Small data solutions of the Vlasov-Poisson system and the vector field method
aa r X i v : . [ m a t h . A P ] A p r Small data solutions of the Vlasov-Poissonsystem and the vector field method
Jacques Smulevici ∗ August 20, 2018
Abstract
The aim of this article is to demonstrate how the vector field method of Klain-erman can be adapted to the study of transport equations. After an illustrationof the method for the free transport operator, we apply the vector field methodto the Vlasov-Poisson system in dimension 3 or greater. The main results areoptimal decay estimates and the propagation of global bounds for commutedfields associated with the conservation laws of the free transport operators, un-der some smallness assumption. Similar decay estimates had been obtainedpreviously by Hwang, Rendall and Velázquez using the method of characteris-tics, but the results presented here are the first to contain the global boundsfor commuted fields and the optimal spatial decay estimates. In dimension 4or greater, it suffices to use the standard vector fields commuting with the freetransport operator while in dimension 3, the rate of decay is such that thesevector fields would generate a logarithmic loss. Instead, we construct modifiedvector fields where the modification depends on the solution itself.The methods of this paper, being based on commutation vector fields andconservation laws, are applicable in principle to a wide range of systems, in-cluding the Einstein-Vlasov and the Vlasov-Nordström system.
Contents ∗ Laboratoire de Mathématiques, Université Paris-Sud 11, bât. 425, 91405 Orsay, France. Proof when n ≥ n ≥ Y α ( ϕ ) Y β ( f ) . . . . . . . . . . . . . . . . . 356.6 Improving the bootstrap assumptions . . . . . . . . . . . . . . . . . . . . 40 A standard approach to the study of asymptotic stability of stationary solutions ofnon-linear evolution equations consists in an appropriate linearization of the sys-tem together with1. a robust method for proving decay of solutions to the linearized equations,2. an appropriate set of estimates for the non-linear terms of the original system,using the linear decay estimates obtained previously.For systems of non-linear wave equations such as the Einstein vacuum equations Ri c ( g ) =
0, several methods for proving decay of solutions to the linear wave equa-tion ä ψ = ä = − ∂ t + n X i = ∂ x i is the wave operator of the flat Minkowski space are a priori available. One of the classical methods to derive decay estimates is to usean explicit representation of the solutions, such as the Fourier representation, to-gether with specific estimates for singular or oscillatory integrals. While this methodprovides very precise estimates on the solutions, it does not seem sufficiently robustto be applicable to quasilinear system of wave equations such as the Einstein equa-tions, and the method of choice for proving decay in view of such applications is thecommutation vector field method of Klainerman [7] and its extensions using mul-tiplier vector fields, see for instance [13, 14, 3]. The method of Klainerman is basedon 1. A coercive conservation law: the standard energy estimate in the case of thewave equation.2. Commutation vector fields: these are typically associated with the symmetriesof the equations. In the case of the wave equation, these are the Killing andconformal Killing fields of the Minkowski space. A third ingredient not needed in the present case is that of modulation theory, see for instance [8] foran application of modulation theory in the context of the Vlasov-Poisson system. In case of perbutations around a non-flat solution with metric g , the operator ä would naturally bereplaced by ä g , the wave operator of the metric g . More recently, a mix of microlocal and vector field methods have also been successfully developped,in particular to handle complex geometries involving trapped trajectories, see for instance [12] for anapplication of these tools.
2. Weighted vector field idendities and weighted Sobolev inequalities: the usualvector fields ∂ t , ∂ x i are rewritten in terms of the commutation vector fields.The coefficients involved in these decompositions contain weights in t and | x | and the presence of these weights leads to weighted Sobolev inequalities,that is to say decay estimates.The typical method used in the study of the Vlasov-Poisson and other systemsof transport equations such as the Vlasov-Nordström system is the method of char-acteristics. This is an explicit representation of the solutions and thus, in our opin-ion, should be compared with the Fourier representation for solutions of the waveequation. What would then be the analogue of the vector field method for trans-port equations? The aim of this article is twofold. First, we will provide a vectorfield method for the free transport operator. In fact, in a joint work with J. Joudiouxand D. Fajman, we have developped a vector field approach to decay of averagesnot only for the free (non-relativistic) transport operator but also for the massiveand massless relativistic transport operators, see [4]. In this paper, we will give twodifferent proofs of Klainerman-Sobolev inequalities. The easier proof will give usa decay estimate for velocity averages of sufficiently regular distribution functions,i.e. quantities such as R v ∈ R n f ( t , x , v ) d v . However, this proof fails in the case of ve-locity averages of absolute values of distribution functions, i.e. quantities such as R v ∈ R n | f | ( t , x , v ) d v , because higher derivatives of | f | will typically not lie in L evenif f is in some high regularity Sobolev space. On the other hand, the decay estimateobtained via the method of characteristics can be applied equally well to f and | f | .We shall therefore give a second proof of Klainerman-Sobolev inequalities for veloc-ity averages which will be applicable to absolute values of regular distribution func-tions. The first approach, which is closer to the standard proof of the Klainerman-Sobolev inequality for wave equations, consists essentially of two steps, a weightedSobolev type inequality for functions in L x and an application of this inequality tovelocity averages, exploiting the commutation vector fields. The improvement inthe second approach comes from mixing the two steps together.In the second part of this paper, we will apply our method to the Vlasov-Poissonsystem in dimension n ≥ ∂ t f + v . ∇ x f + µ ∇ x φ . ∇ v f =
0, (1) ∆ φ = ρ ( f ), (2) f ( t = = f . (3)where µ = ± ∆ = − n X i = ∂ x i , f = f ( t , x , v ) with t ∈ R , x , v ∈ R n , f is a sufficientlyregular function of x , v and ρ ( f ) is given by ρ ( f )( t , x ) : = Z v ∈ R n f ( t , x , v ) d n v .Our main result can be summarized as follows (a more precise version is givenin Section 4.2). Theorem 1.1.
Let n ≥ and N ≥ n + if n ≥ and N ≥ if n = . Let < δ < n − n + .Then, there exists ǫ > such that for all ≤ ǫ ≤ ǫ , if E N , δ [ f ] ≤ ǫ , where E N , δ [ f ] isa norm containing up to N derivatives of f , then the classical solution f ( t , x , v ) of See Section 4.1 for a precise definition of the norms. The δ encodes some additional integrabilityproperties of the solutions. - (3) exists globally in time and satisfies the estimates, ∀ t ∈ R and ∀ x ∈ R n ,1. Global bounds E N , δ [ f ]( t ) ≤ ǫ . (4) Space and time pointwise decay of averages of ρ ( f ) for any multi-index α with | α | ≤ N − n , | ρ ( Z α f )( t , x ) | ≤ C N , n , δ ǫ (1 + | t | + | x | ) n , where Z α is a differential operator of order α obtained as a combination of | α | commuting vector fields and C N , n , δ > is a constant depending only on N , n , δ .3. Improved decay estimates for derivatives of ffor any multi-index α with | α | ≤ N − n , | ρ ( ∂ α x f )( t , x ) | ≤ C N , n , δ ǫ (1 + | t | + | x | ) n +| α | . Boundedness of the L + δ norms of ∇ φ and ∇ Z α φ for any multi-index α with | α | ≤ N , ||∇ Z α φ ( t ) || L + δ ( R n ) ≤ C N , n , δ ǫ . Space and time decay of the gradient of the potential and its derivatives for any multi-index α with | α | ≤ N − (3 n /2 + |∇ Z α φ ( t , x ) | ≤ C N , n , δ ǫ t ( n − (1 + | t | + | x | ) n /2 , as well as the improved decay estimates | ∂ α x ∇ φ ( t , x ) | ≤ C N , n , δ ǫ t ( n − (1 + | t | + | x | ) n /2 +| α | . Remark 1.1.
Stronger bounds can be propagated by the equations provided the dataenjoy additional integrability conditions. More precisley, the improved decay esti-mates for derivatives of ρ ( f ) can be improved to | ρ ( ∂ τ t ∂ α x f )( t , x ) | ≤ C N ǫ (1 + | t | + | x | ) n +| α |+ τ and the improved decay estimates for derivatives of the gradient of φ can be improvedto | ∂ τ t ∂ α x ∇ φ ( t , x ) | ≤ C N ǫ t ( n − (1 + | t | + | x | ) n /2 +| α |+ τ , the point being that additional t derivatives now bring additional decay in t and | x | .These stronger estimates hold provided the initial data have stronger decay in x , vthan what is needed in the proof of Theorem 1.1. Similarly, one can propagates L p norms with p ≥ for ∇ φ and ∇ Z α φ provided additional v decay of the initial data isassumed. Under some mild conditions on the initial data, global existence is already guaranteed from the works[17, 11], so the main points of the theorem, apart from providing an illustration of our new method, arethe propagation of the global bounds and the optimal space and time decay estimates for the solutions. emark 1.2. Similar time decay estimates have been obtained in [6] for derivativesof ρ ( f ) and φ using the method of characteristics under different assumptions on theinitial data. On the other hand, the optimal decay rates in space and the propagationof the global bounds (4) were, as far as we know, not known prior to our work. Remark 1.3.
