇ϕ_∞)(p) |
       ≲ℱ_∞^3|X(s,T_f,x,p) |/s^3
    		+
    		ℱ_∞^N+2k+4/s^5/2
    		+
    		ℱ_∞^3 (B + log s ℱ_∞^3)/s^3.

	Integrating the equations (<ref>) backwards from time T_f, the fact that
	
    ∫_t^T_f∫_s^T_f|X(s',T_f,x,p) |/(s')^3 ds' ds
    		=
    		∫_t^T_f (s-t) |X(s,T_f,x,p) |/s^3 ds,

	then implies that
	
    |X(t,T_f,x,p) |   ≲|∇ϕ_∞(p) |
    		+
    		ℱ_∞^N+2k+4/t^1/2
    		+
    		ℱ_∞^3 (B + log t ℱ_∞^3)/t
    		+
    		ℱ_∞^3∫_t^T_f|X(s,T_f,x,p) |/s^2 ds
    		
       ≤3/2ℱ_∞^3
    		+
    		Cℱ_∞^3∫_t^T_f|X(s,T_f,x,p) |/s^2 ds
    		,

	if T_0 is suitably large.
	The Grönwall inequality, Proposition <ref>, then gives
	
    |X(t,T_f,x,p) | + t |P(t,T_f,x,p) |≤3/2ℱ_∞^3
    		+
    		C(ℱ_∞^3)^2 ∫_t^T_f1/s^2 e^∫_t^s Cℱ_∞^3/(s')^2ds' ds
    		≤3/2ℱ_∞^3
    		+
    		C(ℱ_∞^3)^2 e^Cℱ_∞^3/t/t.

	It then follows that
	
    |
    		X(t,T_f,x,p) - t p + log t ∇ϕ_∞(p) )
    		|
    		+
    		t
    		|
    		P(t,T_f,x,p) - p
    		|≤
    		2ℱ_∞^3 + B,

	which, by (<ref>), completes the proof.







 §.§ Estimates for remainders




In this section the remainder quantities f̌_[k], ϱ̌_[k], ϕ̌_[k] are estimated.  The main result of this section is the following estimate for f̌_[k].



 
	Under the assumptions of Theorem <ref>, if k is sufficiently large (k> (C+1) ℱ_∞^N+1-1 suffices, where C is the constant from (<ref>)), and T_0 is sufficiently large (depending on k and ℱ_∞^N+2k+4) then, for all T≤ t ≤ T_f,
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J f̌_[k] (t,·,·) ‖_L^2_x L^2_p≤
    		C_k
    		(ℱ_∞^N+2k+4)^2
    		 (log t)^1+k/t^1+ k.



The reader is again referred to Section <ref> for an overview of the proof of Proposition <ref>.
The proof requires the following preliminary results.

 
	For any multi-index I such that | I |≤ N, for all k ≥ 0 and all T≤ t ≤ T_f,
	
    ‖∇ (t∂_x)^I ϕ̌_[k] (t,·) ‖_L^2≲
    		t ‖ (t∂_x)^I ϱ̌_[k] (t,·) ‖_L^2.




	Recall that
	
    Δ_ℝ^3ϕ̌_[k]
    		=
    		ϱ̌_[k].

	The proof follows from Proposition <ref> and the fact that | x |≤ 3Bt in supp(ϱ̌_[k]) (see Proposition <ref>, and the property (<ref>)), since t ∂_x^i commutes with Δ_ℝ^3.




 
	For any multi-index I such that | I |≤ N, for all k ≥ 0 and all T≤ t ≤ T_f,
	
    ‖ (t∂_x)^I ϱ̌_[k] (t,·) ‖_L^2_x≲
    		t^-3/2‖ L^I f̌_[k] (t,·,·) ‖_L^2_x L^2_p
    		+
    		C_kℱ_∞^N+2k+3(log t)^1+k/t^5/2 + k.




	Recall (see Proposition <ref>) that
	
    ϱ̌_[k](t,x) 
    		=
    		∫_ℝ^3f̌_[k](t,x,p) dp
    		+
    		P_[k],

	where P_[k] satisfies (<ref>).
	Since
	
    ∫_ℝ^3∂_p^i L^J f̌_[k] (t,x,p) dp = 0,

	for i=1,2,3 and each multi-index J, it follows that, for any multi-index I,
	
    (t∂_x)^I ϱ̌_[k] (t,x) = ∫_ℝ^3 L^I f̌_[k] (t,x,p) dp
    		+
    		(t∂_x)^I P_[k] (t,x).

	Hence
	
    ‖ (t∂_x)^I ϱ̌_[k] (t,·) ‖_L^2_x≲( ∫_ℝ^3( ∫_ℝ^3 L^I f̌_[k] (t,x,p) dp )^2 dx )^1/2
    		+
    		‖ (t∂_x)^I P_[k] (t,·) ‖_L^2_x
    ≲( 
    		∫_ℝ^3∫_ℝ^31_f̌_[k] (t,x,p) dp
    		∫_ℝ^3| L^I f̌_[k] (t,x,p) |^2 dp dx
    		)^1/2
    		+
    		‖ (t∂_x)^I P_[k] (t,·) ‖_L^2_x.

	Note now that
	
    ∫_ℝ^31_(f̌_[k]) (t,x,p) dp ≲1 + (ℱ_∞^3)^3/t^3,

	which follows from the fact that
	
    |x/t
    		+
    		log t/t∇ϕ_∞( x/t)
    		- p
    		|≤3ℱ_∞^3 + 2B/t  in  (f̌_[k]).

	See Proposition <ref> (and cf. Proposition <ref>).
	The proof then follows from the estimate (<ref>).



In the following proposition, the terms arising from commuting the equation (<ref>) for f̌_[k] with the vector fields of Section <ref> are estimated, using the bootstrap assumption (<ref>).

 
	If the bootstrap assumption (<ref>) holds, then
	
    ∑_| I | + | J |≤ N‖_ϕ (L^I (t^-1∂_p)^J f̌_[k]) ‖_L^2_x L^2_p≲∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J F_[k]‖_L^2_x L^2_p
    
    		+
    		C_kℱ_∞^N+2k+3 (log t)^2/t^2∑_| I | + | J | =1^N
    		‖ L^I (t^-1∂_p)^Jf̌_[k]‖_L^2_x L^2_p
    		+
    		(ℱ_∞^N+2k+4)^2
    		log t/t^5/2∑_| K | = 1^N+1‖ (t ∂_x)^K ϕ̌_[k]‖_L^2_x
    		.





	Recall Proposition <ref>, which implies that, for i=1,2,3,
	
    [
    		_ϕ
    		,
    		L_i
    		]
    		=
    		   
    		-
    		log t/t^2∂_i ∂_jϕ_∞(p)  
    		L_j
    		+
    		[
    		t^-1( ∂_i ∂_kϕ_∞(p) - ∂_i ∂_kϕ_∞( x/t) )
    		+
    		log t/t^2∂_lϕ_∞(p)  ∂_i ∂_k ∂_lϕ_∞(p)
    		
       
    		+
    		(log t)^2/t^2∂_i ∂_jϕ_∞(p) 
    		∂_j ∂_k ϕ_∞(p)
    		-
    		t ∂_x^i∂_x^kϕ̌_[0](t,x)
    		+
    		log t  ∂_i ∂_k ∂_lϕ_∞(p)  ∂_x^lϕ̌_[0](t,x)
    		]
    		t^-1∂_p^k
    		,

	and
	
    [
    		_ϕ
    		,
    		t^-1∂_p^i]
    		=
    		-
    		1/t^2
    		L_i
    		+
    		log t/t^2∂_i ∂_k ϕ_∞(p)   t^-1∂_p^k
    		.

	Note that, for any smooth h ℝ^3 →ℝ,
	
    L_i ( h(p) - h ( x/t) )
    		=
    		(∂_i h)(p) - (∂_i h) ( x/t)
    		+
    		log t/t∂_i ∂_k ϕ_∞(p) (∂_k h)(p)
    		,

	and so
	
    sup_| y(t,x,p) |≤ B| L_i ( h(p) - h ( x/t) ) |≲‖∇^2 h ‖_L^∞2B + ℱ_∞^3log t/t
    		+
    		‖∇ h ‖_L^∞ℱ_∞^3 log t/t
    		.