As is clear from the proof below and is typical of strategies based on com-mutation formulae and conservation laws, the method is very robust. In particular,we are not using the method of characteristics, nor the conservation of the total energyfor the system (1) - (3) . An illustration of this robustness will be given in [4] where wewill apply a similar approach to the study of the Vlasov-Nordström system. Previous work on the Vlasov-Poisson system and discussion
There exists a large litterature on the Vlasov-Poisson system. We refer to the intro-duction in [15] for a good introduction to the subject and only quote here the mostimportant results from the point of view of this article. In the pioneered work [1],small data global existence in dimension 3 for the Vlasov-Poisson system was es-tablished together with optimal time decay rates for ρ ( f ) and ∇ φ but no decay wasobtained for their derivatives. The optimal time (but not spatial) decay rates forderivatives of ρ ( f ) and ∇ φ has been only much later obtained in [6], covering at thesame time all dimensions n ≥
3. Both these works use decay estimates obtainedvia the method of characteristics. In fact, in [1] and even more in [6], precise es-timates on the deviation of the characteristics from the characteristics of the freetransport operator are needed in order to obtain the desired decay estimates. Par-allely to these works giving information on the asymptotics of small data solutions,let us mention that under fairly weak assumption on the initial data (in particular,no smallness assumption is needed), it is known that global existence holds in di-mension 3 for the solutions of (1)-(3), see [17, 11]. The strongest results concerningthe stability of non-trivial stationnary solutions of (1)-(3) with µ = − .We believe that, once again, the vector field method would be totally appropriate forthe derivation of such decay estimates. Finally, let us mention the celebrated work[16] on Landau damping concerning the stability of stationnary solutions to (1)-(3)with periodic initial data. In view of the present work, it will be interesting to try torevisit this question using vector field methods. Outline of the paper
In Section 2, we introduce the vector fields commuting with the free transport op-erator and the notations that we will use throughout the paper. In Section 3, wepresent and prove decay estimates for velocity averages of solutions to the transportequation. In the following section, we present our results on the Vlasov-Poisson sys-tem. The remaining last two sections are devoted to the proof of these results, firstin dimension n ≥ See for instance [5] for some stability results using the linearization approach in the case of thespherically-symmetric King model. cknowledgements This project was motivated by my joint work with David Fajman and Jérémie Joudiouxon relativistic transport equations. I would like to thank both of them, as well asChristophe Pallard and Frédéric Rousset for many interesting discussions on thesetopics. I would also like to thank Pierre Raphaël for an extremely stimulating con-versation which took place during the conference “Asymptotic analysis of dispersivepartial differential equations” held in October 2014 at Pienza, Italy. Some of this re-search was done during this conference. Finally, I would like to acknowledge partialfunding from the Agence Nationale de la Recherche ANR-12-BS01-012-01 (AARG)and ANR SIMI-1-003-01.
Throughout this article, f will denote a sufficiently regular function of ( t , x , v ) with t ∈ R and ( x , v ) ∈ R n × R n . By sufficiently regular, we essentially mean that f is suchthat all the terms appearing in the equations make sense as distributions and thatall the norms appearing in the estimates are finite. For simplicity, the reader mightjust assume that f is smooth with compact support in x , v (but any sufficient fall-offwill be enough).We will denote by T the free transport operator i.e. T ( f ) : = ∂ t f + n X i = v i ∂ x i f ,where ∂ t f = ∂ f ∂ t and for all 1 ≤ i ≤ n , ∂ x i f = ∂ f ∂ x i . Similarly, for any sufficiently regularscalar function φ , T φ will denote the perturbed transport operator T φ ( f ) : = T ( f ) + µ ∇ x φ . ∇ v f , (5)where µ = ±
1, corresponding to an attractive or repulsive force. Since we are dealingonly with small data solutions, the sign of µ will play no role in the rest of this article.The notation A . B will be used to specify that there exists a universal constant C > A ≤ C B , where typically C will depends only on the number of di-mensions n and a few other fixed constants, such as the maximum number of com-mutations. Consider first the following set of vector fields• Translations in space and time ∂ t , ∂ x i ,• Uniform motion in one spatial direction t ∂ x i ,• Rotations x i ∂ x j − x j ∂ x i ,• Scaling in space n X i = x i ∂ x i , Recall that these vector fields are the generators of the Gallilean transformations of the form x ∈ R n → x + tv i , where v ki = δ ki . t ∂ t + n X i = x i ∂ x i .The above set of vector fields is associated with the Gallilean invariance of macro-scopic fields and equations. We will denote by Γ the set of all such vector fields Γ = ( ∂ t , ∂ x i , t ∂ x i , x i ∂ x j − x j ∂ x i , n X i = x i ∂ x i , t ∂ t + n X i = x i ∂ x i , 1 ≤ i , j ≤ n ) .One easily check that while the translations commute with T , uniform motions,rotations or the scaling in space do not. The correct replacement for these vectorfields is most easily explained using the language of differential geometry; the inter-ested reader may consult [18, 4] for detailed constructions (in the case of the rela-tivistic transport operator). We shall here only present the resulting objects whichare the vector fields• Uniform motions in one direction in microscopic form t ∂ x i + ∂ v i ,• Rotations in microscopic form x i ∂ x j − x j ∂ x i + v i ∂ v j − v j ∂ v i ,• Scaling in space in microscopic form n X i = x i ∂ x i + v i ∂ v i .One can then easily check Lemma 2.1 (Commutation with the transport operator) . • If Z is any of the translations, microscopic uniform motions or microscopic ro-tations, then [ T , Z ] = .• If Z is the microscopic scaling in space, then [ T , Z ] = .• If Z is the scaling in space and time, then [ T , Z ] = T .
Remark 2.1.
From the two scaling commuting vector fields, it follows automaticallythat t ∂ t − P ni = v i ∂ v i also commutes with T in the sense that [ T , t ∂ t − n X i = v i ∂ v i ] = T . This vector field will be used to obtain improved decay for t derivatives of velocityaverages.
To ease the notation, we will denote by Ω xi j : = x i ∂ x j − x j ∂ x i the rotation vec-tor fields in x and by Ω vi j : = v i ∂ v j − v j ∂ v i the rotation vector fields in v . The fullmicroscopic rotation vector fields are thus of the form Ω xi j + Ω vi j . Similarly, we willdenote by S x + S v : = n X i = x i ∂ x i + v i ∂ v i the scaling in space in microscopic form, with S x = n X i = x i ∂ x i and S v = n X i = v i ∂ v i .Let now γ be the set of all the above microscopic vector fields including the trans-lations and the space and time scaling i.e. γ = ( ∂ t , ∂ x i , t ∂ x i + ∂ v i , Ω xi j + Ω vi j , S x + S v , t ∂ t + n X i = x i ∂ x i ,1 ≤ i , j ≤ n ) .7 .2 Multi-index notations Let Z i , i = n + + n ( n − Γ . For any multi-index α , wewill denote by Z α the differential operator of order | α | given by the composition Z α Z α ...In view of the above discussion, to any vector field of Γ , we can associate a uniquevector field of γ . More precisely, ∂ t → ∂ t , ∂ x i → ∂ x i , t ∂ x i → t ∂ x i + ∂ v i , Ω xi j → Ω xi j + Ω vi j , S x → S x + S v , t ∂ t + n X i = x i ∂ x i → t ∂ t + n X i = x i ∂ x i .Thus, to any ordering of Γ we can associate an ordering of γ . We will by a small abuseof notation, denote again by Z i the elements of such an ordering since it will be clearthat, if Z i is applied to a macroscopic quantity, such as a velocity average, then Z i ∈ Γ and if Z i is applied to a microscopic quantity, i.e. any function depending on ( x , v )(and possibly t ), then Z i ∈ γ . Similarly, for any multi-index α , we will also denote by Z α the differential operator of order | α | given by the composition Z α Z α .. obtainedfrom the vector fields of γ .For some of the estimates below, it will be sufficient to only consider a subsetof all the vector fields of Γ and γ . Let us thus denote by Γ s the set of all the macro-scopic vector fields apart from ∂ t and t ∂ t + n X i = x i ∂ x i , which are the only vector fieldscontaining time derivatives and by γ s the corresponding set of microscopic vectorfields, i.e. Γ s = ( ∂ x i , t ∂ x i , x i ∂ x j − x j ∂ x i , n X i = x i ∂ x i , 1 ≤ i , j ≤ n ) , γ = n ∂ x i , t ∂ x i + ∂ v i , Ω xi j + Ω vi j , S x + S v , 1 ≤ i , j ≤ n o .The notation Z α ∈ Γ | α | s (respectively Z α ∈ γ | α | s ) will be used to denote a genericdifferential operator of order | α | obtained as a composition of | α | vector fields in Γ s (respectively in γ s ). The standard notation ∂ α x will also be used to denote a differen-tial operator of order | α | obtained as a composition of | α | translations among the ∂ x i vector fields.The following lemmae can easily be checked. Lemma 2.2 (Commutation within Γ ) . For any Z α ∈ γ | α | , Z α ′ ∈ γ | α ′ | , where α , α ′ aremultindices, we have [ Z α , Z α ′ ] = X | β |≤| α |+| α ′ |− c α , α ′ β Z β , for some constant coefficients c α , α ′ β . Moreover, if Z α , Z α ′ ∈ γ s , then all the Z β of theright-hand side belongs to γ s . emma 2.3 (Commutation of Z α and weights in v ) . Let q > . For any sufficienlyregular function f of ( t , x , v ) and for any Z α ∈ γ | α | where α is a multi-index, we have ¯¯ Z α £ (1 + v ) q /2 f ¤¯¯ . (1 + v ) q /2 X | β |≤| α | ¯¯¯ Z β ( f ) ¯¯¯ . Moreover, if Z α ∈ γ | α | s then all the Z β belong to γ | β | s in the above inequality. For any integrable function f of v ∈ R n , we will denote by ρ ( f ) the quantity ρ ( f ) : = Z v ∈ R n f d n v We have the following lemma.
Lemma 2.4.
For any sufficiently regular function f of ( x , v ) we have• for all ≤ i ≤ n, ∂ x i ρ ( f ) = ρ ( ∂ x i f ), • for all t ∈ R and all ≤ i ≤ n,t ∂ x i ρ ( f ) = ρ ¡¡ t ∂ x i + ∂ v i ¢ f ¢ , • for all ≤ i , j ≤ n , Ω xi j ρ ( f ) = ρ ³³ Ω xi j + Ω vi j ´ ( f ) ´ , • for all t ∈ R and x ∈ R n , à t ∂ t + n X i = x i ∂ x i ! ρ ( f ) = ρ Ãà t ∂ t + n X i = x i ∂ x i ! ( f ) ! , • and finally S x ρ ( f ) = ρ ¡¡ S x + S v ¢ ( f ) ¢ + n ρ ( f ), where S x and S x + S v are the spatial scaling vector fields in macroscopic andmicroscopic forms.Proof. The proof is straigtforward and consist in identifying total derivatives in v .For instance, we have ρ ¡ S v ¢ ( f ) = Z v ∈ R n à n X i = v i ∂ v i f ! d n v = Z v ∈ R n − n f d n v ,where we have integrated by parts in each of the v i . Similarly, in the case of rota-tions, it suffices to note that for any 1 ≤ i , j ≤ n , Ω vi j is an angular derivative in v andthefore, R v ∈ R n Ω vi j ( f ) d n v = (cid:3) In the remainder of this paper, we shall write the preceding lemma as Z ρ ( f ) = ρ ( Z ( f )) + c Z ρ ( f ),where we are using, by a small abuse of notation, the letter Z to denote a genericmacroscopic vector field and its corresponding microscopic version and where c Z = Z is the spatial scaling vector field, in which case c Z = n .9 .4 Vector field identities The following well-known identity will be used later
Lemma 2.5.
For any ≤ j ≤ n, we have | x | ∂ x j = n X i = x i Ω xi j + x j S x , (6) and thus, at any x , , | x | ∂ x j = n X i = x i | x | Ω xi j + x j | x | S x , where the coefficients x i | x | are all homogeneous of degree and therefore uniformlybounded. The following higher order version will be used often in the derivation of theKlainerman-Sobolev inequalities of the next section.
Lemma 2.6.
For any multi-index α , ( t + | x | ) α ∂ α x = X | β |≤| α | , Z β ∈ Γ | β | s C β Z β , where the coefficients C β are all uniformly bounded.Proof. The lemma is a consequence the previous decomposition, the fact t ∂ x i ispart of our algebra of commuted vector fields and that £ ∂ x i , t + | x | ¤ is homogeneousof degree 0. (cid:3) We now turn to the study of the transport operator T φ defined by (5). Many of the es-timates below are only valid provided φ has sufficient regularity. In the applicationsto the Vlasov-Poisson system of this article, we will eventually control the regularityof φ via a bootstrap argument. For all the estimates below, we therefore assume that φ is a sufficiently regular function of ( t , x ) defined on [0, T ] × R nx , for some T > | x | → +∞ .The following lemma can then easily be checked. Lemma 2.7.
Let f be a sufficiently regular function of (t,x,v) and let α be a multi-index. Then, there exists constant coefficients C αβγ such that, [ T φ , Z α ] f = X | β |≤| α |− X | γ |+| β |≤| α | C αβγ ∇ x Z γ φ . ∇ v Z β f . Moreover if Z α ∈ Γ | α | s , then Z γ ∈ Γ | γ | s and Z β ∈ γ | β | s in the above decomposition. Similarly one has For instance, one can assume that φ is a smooth function on [0, T ] × R nx with compact support in x . emma 2.8. Let f be a sufficiently regular function of ( t , x , v ) and let ψ be the solutionto the Poisson equation ∆ ψ = ρ ( f ) . Then, for any multi-index α , Z α ψ is solution toan equation of the form ∆ Z α ψ = X | β |≤| α | C αβ Z β ρ ( f ), (7) where C αβ are constants.Proof. The lemma is an easy consequence of the fact that all macroscopic vectorfields apart from the two scalings commute with ∆ while for the spatial scaling S x and the space-time scaling t ∂ t + S x we have [ ∆ , S x ] = ∆ and [ ∆ , t ∂ t + S x ] = ∆ . (cid:3) We shall use the following (approximate) conservation laws.
Lemma 2.9.
For any sufficiently regular function f of ( t , x , v ) , we have, for all t ∈ [0, T ] , || f ( t ) || L ( R nx × R nv ) ≤ || f (0) || L ( R nx × R nv ) + Z t || T φ ( f )( s ) || L ( R nx × R nv ) d s . Similarly, we have for all p ≥ , for all t ∈ [0, T ] , || f ( t ) || pL p ( R nx × R nv ) ≤ || f (0) || pL p ( R nx × R nv ) + p Z t || f p − T φ ( f )( s ) || L ( R nx × R nv ) d s , and for all q ≥ , || (1 + v ) q /2 f p ( t ) || L ( R nx × R nv ) . || (1 + v ) q /2 f p (0) || L ( R nx × R nv ) + Z t || (1 + v ) ( q − f p − T φ ( f )( s ) || L ( R nx × R nv ) d s + Z t || f p (1 + v ) ( q − ∂ x φ ( s ) || L ( R nx × R nv ) d s . (8) Note in particular that the conclusions of the lemma hold true when T φ = T (i.e. when ∂ x φ = ).Proof. These are classical estimates so we only sketch their proofs.One has (in the sense of distribution), T φ £ (1 + v ) q /2 | f | p ¤ . (1 + v ) q /2 | f | p − | T φ ( f ) | + (1 + v ) ( q − | ∂ x φ || f | p .Using a standard procedure , one can regularize the previous inequality. We shalltherefore neglect regularity issues here. Integrating the previous line in ( t , x , v ) leadsto Z t Z x Z v T φ £ (1 + v ) q /2 | f | p ¤ d vd xd s . Z t Z x Z v £ (1 + v ) q /2 | f | p − | T φ ( f ) | + (1 + v ) ( q − | ∂ x φ || f | p ¤ d vd xd s . (9) For instance, assume first that f has compact support in ( x , v ) with a uniform bound on the supportof f in ( x , v ) for t ∈ [0, T ]. For all ǫ >
0, consider the function f ǫ = q ǫ + f χ ( x , v ), where χ ( x , v ) is asmooth cut-off function which is 1 on the support of f and vanishes for large x and v . Apply then theprevious estimates to f ǫ and take the limit ǫ →
0. A standard density argument deals with the case ofnon-compact support.
11n the other-hand, remembering that T φ = ∂ t + P ni = v i ∂ ix + µ ∇ x φ . ∇ v and integratingby parts in x and v , we obtain Z t Z x Z v T φ £ (1 + v ) q /2 | f | p ¤ d vd xd s = || (1 + v ) q /2 f p ( t ) || L ( R nx × R nv ) − || (1 + v ) q /2 f p (0) || L ( R nx × R nv ) ,which combined with (9) leads to the desired estimate (8). (cid:3) Since the main purpose of this article is to illustrate how the vector field method canlead to robust decay estimates for velocity averages, let us for the sake of comparisonrecall the Bardos-Degond decay estimate and its proof.
Proposition 3.1 ([1]) . Let f be a sufficiently regular solution of T ( f ) = . Then, wehave the estimate, for all t > and all x ∈ R n , | ρ ( f ) | ( t , x ) ≤ t n Z x ∈ R n sup v ∈ R n | f (0, x , v ) | d n x . (10) Proof.
The proof of this classical estimate is based on the method of charateristics.More precisly, if f is a regular solution to T ( f ) =
0, then it follows that f ( t , x , v ) = f (0, x − v t , v ),for all t ∈ R , x , v ∈ R n .We then have | ρ ( f ) | ( t , x ) = ¯¯¯¯Z v ∈ R n f ( t , x , v ) d n v ¯¯¯¯ = ¯¯¯¯Z v ∈ R n f (0, x − v t , v ) d n v ¯¯¯¯ ≤ Z v ∈ R n sup w ∈ R n | f (0, x − v t , w ) | d n v .Applying now the change of coordinates y = v t for t > | ρ ( f ) | ( t , x ) ≤ t n Z y ∈ R n sup w ∈ R n | f (0, x − y , w ) | d n y = t n Z x ∈ R n sup v ∈ R n | f (0, x , v ) | d n x . (cid:3) In the above proof, the two key ingredients are• the explicit representation obtained via the method of characteristics,• the change of variables y = v t . 12ote that in the presence of a perturbation of the free transport operator, to exploit asimilar change of variables would require estimates on the Jacobian associated withthe differential of the characteristic flow, see [1, 6].Let us now show how the vector field method can be used as an alternative toobtain similar decay estimates. As explained in the introduction, we will give twodifferent proofs.The first proof will give us decay estimates for quantities of the form R v ∈ R n f d v .The starting point of this approach is the following Klainerman-Sobolev inequalityusing L ( R n ) norms of commuted fields. Lemma 3.1 ( L Klainerman-Sobolev inequality) . For any sufficiently regular func-tion ψ defined on R nx , we have | ψ | ( x ) . + t + | x | ) n X | α |≤ n , Z α ∈ Γ s || Z α ( ψ ) || L ( R nx ) . Proof.