	It follows from a simple induction argument, along with (<ref>) and the fact that, for any n ≥ 0 and any multi-indices I, J such that | I | + | J |≤ n,
	
    ‖_ϕ (L^I (t^-1∂_p)^J f̌_[k]) ‖_L^2_x L^2_p≤‖ L^I (t^-1∂_p)^J F_[k]‖_L^2_x L^2_p
    		+
    		C
    		ℱ_∞^n+3 (log t)^2/t^2∑_| I ' | + | J'| =1^n
    		‖ L^I' (t^-1∂_p)^J'f̌_[k]‖_L^2_x L^2_p
    
    		+
    		C
    		(1 + ℱ_∞^n+3)
    		 log t/t(
    		∑_| K | = 1^n+1‖ (t ∂_x)^K ϕ̌_[0]‖_L^2_x∑_| I ' | + | J'| =1^⌊n+2/2⌋‖ L^I' (t^-1∂_p)^J'f̌_[k]‖_L^∞_x L^2_p
    
    		+
    		∑_| K | = 1^⌊n+4/2⌋‖ (t ∂_x)^K ϕ̌_[0]‖_L^∞_x∑_| I ' | + | J'| =1^n
    		‖ L^I' (t^-1∂_p)^J'f̌_[k]‖_L^2_x L^2_p)
    		,

	The Sobolev inequality of Proposition <ref> and the fact (<ref>) (since N ≥ 3), implies that the right hand side of (<ref>) is controlled by
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J F_[k]‖_L^2_x L^2_p
    
    		+
    		(
    		ℱ_∞^N+3 (log t)^2/t^2
    		+
    		(1 + ℱ_∞^N+3)
    		 log t/t^5/2∑_| K | = 1^N+1‖ (t ∂_x)^K ϕ̌_[0]‖_L^2_x)
    		∑_| I | + | J | =1^N
    		‖ L^I (t^-1∂_p)^Jf̌_[k]‖_L^2_x L^2_p
    		.

	Note now that ϕ̌_[0] = ϕ̌_[k] + (ϕ_[k] - ϕ_[0]), and that Proposition <ref> and Proposition <ref> imply that
	
    ∑_| I| =1^N+1‖ (t ∂_x)^I (ϕ_[k] - ϕ_[0])(t,·) ‖_L^2≲
    		t^1/2∑_| I |≤ N‖ L^I (f_[k] - f_[0]) (t,·,·) ‖_L^2_x L^2_p≤C_kℱ_∞^N+2k+3log t/t^1/2,

	where (<ref>) has been used.  It follows that
	
    ∑_| I | + | J |≤ N‖_ϕ (L^I (t^-1∂_p)^J f̌_[k]) ‖_L^2_x L^2_p≲∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J F_[k]‖_L^2_x L^2_p
    
    		+
    		(
    		C_kℱ_∞^N+2k+3 (log t)^2/t^2
    		+
    		(1 + ℱ_∞^N+3)
    		 log t/t^5/2∑_| K | = 1^N+1‖ (t ∂_x)^K ϕ̌_[k]‖_L^2_x)
    		∑_| I | + | J | =1^N
    		‖ L^I (t^-1∂_p)^Jf̌_[k]‖_L^2_x L^2_p
    		.

	The proof of (<ref>) follows from inserting the bootstrap assumption (<ref>).
	


The proof of Proposition <ref> can now be given.


	If T_0=T_0(ℱ_∞^N+2k+4) is suitably large so that the coefficient of the final term in Proposition <ref> satisfies
	
    (ℱ_∞^N+2k+4)^2
    		log t/t≤ℱ_∞^N+1,

	then
	Proposition <ref> and Proposition <ref> imply that,
	
    ∑_| I | + | J |≤ N‖_ϕ (L^I (t^-1∂_p)^J f̌_[k]) ‖_L^2_x L^2_p≤   
    		C_k
    		(ℱ_∞^N+2k+3)^2
    		(log t)^1+k/t^2+k
    		+
    		C ℱ_∞^N+1/t^3/2∑_| I| =1^N+1‖ (t ∂_x)^I ϕ̌_[k](t,·) ‖_L^2
       
    		+
    		C_kℱ_∞^N+2k+3 (log t)^2/t^2∑_| I | + | J | =1^N
    		‖ L^I (t^-1∂_p)^Jf̌_[k]‖_L^2_x L^2_p
    		.