This is relatively standard material and we adapt here the presentation givenin [19] Chap.2 to our setting.Fix ( t , x ) ∈ R t × R nx and let e ψ be the function e ψ : R n → R (11) y → e ψ ( y ) : = ψ ( x + ( t + | x | ) y ). (12)Applying a standard Sobolev inequality, we have | ψ ( t , x ) | = | e ψ (0) | . X | α |≤ n || ∂ α y e ψ || L ( B n (0,1/2)) , (13)where B n (0,1/2) denote the ball in R ny of radius 1/2. On the other hand, we have ∂ y i e ψ ( y ) = ( t + | x | ) ∂ x i ψ ( x + ( t + | x | ) y ) and thus, for y ∈ B n (0,1/2), | ∂ α y e ψ ( y ) | . X | β |≤| α | ( t + | x | ) β ¯¯¯ ∂ β x ψ ¡ x + ( t + | x | ) y ¢¯¯¯ , . X | β |≤| α | ( t + | x + ( t + | x | ) y | ) β ¯¯¯ ∂ β x ψ ¡ x + ( t + | x | ) y ¢¯¯¯ , . X | β |≤| α | , Z β ∈ Γ s ¯¯¯ Z β ψ ¡ x + ( t + | x | ) y ¢¯¯¯ ,where we have used Lemma 2.6 in the last step and the fact that t + | x | and t + ¯¯ x + ( t + | x | ) y ¯¯ are comparable for y ∈ B n (0,1/2). Inserting the last line in the Sobolevinequality (13) and applying the change of variables z = ( t +| x | ) y conclude the proofof the lemma. (cid:3) Using Lemma 2.4, we now note that for any vector field Z and any sufficientlyregular function f of x , v , we have || Z ( ρ ( f )) || L ( R nx ) = || ρ ( Z ( f )) + c Z ρ ( f ) || L ( R nx ) and thus, || Z ( ρ ( f )) || L ( R nx ) . || Z ( f ) || L ( R nx × R nv ) + || ρ ( f ) || L ( R nx × R nv ).Combined with the previous Klainerman-Sobolev inequality, we obtain Recall that, while the general W k , p ( R n ) , → L ∞ ( R n ) embedding requires k > np , the special case p = k ≥ n . roposition 3.2 (Global Klainerman-Sobolev inequality for velocity averages) . Forany sufficiently regular function f defined on R nx × R nv , we have, for all t > and allx ∈ R n , | ρ ( f ) | ( x ) . + t + | x | ) n X | α |≤ n , Z α ∈ γ | α | s || Z α f || L ( R x × R v ) . (14)Note that the above inequality cannot be apply to | f | even is f is say a smooth,compactly supported function. Indeed, if | f | is say in W n , p , then | f | is in W p but,unless f has some extra special properties, | f | ∉ W n , p when n ≥
2. On the otherhand, (10) clearly holds both for f and for | f | .Two disctinct steps lead to the proof of (14), the L Klainerman-Sobolev inequal-ity of Lemma 3.1 and the special commutation properties of the velocity averagingoperator as described in Lemma 2.4. To improve upon (14), the strategy is to tryto use at the same time arguments similar to those of Lemma 3.1 and Lemma 2.4,instead of applying them one after the other. This will lead to us to the followingimprovement.
Proposition 3.3 (Global Klainerman-Sobolev inequality for velocity averages of ab-solute values) . For any sufficiently regular function f defined on R nx × R nv , we have,for all t > and all x ∈ R n , | ρ ( | f | ) | ( x ) . + t + | x | ) n X | α |≤ n , Z α ∈ γ | α | s || Z α f || L ( R x × R v ) . (15) Proof.
Let us assume that f is smooth and compactly supported for simplicity.Define e ψ similarly to (11) as e ψ : B n (0,1/2) → R y → e ψ ( y ) : = Z v ∈ R n | f | ( x + ( t + | x | ) y , v ) d n v .Next, recall that for any ψ ∈ W , | ψ | ∈ W and ∂ | ψ | = ψ | ψ | ∂ψ (in the sense of distri-bution, see for instance [9], Chap 6.17) so that in particular ¯¯ ∂ | ψ | ¯¯ ≤ | ∂ψ | . Let us write y = ( y ,.., y n ) and let δ = n . Using a 1 dimensional Sobolev inequality, we have | e ψ (0) | . Z | y |≤ δ ¡¯¯ ∂ y e ψ ( y ,0,..,0) ¯¯ + ¯¯ e ψ ( y ,0,..,0) ¯¯¢ d y . (16)Now, ∂ y e ψ ( y ,0,..,0) = Z v ∈ R n ( t + | x | ) ∂ x | f | ¡ x + ( t + | x | )( y ,0,..,0) ¢ d n v , = Z v ∈ R n t ∂ x | f | ¡ t , x + ( t + | x | )( y ,0,..,0) ¢ d n v + Z v ∈ R n | x | ∂ x | f | ¡ x + ( t + | x | )( y ,0,..,0) ¢ d n v . (17)As before, we can introduce total derivatives in v . For instance, Z v ∈ R n t ∂ x | f | ¡ x + ( t + | x | )( y ,0,..,0) ¢ d n v = Z v ∈ R n ¡ t ∂ x + ∂ v ¢ | f | ¡ x + ( t + | x | )( y ,0,..,0) ¢ d n v , ≤ Z v ∈ R n ¯¯¡ t ∂ x + ∂ v ¢ f ¡ t , x + ( t + | x | )( y ′ ,0,..,0) ¢¯¯ d n v ,14nd, using that ( y ,0,..,0) ∈ B n (0,1/2) if | y | ≤ δ , a similar argument can be usedto handle the second term on the right-hand side of (17). This gives us ¯¯ ∂ y e ψ ( y ,0,..,0) ¯¯ ≤ X Z s ∈ γ s Z v ∈ R n ¯¯ Z f ¡ x + ( t + | x | )( y ,0,..,0) ¢¯¯ d n v .Combined with (16), we have obtained that | e ψ (0) | . X Z s ∈ γ s Z | y |≤ δ Z v ∈ R n ¯¯ Z f ¡ x + ( t + | x | )( y ,0,..,0) ¢¯¯ d n vd y + Z | y |≤ δ Z v ∈ R n ¯¯ f ¡ x + ( t + | x | )( y ,0,..,0) ¢¯¯ d n vd y .We now repeat the same argument varying the variable y . | e ψ (0) | . Z | y |≤ δ Z | y |≤ δ X | α |≤ Z α ∈ γ | α | s Z v ∈ R n ¯¯ Z α f ¡ x + ( t + | x | )( y , y ,0,..,0) ¢¯¯ d n vd y d y .Iterating again this argument until all variables y i appears in the integral of the right-hand side, we obtain | e ψ (0) | . Z y ∈ B n (0,1/2) X | α |≤ n , Z α ∈ γ | α | s Z v ∈ R n ¯¯ Z α f ¡ x + ( t + | x | ) y ¢¯¯ d n vd y ,and the conclusion of the proof follows as in the proof of Lemma 3.1 by the changeof variable z = ( t + | x | ) y . (cid:3) We now recall that if f is a solution to T ( f ) =
0, then, in view of the commutationproperties of Lemma (2.1) and standard properties of differentiation of the absolutevalue, so are | f | and the commuted fields | Z α f | . Thus, from Lemma 2.9, all thenorms on the right-hand side of (15) are preserved by the flow and we have obtainedthe decay estimate Proposition 3.4 (Decay estimates for velocity averages) . For any sufficiently regularsolution f to T ( f ) = , we have, for all t > and all x ∈ R n , | ρ ( | f | ) | ( t , x ) . + t + | x | ) n X | α |≤ n , Z α ∈ γ | α | s || Z α f ( t = || L ( R x × R v ) . (18) Remark 3.1.
If we restrict the set of vector fields only to the uniform motions t ∂ x i + ∂ v i ,then we still obtain the time decay estimate, | ρ ( | f | ) | ( t , x ) . t n X | α |≤ n , Z α i = t ∂ x j + ∂ v j || Z α f ( t = || L ( R x × R v ) . (19) Now, note that the vector fields of the form t ∂ x j + ∂ v j degenerate to ∂ v j when evalu-lated at t = . Since from the Sobolev inequality, we have that || f ( x ,.) || L ∞ ( R nv ) . X | α |≤ n || ∂ α v f ( x ,.) || L ( R v ) ,15 t follows that (19) is striclty weaker than the inequality (10) . In some sense, the factthat our method gives us a slightly worse estimate reflects its robustness and thus itsappropriateness to deal with non-linear problems. Remark 3.2.
Estimates similar to (18) hold for the relativistic transport operator T m = p m + | v | ∂ t + P ni = v i ∂ x i , where m = for massless particles and m > for massiveparticles. These estimates are presented in [4] together with non-linear applicationsto the Vlasov-Nordström system. Interestingly, in the case of the relativistic transportoperator, the estimates obtained have additional benefits. Decay estimates for rela-tivistic operators in the style of the Bardos-Degond estimate (10) typically require theextra assumptions of compact support in v of the solution, while for the estimatesobtained via the vector field method, only the finiteness of the L x , v norms of the com-muted fields are required. See [4]. As is classical, derivatives enjoy better decay properties as follows Proposition 3.5 (Improved decay estimates for derivatives) . For any sufficiently reg-ular function f defined on R nx × R nv , we have, for all x ∈ R n and all multi-index α | ρ ( ∂ α x ( f )) | ( x ) . + t + | x | ) n +| α | X | β |≤ n +| α | , Z β ∈ γ | β | s || Z β f || L ( R x × R v ) , Similarly, for any sufficiently regular function f defined on R t × R nx × R nv , we have, forall t ≥ all x ∈ R n and all τ ∈ N , | ρ ( ∂ α x ∂ τ t ( f )) | ( t , x ) . + t + | x | ) n +| α | (1 + t ) τ X | β |≤ n + τ +| α | , Z β ∈ γ | β | || Z β f || L ( R x × R v ) , Proof.
The first part of the proposition is a consequence of Lemma 2.6, Lemma 2.4and the global Klainerman Sobolev inequality (14).The second part of the proposition follows similarly, using that t ∂ t ρ ( f ) = ρ ( t ∂ t f ) = ρ " t ∂ t f − n X i = v i ∂ v i f − n f with t ∂ t − P ni = v i ∂ v i being a linear combination of commuting vector fields as ex-plained in Remark 2.1. (cid:3) Remark 3.3.
In the above proposition, additional t derivatives yield only additionalt decay and no improvement in terms of | x | decay. This improvement can also beachieved assuming stronger decay in v of the initial data. More precisely, if f is asolution to T ( f ) = then ∂ t f = − n X i = v i ∂ x i f = n X i = ∂ x i ( − v i f ). Note now that if f is a solution, so is v i f , for any v i . Thus, for all i , R v ∂ x i ( − v i f ) d venjoys the additional decay ( t + | x | ) as stated in Proposition 3.5, provided that the L norms of Z α ( v i f ) are finite for | α | ≤ n + . Iterating the procedure, we obtain thateach ∂ t derivatives gives an additional decay of ( t + | x | ) . In that case, the decay estimates are worse near the cone t = | x | , as for the wave equation, see again[4]. Note that however, quantities such as ρ ¡¯¯ ∂ x f ¯¯¢ do not typically enjoy any additional decay. L based Klainerman-Sobolev inequality to estimate ∇ Z α φ pointwise, for φ asolution to the Poisson equation (2). The estimate that we will used is contained inthe following lemma. Lemma 3.2.
For any sufficiently regular function of ψ defined on R nx , we have, for allt ≥ , | ψ ( x ) | . + t + | x | ) n /2 X | α |≤ ( n + Z α ∈ Γ s || Z α ( ψ ) || L ( R nx ) . Proof.
The proof is similar to that of Lemma 3.1, considering the function e ψ : y → ψ ( x + ( t +| x | ) y ) and replacing the L Sobolev inequality with the L Sobolev inequal-ity | ψ ( x ) | = | e ψ (0) | . X | α |≤ ( n + Z B n (0,1/2) | ∂ α ( e ψ ) | d y . (cid:3) The vector fields presented in Section 2.1 and the resulting decay estimates of theprevious section will be sufficient to prove global existence of solutions to the Vlasov-Poisson system and derive their asymptotics for all dimension n ≥
4. On top of theboundedness of the L norms of commuted fields Z α ( f ), which are needed in orderto obtain pointwise decay from the vectorfield method, we will also need a little bitof additional integrability to prove the L p boundedness of the gradient of the com-muted potentials ∇ Z α φ .With this in mind, for any n ≥ N ∈ N and δ >
0, let us consider, for any suffi-ciently regular function g of ( x , v ), the norm E N , δ defined by E N , δ [ g ] : = X | α |≤ N , Z α ∈ γ | α | s || Z α ( g ) || L ( R nx × R nv ) + X | α |≤ N , Z α ∈ γ | α | s || (1 +| v | ) δ ( δ + n )2(1 + δ ) Z α ( g ) || L + δ ( R nx × R nv ) .As we shall see below, applying a similar strategy would fail in dimension 3 dueto the lack of sufficiently strong decay. In order to close the estimates, it will thus benecessary to improve the commutation relation between our commutation vectorfields and our perturbed transport operator. This led us to the introduction of mod-ified vector fields, denoted Y (and Y α for a combination of | α | such vector fields)below. Our main results are then similar to the n ≥ Z vectorfields by the Y ones. In dimension 3, the norm E N , δ will therefore be defined as E N , δ [ g ] : = X | α |≤ N , Y α ∈ γ | α | m , s || Y α ( g ) || L ( R x × R v ) + X | α |≤ N , Y α ∈ γ | α | m , s || (1 +| v | ) δ ( δ + + δ ) Y α ( g ) || L + δ ( R x × R v ) .(20)The precise definitions of the modified vector fields Y and of the algebra γ m , s aregiven in Section 6.2. 17 emark 4.1. In order to close our main estimates, we will not need to commute withany vector field containing t derivatives, i.e. commuting with vector fields in γ s ifn ≥ (respectively γ m , s if n = ) will be sufficient. Commutations with vector fieldscontaining t derivatives are of course usefull if one wants to obtain decay estimatesof t-derivatives of ρ ( f ) and ∇ φ . In that case, one would simply modify the norms,replacing the algebra γ s (respectively γ m , s ) by the algebra γ (respectively γ m ). Theinterested reader can then verify that all the arguments below still hold. For thesereasons, we will sometimes omit in the following section to specify whether the vectorfields considered lie in γ (respectively γ m ) or in γ s (respectively γ m , s ). Our main results are the following
Theorem 4.1.
Let n ≥ , < δ < n − n + , and N ≥ n /2 + if n ≥ , N ≥ if n = . Then,there exists ǫ > such that for all < ǫ < ǫ , if E N , δ [ f ] ≤ ǫ , then the classical solutionf ( t , x , v ) of (1) - (3) exists globally in time and satisfies the estimates1. Global bounds ∀ t ∈ R , E N , δ [ f ( t )] . ǫ .
2. Space and time pointwise decay of averages of ffor any multi-index α of order | α | ≤ N − n , | ρ ( Z α f )( t , x ) | . ǫ (1 + | t | + | x | ) n , as well as the improved decay estimates | ρ ( ∂ α x f )( t , x ) | . ǫ (1 + | t | + | x | ) n +| α | .
3. Boundedness for L + δ norms of ∇ Z α φ for any multi-index α of order | α | ≤ N , ||∇ Z α φ || L + δ ( R n ) . ǫ .
4. Space and time decay of the potential and its derivativesfor any multi-index α with | α | ≤ N − (3 n /2 + | Z α ∇ φ ( t , x ) | . ǫ t ( n − (1 + | t | + | x | ) ( n )/2 as well as the improved decay estimates | ∂ α x ∇ φ ( t , x ) | . ǫ t ( n − (1 + | t | + | x | ) n /2 +| α | . Finally, all the constants in the above inequalities depend only on N , n , δ . As explained above, when n ≥
4, the proof is easier and can be performed usingonly the commuting vector fields of the free transport operator while in the n = n ≥ but are not necessary there. In order tobetter explain the general framework, we will treat first the dimension n ≥ n = An interesting question left open by our work is to understand what happen in dimension 1 and 2where decay is even sparser than in dimension 3. We believe that at least in dimension 2, a carefullanalysis using modified vector fields would lead to similar conclusions. We hope to treat this in futurework. Proof when n ≥ In this section, we assume that n ≥
4, that the assumptions of Theorem 4.1 are sat-isfied for some initial data f and we denote by f the classical solution of (1)-(3)arising from f . We will assume the following bootstrap assumption on the norm of the solution E N , δ [ f ( t )]. Let T ≥ ∀ t ∈ [0, T ], E N , δ [ f ( t )] ≤ ǫ . (21)It follows from the smallness assumptions on E N , δ [ f ] and a continuity argu-ment that T > Applying our decay estimate (18) to ¯¯ Z α f ¯¯ and ¯¯ Z α f ¯¯ p (1 + v ) q /2 as well as the im-proved decay estimates of Proposition 3.5, we automatically obtain from our boot-strap assumption (21) Lemma 5.1.
For any multi-index α of order | α | ≤ N − n and for all t ∈ [0, T ] , ¯¯ ρ ¡¯¯ Z α f ¯¯¢ ( t , x ) ¯¯ ≤ C n ǫ (1 + | t | + | x | ) n , (22) ¯¯¯ ρ ³¯¯ Z α f ¯¯ + δ (1 + v ) δ ( δ + n )2 ´ ( t , x ) ¯¯¯ ≤ C n ǫ + δ (1 + | t | + | x | ) n , (23) for some constant C n > depending on n, as well as the improved decay estimates, ¯¯ ρ ¡ ∂ α f ¢ ( t , x ) ¯¯ ≤ C n ǫ (1 + | t | + | x | ) n +| α | . Z α ( φ )From standard elliptic estimates, we can also bound an L p norm of ∇ Z α φ Lemma 5.2.
For any multi-index α with | α | ≤ N and for all t ∈ [0, T ] , ||∇ Z α φ ( t ) || L + δ ( R n ) ≤ C N , n , δ ǫ , where C N , n , δ > is a constant depending on N , n and δ . Remark 5.1.
There is obviously no difficulty in propagating higher L p norms of ∇ Z α φ provided the initial data for f satisfy additional integrability and decay in v. We willnot need them to close the estimates of our main theorem, which is why we did notassume the initial bounds on these L p norms.Proof. Let p = + δ . From the commuted equation for Z α φ and the Calderón-Zygmund inequality, we have ||∇ Z α φ || L p ( R n ) . || Z α ( ρ ( f )) || L p ( R n ) , . X | β |≤| α | || ρ ³¯¯¯ Z β ( f ) ¯¯¯´ || L p ( R n ) ∇ Z α φ follow if we can prove L p bounds onthe ρ ¡¯¯ Z β f ¯¯¢ . For this, let us note that for any weight function χ ( v ), we have, usingthe Hölder inequality with 1/ p + q = Z x µZ v | Z β ( f ) | d v ¶ p d x ≤ Z x µZ v χ ( v ) d v ¶ p / q d v Z v χ ( v ) p / q | Z β ( f )) | p d vd x .Thus, we need χ ( v ) to be integrable in v and we choose χ ( v ) = (1 + | v | ) ( δ + n )/2 . Thelemma then follows from the bound on E N , δ [ f ( t )] noting that with p = + δ , we have p / q = δ . (cid:3) Applying the Gagliardo-Niremberg inequality, we have immediately
Corollary 5.1.
Let q = n (1 + δ ) n − (1 + δ ) . For all t ∈ [0, T ] , ||∇ Z α φ ( t ) || L q ( R n ) ≤ C N , n , δ ǫ .Applying L estimates for solutions to the Poisson equation (7) and the previouspointwise estimates on ρ ¡¯¯ Z α ( f ) ¯¯¢ , we can also obtain L decay estimates for ∇ Z α φ provided | α | is not too large. Lemma 5.3.
For all multi-index α such that | α | ≤ N − n ||∇ Z α φ || L ( R n ) . ǫ t ( n − . Proof.