	Now Proposition <ref> and Proposition <ref> imply that
	
    ∑_| I| =1^N+1‖ (t ∂_x)^I ϕ̌_[k](t,·) ‖_L^2≲
    		t^1/2∑_| I |≤ N‖ L^I f̌_[k] (t,·,·) ‖_L^2_x L^2_p
    		+
    		C_kℱ_∞^N+2k+3(log t)^1+k/t^1/2 + k.

	Thus, if T_0 = T_0(ℱ_∞^N+2k+3,k) is suitably large so that
	
    C_kℱ_∞^N+2k+3 (log t)^2/t≤ℱ_∞^N+1,

	then
	
    ∑_| I | + | J |≤ N‖_ϕ (L^I (t^-1∂_p)^J f̌_[k]) ‖_L^2_x L^2_p≤C ℱ_∞^N+1/t∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J f̌_[k]‖_L^2_x L^2_p
    		+
    		C_k
    		(ℱ_∞^N+2k+4)^2
    		(log t)^1+k/t^2+k.

	Defining
	
    ℰ̌_[k](t)
    		: =
    		∑_| I | +| J | = 0^N ‖ L^I (t^-1∂_p)^J f̌_[k](t,·,·) ‖_L^2_x L^2_p,

	and noting that the vanishing of f̌_[k] at time T_f (see (<ref>)) implies that ℰ̌_[k](T_f) = 0,
	Proposition <ref> then implies that, for all T ≤ t ≤ T_f,
	
    ℰ̌_[k](t)
    		≤
    		C_k
    		(ℱ_∞^N+2k+4)^2
    		(log t)^1+k/t^1+k
    		+
    		C ℱ_∞^N+1∫_t^T_fℰ̌_[k](s)/s
    		ds.

	Defining
	
    a(t)
    		=
    		C ℱ_∞^N+1/t,
    		    
    		b(t) = C_k
    		(ℱ_∞^N+2k+4)^2
    		(log t)^1+k/t^1+k,

	it follows that
	
    e^∫_t^s a(s')ds' = ( s/t)^C ℱ_∞^N+1.

	Thus, if k> C ℱ_∞^N+1 - 1,
	
    ∫_t^T_f a(s) b(s) e^∫_t^s a(s')ds'ds
    		   ≤C_k ℱ_∞^N+1
    		(ℱ_∞^N+2k+4)^2
    		/t^Cℱ_∞^3∫_t^T_f(log s)^1+k/s^2 + k - Cℱ_∞^N+1ds
    		≤C_k ℱ_∞^N+1
    		(ℱ_∞^N+2k+4)^2 (log t)^1+k/t^1+ k,

	and the Grönwall inequality, Proposition <ref>, implies that
	
    ℰ̌_[k](t)
    		≤
    		C_k
    		(ℱ_∞^N+2k+4)^2(1+ℱ_∞^N+1)
    		 (log t)^1+k/t^1+ k
    		,

	which completes the proof of the estimate (<ref>) (after increasing the elements of the sequence {𝔫(n)} appropriately).
	


This section is ended with the following proposition, which will be used to recover the bootstrap assumption (<ref>).

 
	Under the assumptions of Proposition <ref>, for all T≤ t ≤ T_f,
	
    ∑_| K | = 1^N+1‖ (t ∂_x)^K ϕ̌_[0]‖_L^2_x≲C_kℱ_∞^N+2k+4log t/t^1/2.




	Proposition <ref> and Proposition <ref> imply that
	
    ∑_| I| =1^N+1‖ (t ∂_x)^I (ϕ̌_[0])(t,·) ‖_L^2≲
    		t^1/2∑_| I |≤ N‖ L^I f̌_[0] (t,·,·) ‖_L^2_x L^2_p
    ≤
    		t^1/2∑_| I |≤ N(
    		‖ L^I f̌_[k] (t,·,·) ‖_L^2_x L^2_p
    		+
    		‖ L^I (f_[k] - f_[0]) (t,·,·) ‖_L^2_x L^2_p)
    		≤C_kℱ_∞^N+2k+4log t/t^1/2,

	where the final inequality follows from Proposition <ref> and Theorem <ref> (see (<ref>)).







 §.§ Estimates for the finite problems: the proof Theorem <ref>


The proof of Theorem <ref> can now be given.


	The estimate (<ref>) follows from Proposition <ref>.  The estimate (<ref>) follows from Proposition <ref> and the estimate (<ref>), and the estimate (<ref>) follows from Proposition <ref> and the estimate (<ref>).
	
	The bootstrap assumption (<ref>) clearly holds with the right hand side replaced by 1/2 (ℱ_∞^N+2k+4)^2 if T_0 is sufficiently large.  For the improvement of (<ref>), note that Proposition <ref> and the Sobolev inequality of Remark <ref> imply that, for all T ≤ t ≤ T_f,
	
    sup_x∈ℝ^3|∇ϕ̌_[0](t,x) |≤C/t^5/2∑_| I | = 1^N+1‖ (t ∂_x)^I ϕ̌_[0]‖_L^2_x≤C_kℱ_∞^N+2k+4log t/t^3,

	and so, if T_0 is sufficiently large, (<ref>) holds with the right hand side replaced by ℱ_∞^N+2k+4/2t^5/2.
	
	Finally, the property (<ref>) follows from Proposition <ref>.









§ THE PROOF OF THE MAIN RESULTS



In this section the proofs of Theorem <ref>, Theorem <ref>, and Theorem <ref> are given.  The following (non-optimal) local well posedness theorem will be used.


 
	There is k such that, for any t_0∈ℝ and any smooth compactly supported function f_0 ℝ^3_x ×ℝ^3_p → [0,∞), there exists T=T(‖ f_0 ‖_H^k_xH^k_p) > 0 and a unique smooth solution (f, ϱ, ϕ) of the Vlasov–Poisson system (<ref>)–(<ref>) on (t_0-T,t_0+T) ×ℝ^3_x ×ℝ^3_p such that
	
    f(t_0,x,p) = f_0(x,p),

	for all (x,p) ∈ℝ^3 ×ℝ^3.



Let T_0 ≫ 1 be sufficiently large so as to satisfy the assumptions of Theorem <ref> and Theorem <ref>.   Before the proofs of Theorem <ref>, Theorem <ref>, and Theorem <ref> are given, Theorem <ref> is used to establish, for each T_f>T_0, the existence of a solution of Vlasov–Poisson on the interval [T_0,T_f] with trivial remainder f̌_[k] at t=T_f.



 
	Let k>k_* be as in Theorem <ref>.  For any T_f > T_0, there exists a solution f [T_0,T_f] ×ℝ^3×ℝ^3 → [0,∞) of Vlasov–Poisson (<ref>)–(<ref>) such that
	
    f|_t=T_f = f_[k](T_f,·,·),

	which satisfies the estimates (<ref>)–(<ref>) and the support property (<ref>) for all t∈ [T_0,T_f].



	Let ℬ⊂ [T_0,T_f] be the set of times T∈ [T_0,T_f] such that a solution f [T,T_f] ×ℝ^3×ℝ^3 → [0,∞) of Vlasov–Poisson (<ref>)–(<ref>) exists on [T,T_f] satisfying the final condition (<ref>) and the estimates (<ref>)–(<ref>).
	
	The set ℬ is a non-empty, closed subset of [T_0,T_f], by Theorem <ref> and the fact that f̌_[k](T_f,·,·) ≡ 0 and
	
    sup_x∈ℝ^3|∇ϕ̌_[0](T_f,x) |≤C_kℱ_∞^N+2k+4log T_f/(T_f)^3≤ℱ_∞^N+2k+4/2(T_f)^5/2,

	for T_0 as in Theorem <ref> (see, for example, the estimate (<ref>)).  If T∈ℬ satisfies T>T_0, Theorem <ref> implies that there is δ>0 such that the solution extends to the interval [T-δ,T_f].  Theorem <ref> implies that the inequalities (<ref>)–(<ref>) in fact hold with better constants and thus, by continuity, continue to hold for all t∈ [T - δ,T_f] if δ is sufficiently small.  The set ℬ is therefore open, and hence equal to [T_0,T_f], and the proof follows from Theorem <ref>.



The proofs of Theorem <ref> and Theorem <ref> can now be given.  It is convenient to give both together.


	The proof of Theorem <ref> and Theorem <ref> is divided into several steps.  First, the main part of the proof — namely the existence of the solution (f,ϱ,ϕ) — is given.  Then properties of the solution, including the expansions of Theorem <ref> and the support property (<ref>), are established.  Finally, the part of Theorem <ref> concerning the fact that the solution attains the scattering data f_∞ — in the sense that the estimate (<ref>) holds — is shown.
	