Multiply the Poisson equation satisfied by Z α φ by Z α φ and integrate by partsto obtain ||∇ Z α ( φ ) || L ( R n ) = − Z x ∈ R n Z α ( φ ) Z α ( ρ ( f )) d x ≤ || Z α ( φ ) || L nn − ( R n ) || Z α ( ρ ( f )) || L nn + ( R n ) .Using the Gagliardo-Nirenberg inequality || ψ || L nn − ( R n ) . ||∇ ψ || L ( R n ) and Lemma 2.4,we obtain ||∇ Z α ( φ ) || L ( R n ) . || Z α ( ρ ( f )) || L nn + ( R n ) . X | β |≤| α | || ρ ( Z β ( f )) || L nn + ( R n ) (24)Since Z x ∈ R n d x (1 + | x | + t ) n n + . t − ³ n n + − n ´ Z x ∈ R n d ( x / t )( | x / t | + n n + . t − ³ n n + − n ´ ,the lemma now follows by estimating the right-hand side of (24) using the pointwiseestimates (22) on ρ ( Z β ( f )). (cid:3) The previous lemma combined with the L based Klainerman-Sobolev inequal-ity of Lemma 3.2 gives the pointwise estimates on ∇ Z α ( φ ) claimed in the theorem. Corollary 5.2.
For any multi-index | α | ≤ N − (3 n /2 + and Z α ∈ Γ | α | s , we have for allt ∈ [0, T ], ¯¯ ∇ Z α φ ¯¯ . ǫ (1 + | x | + t ) n /2 t ( n − .20 .3 Improving the global bounds L estimates of Z α f We first consider the L estimate on Z α ( f ). From Lemma 2.9, we have for all multi-index | α | ≤ N , and all t ∈ [0, T ], || Z α f ( t ) || L x , v ≤ || Z α f (0) || L x , v + Z t || T φ ( Z α ( f )) || L x , v .Thus, we only need to prove that the term below the integral is integrable in t .Now, from the commutation formula of Lemma 2.7, we know that || T φ ( Z α ( f )) || L x , v ≤ C N X | β |≤| α |− X | γ |+| β |≤| α | | C αβγ |||∇ x Z γ φ ∇ v Z β f || L x , v .Note that we have so far no estimates on v derivatives of Z β f . To circumvent thisdifficulty, let us rewrite any v derivative as ∂ v i Z β ( f ) = ( t ∂ x i + ∂ v i ) Z β ( f ) − t ∂ x i Z β ( f ).Since both t ∂ x i + ∂ v i and ∂ x i are part of our algebra of commuting vector fields, wehave || T φ ( Z α ( f )) || L x , v ≤ C N (1 + t ) X | β |≤| α | , | γ |≤| α || γ |+| β |≤| α |+ | C αβγ |||∇ x Z γ ( φ ) Z β ( f ) || L x , v . (25)Now, since N ≥ n /2 + | γ | + | β | ≤ | α | + ≤ N +
1, we always have at leasteither | γ | ≤ N − (3 n /2 +
1) or | β | ≤ N − n .Case 1: | γ | ≤ N − (3 n /2 + t ∈ [0, T ], |∇ Z β φ ( t , x ) | ≤ C N ǫ (1 + t ) n − .Thus, we get the estimates ||∇ x Z γ ( φ ) Z β ( f ) || L x , v ≤ C N ǫ (1 + t ) n − X | β |≤| α | || Z β f || L x , v , (26)and we see that if n ≥
4, then the error is integrable.Case 2: | β | ≤ N − n In that case, we estimate the error term as follows. First, ||∇ x Z γ ( φ ) Z β ( f ) || L x , v = Z x Z v |∇ x Z γ ( φ ) || Z β ( f ) | d xd v = Z x |∇ x Z γ ( φ ) | ρ ³ | Z β ( f ) | ´ d x ,since Z γ ( φ ) is independent of v . Now applying the Hölder inequality with 1/ p + q =
1, we obtain ||∇ x Z γ ( φ ) Z β ( f ) || L x , v ≤ ||∇ x Z γ ( φ ) || L qx ¯¯¯¯¯¯ ρ ³ | Z β ( f ) | ´¯¯¯¯¯¯ L px
21e would like to take q =
2, since the L bounds are so easy to obtain for solutions tothe Poisson equation. Using the pointwise bounds to estimate || ρ ¡ | Z β ( f ) | ¢ || L x , wewould obtain ¯¯¯¯¯¯ ρ ³ | Z β ( f ) | ´¯¯¯¯¯¯ L x ≤ C N ǫ t − n /2 .On the other hand, we would get no decay a priori on ||∇ x Z γ ( φ ) || L x , since | γ | is apriori too large to have access to decay estimates for the source term of the Poissonequation satisfied by Z γ ( φ ). With the extra weight of t in (25), we see that if n = t decay and we would get a logarithmic loss.To avoid this problem, we want to take q <
2. Recalling the L q bounds of Corol-lary 5.1, we see that since δ < n − n + , we have q = n (1 + δ ) n − (1 + δ ) < || ρ ³ | Z β ( f ) | ´ || L px ≤ C N ǫ (1 + t ) n /2 − + σ ,for some σ >
0, which is now integrable in t . Putting everything together, we haveobtain || T φ ( Z α ( f )) || L x , v ≤ C N ǫ (1 + t ) n − E N , δ [ f ( t )] + C N ǫ (1 + t ) n /2 − + σ . C N ǫ (1 + t ) n /2 − + σ ,using the bootstrap assumptions to bound E N , δ [ f ( t )]. v weighted L p estimates of Z α f Let p = + δ and q = δ ( δ + n ). Recall from (8) the inequality, ¯¯¯¯ (1 + v ) q /2 | Z α f ( t ) | p ¯¯¯¯ L ( R nx × R nv ) . ¯¯¯¯ (1 + v ) q /2 | Z α f (0) | p ¯¯¯¯ L ( R nx × R nv ) + Z t ¯¯¯¯ (1 + v ) q /2 | Z α f | p − T φ ( Z α f ) ¯¯¯¯ L ( R nx × R nv ) d s + Z t ¯¯¯¯ | Z α f | p (1 + v ) ( q − ∂φ ¯¯¯¯ L ( R nx × R nv ) d s , . ¯¯¯¯ (1 + v ) q /2 | Z α f (0) | p ¯¯¯¯ L ( R nx × R nv ) + I + I ,where I = Z t ¯¯¯¯ (1 + v ) q /2 | Z α f | p − T φ ( Z α f ) ¯¯¯¯ L ( R nx × R nv ) and I = Z t ¯¯¯¯ | Z α f | p (1 + v ) ( q − ∂φ ¯¯¯¯ L ( R nx × R nv ) .To estimate the error term I , we proceed as in the previous section, replacing T φ ( Z α f ) using the commutation formula of Lemma 2.7 and rewriting the terms ofthe form ∂ iv Z β f as ¡ t ∂ x i + ∂ v ¢ Z β ( f ) − t ∂ x i Z β f . We are then left with error terms ofthe form (1 + t ) Z x Z v | ∂ x Z γ φ || Z β f | (1 + v ) q /2 | Z α f | p − d xd v , (27)which need to be integrable in t . Using Young inequality, we have that Z v | Z β f | (1 + v ) q /2 | Z α f | p − d v . Z v | Z β f | p (1 + v ) q /2 d v + Z v | Z α f | p (1 + v ) q /2 d v .22ote moreover that, as in the previous section, if | α | ≤ N − n , we have access to thepointwise estimates (23). With this in mind, all the error terms coming from I canthen be estimated as in the previous section.The estimates on the error term I are easier and rely on the pointwise estimatesof ∂φ of Lemma 5.2. n ≥ case Thus, we have obtained E N , δ ( t ) ≤ E N , δ (0) + C N Z t ǫ (1 + s ) n /2 + σ d s ,for some σ >
0. Using the smallness assumption E N , δ (0) ≤ ǫ , it follows that E N , δ ( t ) ≤ ǫ + C ǫ ≤ ǫ , provided ǫ is sufficiently small, which improves our original bootstrapassumption (21) and concludes the proof. Repeating the previous argument in the n = d , and it does not seem possible to close the estimates allowing the norms to growslowly, using some hierarchy in the equations, as it sometimes happen for some sys-tem of non-linear evolution equations (for instance in [10]). Thus, it seems that oneis forced to try to improve the commutation relations so as to remove the most prob-lematic error terms. This will be done using modified vector fields. In hindsight, thisstrategy is reminiscent of the strategy of [2] where modified vector fields are con-structed by solving transport equations along null cones.To understand and motivate the definitions of the modified vector fields that wewill use here, let us consider a solution f to the transport equation T φ ( f ) = Z i = t ∂ x i + ∂ v i . We obtain T φ ( Z i ( f )) = T ( Z i ( f )) + µ ∇ x φ . ∇ v Z i ( f ) = Z i ( T ( f )) + µ Z i ( ∇ x φ . ∇ v f ) − µ ∇ x ( Z i φ ). ∇ v f = Z i ( T φ ( f )) − µ n X j = ∂ x j Z i ( φ ).