	The existence of the solution:
	Consider some T_f'>T_f >T_0.  By Theorem <ref>, there exist corresponding solutions
	
    f^(T_f') [T_0,T_f'] ×ℝ^3 ×ℝ^3 → [0,∞),
    		    
    		f^(T_f) [T_0,T_f] ×ℝ^3 ×ℝ^3 → [0,∞),

	of the Vlasov–Poisson system, which each satisfy the estimates (<ref>)–(<ref>) and the support property (<ref>).  The differences
	
    f := f^(T_f') - f^(T_f),
    		    ϕ := ϕ^(T_f') - ϕ^(T_f),
    		    ϱ := ϱ^(T_f') - ϱ^(T_f),

	satisfy the system
	
    _ϕ'f = ∂_x^iϕ ∂_p^i f^(T_f),
    	    Δ_ℝ^3ϕ = ϱ,

	on [T_0,T_f], where ϕ' = ϕ^(T_f'), along with the final condition
	
    f(T_f,x,p) = f̌_[k]^(T_f')(T_f,x,p),

	for all x,p∈ℝ^3.
	
	The proof proceeds by estimating this difference on [T_0,T_f], in a manner similar to the proof of Theorem <ref>.  Indeed, using the latter of (<ref>) and the fact that f satisfies the support property (<ref>), minor adaptations of Proposition <ref> and Proposition <ref> give, for any multi-index I, and any T_0 ≤ t ≤ T_f,
	
    ‖∇ (t∂_x)^I ϕ (t,·) ‖_L^2≲
    		t ‖ (t∂_x)^I ϱ (t,·) ‖_L^2≲
    		t^-1/2‖ L^I f (t,·,·) ‖_L^2_x L^2_p.

	Thus, minor adaptations of the proofs of Proposition <ref> and Proposition <ref>, using the fact that f^(T_f') and f^(T_f) satisfy the estimates (<ref>)–(<ref>), give
	
    ∑_| I | + | J |≤ N‖_ϕ (L^I (t^-1∂_p)^J f) ‖_L^2_x L^2_p≲   ℱ_∞^N+1/t^3/2∑_| I| =1^N+1‖ (t ∂_x)^I ϕ(t,·) ‖_L^2
    		+
    		ℱ_∞^N+3 (log t)^2/t^2∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^Jf‖_L^2_x L^2_p
    ≲   ℱ_∞^N+1/t∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^Jf‖_L^2_x L^2_p
    		.

	Thus, using the estimate (<ref>) for the final condition (<ref>), Proposition <ref> gives that
	
    ℰ(t) : = ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^Jf (t,·,·) ‖_L^2_x L^2_p

	satisfies, for all T_0 ≤ t ≤ T_f,
	
    ℰ(t)
    		≤
    		C_k
    		(ℱ_∞^N+2k+4)^2
    		(log T_f)^1+k/(T_f)^1 + k
    		+
    		∫_t^T_fCℱ_∞^N+1/sℰ(s)
    		ds.

	The Grönwall inequality, Proposition <ref>, with 
	
    a(t)
    		=
    		Cℱ_∞^N+1/s
    		,
    		    
    		b(t) ≡ C_k
    		(ℱ_∞^N+2k+4)^2
    		(log T_f)^1+k/(T_f)^1 + k

	then gives
	
    ℰ(t)
    		≤
    		C_k
    		(ℱ_∞^N+2k+4)^2
    		(log T_f)^1+k/(T_f)^1 + k(
    		1
    		+
    		Cℱ_∞^N+1/t^Cℱ_∞^N+1∫_t^T_f
    		s^Cℱ_∞^N+1 - 1
    		ds
    		).

	Thus, provided k ≥ Cℱ_∞^N+1 + 1,
	
    sup_T_0 ≤ t ≤ T_f∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J (f^(T_f') - f^(T_f)) (t,·,·) ‖_L^2_x L^2_p≤C_k
    		(ℱ_∞^N+2k+4)^2
    		/T_f.