( t ∂ x j + ∂ v j ) f + µ n X j = ∂ x j Z i ( φ ). t ∂ x j f Thus, the error terms on the right-hand side are of two forms. The good terms are ofthe form ∂ x Z i ( φ ) Z ′ f , where the Z ′ are some of the commuting vector fields. Thesehave enough decay so that they can be estimated as before. The bad terms are of theform t ∂ x Z i ( φ ) Z ′ f where, in view of the extra t factor, the previous arguments wouldlead to logarithmic growth.The aim of the modified vector fields will be to avoid the introduction of suchbad terms. Note that on the other hand, commutations with vector fields such as ∂ t or ∂ x i would be better, because ∂∂φ enjoys improved decay. As a consequence, wewill only need to modify the homogeneous vector fields.23et us consider, for all 1 ≤ i ≤ n , vector fields of the form Y i = t ∂ x i + ∂ v i − n X j = Φ ji ( t , x , v ) ∂ x j ,where the coefficients Φ ji ( t , x , v ) are sufficiently regular functions to be specified be-low.Since ∂ x j commute with the free transport operator, we have the commutationformula [ T φ , Y i ]( f ) = − µ n X j = ∂ x j Z i ( φ ) Z j ( f ) + µ n X j = Φ ji ∂ x j ( ∇ x φ ). ∇ v f (28) − n X j = T φ ( Φ ji ) ∂ x j f + µ n X j = ∂ x j Z i ( φ ) t ∂ x j f where Z j = t ∂ x j + ∂ v j . The first term on the right-hand can be handled as before asit does not have v derivatives leading to the extra power of t . Note moreover thatsince t ∂ x j is part of the algebra of macroscopic commuting vector field, the secondterm can be rewritten as n X j = Φ ji t − ∇ x Z j φ ∇ v f and thus is expected to be integrable by the previous arguments provided Φ ji areuniformly bounded by any power of t strictly less 1.The key idea is then to choose appropriately the Φ ji so as to be able to cancel thelast two terms in (28). For this, we impose that each Φ ji is obtained as the uniquesolution to the inhomogeneous transport equation T φ ( Φ ji ) = µ t ∂ x j Z i ( φ )with 0 initial data. Note that the right-hand side of this equation only decay like 1/ t ,so we expect Φ ji to actually grow logarithmically in t .Assuming that this holds, we see that all the error terms in (28) are now inte-grable and thus, we should expect to control our norms E N , δ [ f ( t )] provided all theoriginal commutation vector fields Z are replaced by their modified form Y , hencethe definition of the norm in the 3d case given by (20). A few difficulties remain1. Since the Φ ji (and in fact all the coefficients involved in the contruction of themodified vector fields) are growing logarithmically, it is apriori not clear howto exploit the new energies obtained after commutation with modified vectorfields to obtain pointwise decay. The idea is again to rewrite the extra termssuch as Φ ji ∂ x j ( f ) in the form Φ ji ∂ x j ( f ) = Φ ji t ¡ t ∂ x j + ∂ v j ¢ ( f ) − Φ ji t ∂ v j ( f )and to use an integration by parts in v for the last term to push the v deriva-tives on the Φ ji coefficient. See Section 6.4 below.24. The Poisson equation ∆ φ = ρ ( f ) cannot be commuted with modified vectorfields, since the coefficients in the modified vector fields depend on v , while φ and ρ ( f ) are macroscopic quantities depending only on ( t , x ). Thus, we keepcommuting this equation with non-modified vector fields. Quantities such as Z α ρ ( f ) are then rewritten as ρ ( Y α ( f )) plus error terms. The structure of theseerror terms is the subject of Lemma 6.3.3. As is seen from the statement of Lemma 6.3, some of the error terms will de-pend on Y α ( ϕ ), where ϕ is a coefficient obtained by solving a transport equa-tion of the form T φ ( ϕ ) = t ∂ x Z φ and Y α is a composition of | α | modified vectorfields. Commuting the transport equation satisfied by ϕ by | α | vector fields,we see that we need to control tY α ∂ x Z ( φ ). This poses a problem at the toporder, when | α | = N , since it looks as if one needs to control ∂ Z α Z ( φ ), whichwould a priori require to commute the Poisson equation by N + Z α ( φ ), we may hope to control ∂ Z α ( φ ). To exploit this fact and close the top order estimates, we devote sometime to describe the structure of the top order terms of Lemma 6.3 in Lemma6.4.4. Finally, numerous terms of the form Y α ( ϕ ) Y β ( f ), where ϕ is a coefficient ob-tained by solving a transport equation as above, will appear in the equations.This creates another difficulty to close the top order estimates, since we do nothave access to pointwise estimates on Y α ( ϕ ) when | α | is large, nor do we haveaccess to L px , v estimates on Y α ( ϕ ) because the source terms in their transportequations are not integrable in v . We will circumvent this difficulty by consid-ering directly a transport equation satisfied by the product Y α ( ϕ ) Y β ( f ). SeeSection 6.5 below.We now turn to the details of the proof. As explained above, our new algebra of microscopic vector fields will consist in1. Standard translations ∂ t , ∂ x i ,2. Modified uniform motions Y i : = t ∂ x i + ∂ v i − P nk = Φ ki ∂ x k ,3. Modified rotations Ω xi , j + Ω vi , j − P nk = ω ki , j ∂ x k ,4. Modified scaling in space S x + S v − P nk = σ k ∂ x k ,5. Modified scaling in space and time t ∂ t + P ni = x i ∂ x i − P nk = θ k ∂ x k ,where the coefficients Φ ki , ω ki , j , σ k , θ k are solutions of the following inhomogeneousequations with 0 data at t = T φ ( Φ ki ) = µ t ∂ x k £ Z i ( φ ) ¤ , T φ ( ω ki , j ) = µ t ∂ x k h Ω xi , j ( φ ) i , T φ ( σ k ) = µ t ∂ x k £ S x ( φ ) − φ ¤ , T φ ( θ k ) = µ t ∂ x k "Ã t ∂ t + n X i = x i ∂ x i ! ( φ ) .25 .2.1 Further notations Since the sign of µ will play no role in the analysis to come, we will asume withoutloss of generality that µ = M the set of all the coefficients Φ ki , ω ki , j , σ k , θ k and by ϕ ageneric coefficient among them. Similarly to the set γ and γ s , we define the sets γ m and γ m , s where γ m is the set of all modified vector fields (including the translations)and γ m , s is the set of all modified vector fields minus the time translation and thespace-time scaling.If Z is an original, non-modified vector field, we will sometimes write schemat-ically Y = Z + ϕ∂ x to denote the associated modified vector field, where by conven-tion ϕ∂ x = Z is any of the translations, ϕ∂ x = n X k = Φ ki ∂ x k if Z is one of the uniformmotions and similarly for the other vector fields.Finally, we will use the notation P ( ¯ ϕ ) to denote a function depending on all thecoefficients in γ m or γ m , s . With these definitions, we now have the following improved commutation formula . Lemma 6.1.
For any Y ∈ γ m and any sufficiently regular function g of ( t , x , v ) , [ T φ , Y ]( g ) can be written as a linear combination with constant coefficients of terms of the form ∂ x Z ( φ ) Y ( g ) , ϕ∂ x Z ( φ ) Y ( g ) or ϕ ∂ x Z ( φ ) Y ( g ) where ϕ ∈ M , ϕ denotes a genericproduct of two coefficients in M , Z ∈ Γ and Y ∈ γ m .Proof. If Y is a translation, for instance Y = ∂ x k , then[ T φ , Y ]( g ) = − ( ∂ x k ∇ x φ ). ∇ v g , = − n X i = ∂ x i ∂ x k φ . Ã t ∂ x i g + ∂ v i g − n X j = Φ ji ∂ x j g ! + n X i = ∂ x i ∂ x k φ . Ã t ∂ x i g − n X j = Φ ji ∂ x j g ! = − n X i = ∂ x i ∂ x k φ . Y i ( g ) + n X i = ∂ x i t ∂ x k φ . ∂ x i g − n X i = n X j = Φ ji ∂ x i ∂ x k ( φ ). ∂ x j g ,which is of the desired form since t ∂ x k ∈ Γ . If Y = Y i = t ∂ x i + ∂ v i − P nj = Φ ji ∂ x j , thenit follows from (28) and the definition of the coefficients Φ ji that[ T φ , Y i ]( g ) = − n X j = ∂ x j Z i ( φ ) Z j ( g ) + n X j = Φ ji ∂ x j ( ∇ x φ ). ∇ v g . (29)where Z i ( φ ) = t ∂ x i φ and Z j ( g ) = t ∂ x i g + ∂ v j g . Since Z j ( g ) = Y j ( g ) + P ni = Φ ij ∂ x i ( g ),the first term term on the right-hand side of (29) is of the desired form while for the Once again, in all the formulae of this section, we assume that φ is a sufficiently regular function of( t , x ) defined on [0, T ] × R and we will eventually control the regularity of φ through a bootstrap argu-ment. n X j = Φ ji ∂ x j ( ∇ x φ ). ∇ v g = n X j = n X k = Φ ji ∂ x j ∂ x k φ . ∂ v k g = n X j = n X k = Φ ji ∂ x j ∂ x k φ . Y k ( g ) − n X j = n X k = Φ ji ∂ x j ∂ x k φ . Ã t ∂ x k g − n X l = Φ li ∂ x l g ! , = n X j = n X k = Φ ji ∂ x j ∂ x k φ . Y k g − n X j = n X k = Φ ji ∂ x j ( t ∂ x k φ ). ∂ x k g + n X j = n X k = n X l = Φ li Φ ji ∂ x j ( ∂ x k φ ) ∂ x l g ,which is of the desired form. The commutation with the other modified vector fieldscan be computed similarly. Note that in the case of the spatial vector field, we have[ T φ , S x + S v ] = −∇ x S x ( φ ). ∇ v + ∇ x φ . ∇ v , which explains the extra term in the defini-tion of the coefficients σ k compared to the other vector fields. (cid:3) If we apply several times the previous formula in order to compute [ T φ , Y α ],many products of the form Y ρ ( ϕ ) Y ν ( ϕ ′ ) will appear. In order to simplify the pre-sentation below, it will be usefull to use the following definition. Definition 6.1.