	Returning to (<ref>), one has similar estimates for ϕ^(T_f') - ϕ^(T_f) and ϱ^(T_f') - ϱ^(T_f).  It follows that, for any increasing sequence T_f^n →∞, the sequence { (f^(T_f^n), ϕ^(T_f^n), ϱ^(T_f^n))} is Cauchy on their common domains of the elements, in the above norms, and thus converges to a unique limit (f,ϕ, ϱ) (independent of the sequence {T_f^n} chosen) which moreover satisfies the estimates (<ref>)–(<ref>) and the support property (<ref>) on [T_0,∞) and solves the Vlasov–Poisson system (<ref>)–(<ref>) on [T_0,∞).
	
	Properties of the solution:
	The fact that f satisfies the support property (<ref>) follows from (<ref>), along with the fact that each f^(T_f) satisfies (<ref>).
	
	Consider now the estimates (<ref>)–(<ref>) and recall k_* from Theorem <ref>.  For K ≥ k_* the estimates (<ref>)–(<ref>) follow immediately from the fact that (f,ϱ,ϕ) satisfy (<ref>)–(<ref>).  For 0 ≤ K < k_*, note that
	
    f̌_[K](t,x,p)
    		=
    		f̌_[k_*](t,x,p)
    		-
    		∑_k=K+1^k_*∑_l=0^k(log t)^l/t^k f_k,l(x-tp + log t  ∇ϕ_∞ ( p), p ),

	and thus (<ref>) and (<ref>) imply that
	
    ∑_| I | + | J |≤ N‖ L^I (t^-1∂_p)^J f̌_[K] (t,·,·) ‖_L^2_x L^2_p≤
    		C_k_*(
    		(ℱ_∞^N+2k_*+4)^2
    		 (log t)^1+k_*/t^1+ k_*
    		+
    		∑_k=K+1^k_*(log t)^k/t^k (ℱ_∞^N+k)^2
    		),
 
	and (<ref>) follows if T_0 is suitably large.  Similarly for (<ref>)–(<ref>).
	
	The L^∞ estimates (<ref>) follow from the L^2 estimates (<ref>) and the Sobolev inequality of Proposition <ref>, and the L^∞ estimates (<ref>)–(<ref>) follow from the L^2 estimates (<ref>)–(<ref>) and the Sobolev inequality of Remark <ref>.
	
	The scattering data is attained:
	In order to see that the data f_∞ is attained — in the sense that the estimate (<ref>) holds — first note that, for
	
    y(t,x,p) = x - tp + log t ∇ϕ_∞(p),
    		    ỹ(t,x,p) = x + tp - log t ∇ϕ_∞(p),

	one has
	
    ∫∫| f(t,ỹ(t,x,p),p) - f_∞(x,p) |^2 dp dx
    		=
    		∫∫| f(t,x,p) - f_∞(y(t,x,p),p) |^2 dp dx,

	and so the zeroth order part of the estimate (<ref>) follows from setting K=0 in (<ref>).  For the terms involving first order derivatives, recall first Remark <ref> (considering, in particular, (<ref>) and (<ref>) with h =f_∞) and note also that 
	
    ∂_x^i( f(t,ỹ(t,x,p),p) )
    		=
    		   
    		(∂_x^i f) (t,ỹ(t,x,p),p),
    	
    ∂_p^i( f(t,ỹ(t,x,p),p) )
    		=
    		   ( δ_ij - log t/t∂^2_i,jϕ_∞(p) )(L_i f) (t,ỹ(t,x,p),p)
    		
       
    		+
    		(log t)^2/t∂_i,j^2 ϕ_∞(p) ∂_j,k^2 ϕ_∞(p) (t^-1∂_p^k f) (t,ỹ(t,x,p),p),

	and
	
    (∂_x^if_∞)(y(t,x,p),p)
    		=
    		   
    		-
    		(t^-1∂_p^i) ( f_∞(y(t,x,p),p) )
    		
       
    		+
    		log t/t∂_k ∂_i ϕ_∞(p) (∂_x^kf_∞)(y(t,x,p),p)
    		+
    		t^-1 (∂_p^if_∞)(y(t,x,p),p)
    		
    
    		(∂_p^if_∞)(y(t,x,p),p)
    		=
    		   
    		L_i ( f_∞(y(t,x,p),p) )
    		
       
    		-
    		(log t)^2/t∂_k ∂_i ϕ_∞(p) (∂_x^kf_∞)(y(t,x,p),p)
    		-
    		log t/t∂_k ∂_i ϕ_∞(p) (∂_p^kf_∞)(y(t,x,p),p)
    		.

	It follows that
	
    (∂_x^if)(t,x,p) - (∂_x^if_∞)(y(t,x,p),p)
    		=
    		-
    		(t^-1∂_p^i) ( f(t,x,p) - f_∞(y(t,x,p),p) )
    		
       
    		+
    		1/t( ∂_x^i + t ∂_p^i) f(t,x,p)
    		-
    		log t/t∂_k ∂_i ϕ_∞(p) (∂_x^kf_∞)(y(t,x,p),p)
    		-
    		t^-1 (∂_p^if_∞)(y(t,x,p),p).

	Hence
	
    ∫∫|∂_x^i( f(t,ỹ(t,x,p),p) - f_∞(x,p) ) |^2 dp dx
    		=
    		∫∫| (∂_x^i f)(t,x,p) - (∂_x^i f_∞) (y(t,x,p),p) |^2  dp dx
    		
               ≲∫∫| t^-1∂_p^i( f(t,x,p) -  f_∞ (y(t,x,p),p) ) |^2 dp dx
    		+
    		ℱ_∞^3 log t/t≲ℱ_∞^N+4log t/t
    		,

	and similarly,
	
    ∫∫|∂_p^i( f(t,ỹ(t,x,p),p) - f_∞(x,p) ) |^2 dp dx
    		
               ≲∫∫| L_i ( f(t,x,p) -  f_∞ (y(t,x,p),p) ) |^2 dp dx
    		+
    		ℱ_∞^3 (log t)^2/t≲ℱ_∞^N+4 (log t)^2/t
    		,

	by (<ref>) with K=0.  The part of (<ref>) concerning higher order derivatives follows similarly.
	The L^∞ estimate (<ref>) follows from (<ref>) and the Sobolev inequality of Proposition <ref>.
	
	
	Finally, the proof of Theorem <ref> is given.
	
	
	Consider the solution (f,ϱ,ϕ) of Theorem <ref>, and
	suppose that (f',ϱ',ϕ') is another solution which satisfies (<ref>) for some K=K(ℱ_∞^N+1) and for all t sufficiently large, and f' satisfies the support property (<ref>).  It follows that the difference of the two solutions satisfies the system
	
    _ϕ' (f-f') = ∂_x^i(ϕ - ϕ ')  ∂_p^i f,
    	    Δ_ℝ^3 (ϕ - ϕ ') = ϱ - ϱ'.

	Since
	
    sup_x∈ℝ^3‖∂_p^i f (t,x,·) ‖_L^2_p≤Cℱ_∞^N/t^1/2,
    		    ‖∇ (ϕ - ϕ ') (t,·) ‖_L^2≲
    		t ‖ (ϱ - ϱ ') (t,·) ‖_L^2≲
    		t^-1/2‖ (f-f') (t,·,·) ‖_L^2_x L^2_p.

	where the latter follows from the support property (<ref>) (see Proposition <ref> and Proposition <ref>), it follows that
	
    ‖_ϕ' (f-f')(t,·,·)‖_L^2_x L^2_p≤Cℱ_∞^N/t‖ (f-f')(t,·,·)‖_L^2_x L^2_p.

	Proposition <ref> and the Grönwall inequality, Proposition <ref>, with a(t) = t^-1Cℱ_∞^N, b(t) ≡‖ (f-f')(T,·,·)‖_L^2_x L^2_p then imply that, for any T large and t≤ T,
	
    ‖ (f-f')(t,·,·)‖_L^2_x L^2_p   ≤‖ (f-f')(T,·,·)‖_L^2_x L^2_p(
    		1
    		+
    		Cℱ_∞^N/t^Cℱ_∞^N∫_t^T s^C ℱ_∞^N - 1 ds
    		)
    		
       ≤
    		C_K
    		(
    		1
    		+
    		(ℱ_∞^N+2K+4)^2
    		)
    		 (log T)^1+K/T^1+ K 
    		T^Cℱ_∞^N.

	where the latter follows from (<ref>) and the assumption (<ref>).  If K ≥ C ℱ_∞^N, letting T→∞, it follows that the left hand side vanishes for all t, and thus (f',ϱ ',ϕ ') ≡ (f,ϱ,ϕ).





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