We will say that P ( ¯ ϕ ) is a multilinear form of degree d and signatureless than k if P ( ¯ ϕ ) is of the formP ( ¯ ϕ ) = X ρ ∈ I d , ρ = ( ρ ,.., ρ d ) C ρ Y j = n , ϕ ∈ M Y ρ j ( ϕ ), where I denotes the set of all multi-indices (thus ρ j is a multi-index for each j ), foreach ρ in the above formula d X i = | ρ j | ≤ k and where the C ρ are constants. From Lemma 6.1, we now obtain
Lemma 6.2.
For any multi-index α , we have [ T φ , Y α ] = | α |+ X d = n X i = X | γ |≤| α | , | β |≤| α | P α , id γβ ( ¯ ϕ ) ∂ x i Z γ ( φ ) Y β , (30) where the P α , id γβ ( ¯ ϕ ) are multilinear forms of degree d and signature less than k suchthat k ≤ | α | − and k + | γ | + | β | ≤ | α | + .Proof. This is a classical proof by induction for which we will just sketch the details.Lemma 6.1 shows that (30) holds when | α | =
1. Assume that (30) holds for somemulti-index α and let Y ∈ γ be an arbitrary modified vector field. We have[ T φ , Y Y α ] = [ T φ , Y ] Y α + Y [ T φ , Y α ].27ne easily see that the first term on the right-hand side has the correct form usingLemma 6.1. The second term will generate three types of terms. The terms of theform Y ³ P α , id γβ ´ ∂ x i ¡ Z γ ( φ ) ¢ Y β are of the correct form, the multilinear form being of the same degree and its signa-ture being increased by 1 at most. For the terms of the form P α , id γβ Y ¡ ∂ x i ( Z γ ( φ )) ¢ Y β ,we recall that Y is schematically of the form Z + ϕ∂ x . Thus, P α , id γβ Y ( ∂ x i Z γ ( φ )) Y β = X | γ ′ |≤| γ |+ P ′ γ ′ ∂ x i Z γ ′ ( φ )) Y β ,where P ′ γ ′ are multilinear forms of degree at most d + P α , id γβ ∂ x i ¡ Z γ ( φ ) ¢ Y Y β , which clearly satis-fied the required properties. (cid:3) As explained above, we also need to revisit our commutation relations for thePoisson equation satisfied by the potential φ . Contrary to f , we cannot commutewith a modified vector field, because the coefficients of the modified vector fieldsdepend on v , while φ is a macroscopic quantity and depends only on ( t , x ). Thus,we keep commuting with Z α . We have Lemma 6.3.
Let g be a sufficiently regular function of ( t , x , v ) and let φ g solves ∆ φ g = ρ ( g ). For any multi-index α with | α | ≤ N , we have, for all < t ≤ T , ρ ( Z α ( g )) = | α | X j = | α |+ X d = X | β |≤| α | t j ρ ³ P α , jd β ( ¯ ϕ ) Y β ( g ) ´ + | α | X d = X | β |≤| α | ρ ³ Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) ´ (31) and ∆ Z α ( φ ) = | α | X j = | α |+ X d = X | β |≤| α | t j ρ ³ ˜ P α , jd β ( ¯ ϕ ) Y β ( g ) ´ + | α | X d = X | β |≤| α | ρ ³ ˜ Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) ´ , (32) where P α , jd β ( ¯ ϕ ) and ˜ P α , jd β ( ¯ ϕ ) are multilinear forms of degree d and signature less thank satisfying k ≤ | α | , k + | β | ≤ | α | and where the Q α d β ( ∂ x ¯ ϕ ) and ˜ Q α d β ( ∂ x ¯ ϕ ) are all multilinear forms of degree d of theform Q ( ∂ x ¯ ϕ ) = X ρ ∈ I d , ρ = ( ρ ,.., ρ d ) C ρ Y j = n , ϕ ∈ M Y ρ j ( ∂ x rj ϕ ), (33) where the C ρ are constants, ≤ r j ≤ , and such that k ′ : = d X j = | ρ j | satisfiesk ′ ≤ | α | − d + k ′ + | β | ≤ | α | .28 roof. First, recall that if Z ∈ Γ , then ∆ Z ( φ g ) = Z ∆ φ g + d Z ∆ φ g ,where d Z = Z is one of the two scaling vector fields, in which case d Z = Z ( ρ ( g )) = ρ ( Z ( g )) + c Z ρ ( g ),where c Z = Z is the spatial scaling vector field, in which case c Z =
3. Since ∆ φ g = ρ ( g ), it follows that (31) implies (32).In the next lines of computations, given Z ∈ γ and Y the modified vector fieldcorresponding to Z , we will use the schematic notations Y = Z − ϕ∂ x and Z = Y + ϕ∂ x instead of any of the lengthy formulae given at the beginning of Section 6.2,such as Y i = t ∂ x i + ∂ v i − P nk = Φ ki ∂ x k . We will also use the notation t ∂ x + ∂ v − ϕ∂ x todenote a generic vector field among the Y i , the letter Y ′ to denote a generic modifiedvector field and the letter ϕ ′ to denote a generic coefficient belonging to M .We now compute, for any Z ∈ γ Z v Z ( g ) d v = Z v ¡ Z + ϕ∂ x − ϕ∂ x ¢ ( g ) d v = Z v Y ( g ) d v − Z v ϕ∂ x g d v = Z v Y ( g ) d v − Z v ϕ t ¡ t ∂ x g + ∂ v g − ϕ∂ x g − ∂ v g + ϕ∂ x g ¢ d v = Z v Y ( g ) d v − t Z v ϕ ¡ Y ′ ( g ) + ϕ∂ x g ¢ d v + Z v ϕ t ∂ v g d v The first and second terms on the right-hand side of the last line have the correctforms. For the last term, we integrate by parts in v − Z v ϕ t ∂ v g d v = t Z v ∂ v ϕ g d v = t Z v ¡ t ∂ x + ∂ v − ϕ ′ ∂ x − t ∂ x + ϕ ′ ∂ x ¢ ( ϕ ) g d v = t Z v ¡ Y ′ + ϕ ′ ∂ x ¢ ( ϕ ) g d v − Z v ∂ x ( ϕ ) g d v ,where now all terms are of the correct forms. This prove (31) when | α | =
1. Wenow assume that (31) is true for some α . Let Z be a non-modified vector field. Us-ing again that Z ρ ( Z α ( g )) = ρ ( Z Z α ( g )) + c Z ρ ( Z α ( g )), we only need to prove that Z ρ ( Z α ( g )) is of the correct form. Assume first that Z contains no t -derivative ,i.e. Z , ∂ t and Z , t ∂ t + n X i = x i ∂ x i . Using the induction hypothesis and writing Y = Z + ϕ∂ x to denote the associated modified vector field, we have Recall also that commuting with the vector fields containing t -derivatives is not necessary to proveTheorem 4.1 and is only usefull if one wants to obtain improved decay of t -derivatives. ρ ( Z α ( g )) = Z à | α | X j = | α |+ X d = X | β |≤| α | t j ρ ³ P α , jd β ( ¯ ϕ ) Y β ( g ) ´ + | α | X d = X | β |≤| α | ρ ³ Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) ´! = | α | X j = | α |+ X d = X | β |≤| α | t j ρ ³ Z h P α , jd β ( ¯ ϕ ) Y β ( g ) i´ + | α | X d = X | β |≤| α | ρ ³ Z h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ + c Z ρ ( Z α ( g )). (34)If now Z contains t -derivatives, we would get extra terms in (34) which arise when ∂ t hits the t j factors. Since these extra terms are all of the correct forms, so we onlyneed to analyse the terms on the right-hand side of (34).The last term in (34) has already the right form. Next, replacing Z by Y + ϕ∂ x inthe terms ρ ³ Z h P α , jd β ( ¯ ϕ ) Y β ( g ) i´ , one easily see that they also have the desired form.For the terms ρ ³ Z h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ , we have ρ ³ Z h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ = ρ ³ ( Y + ϕ∂ x ) h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ = ρ ³ Y h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ + ρ ³ ϕ∂ x h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ The first term on the right-hand side is easily seen to have the correct form. For thesecond term, we write ϕ∂ x = ϕ t ¡ t ∂ x + ∂ v − ϕ ′ ∂ x + ϕ ′ ∂ x − ∂ v ¢ = ϕ t Y ′ − ϕ t ∂ v + ϕϕ ′ t ∂ x sothat ρ ³ ϕ∂ x h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ = t ρ ³¡ ϕ Y ′ + ϕϕ ′ ∂ x ¢ h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ + t ρ ³ ∂ v ϕ h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ using an integration by parts in v . The first term on the right-hand side has now theright-form. For the second term, we again write ∂ v ϕ = ¡ t ∂ x + ∂ v − ϕ ′ ∂ x ¢ ϕ − t ∂ x ϕ + ϕ ′ ∂ x φ so that1 t ρ ³ ∂ v ϕ h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ = t ρ ³ Y ′ ( ϕ ) h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ − ρ ³ ∂ x ( ϕ ) h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ + t ρ ³ ϕ ′ ∂ x ( ϕ ) h Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) i´ where all terms now have the correct form. (cid:3) Recall that we do not hope to have any good estimate on Y α ( ϕ ) if | α | = N , sinceits transport equation would then contain a source term of the form ∇ Z α Z φ and weonly hope to have estimates for ∇ Z β φ up to β = N . On the other hand, using directlythe Poisson equation satisfied by Z α ( φ ), we will have good estimates on ∂ x ∂ x Z α ( φ ).To take advantage of that fact , we will need later the following technical lemma,which improves upon Lemma 6.3 by describing the structure of the P multilinearforms a little further. Note that these difficulies arise only at the top order | α | = N . An alternative to the approach takenhere would be to allow for the top order estimates to grow slightly in t . emma 6.4. With the notations of Lemma 6.3, the multilinear forms P α , jd β ( ¯ ϕ ) and ˜ P α , jd β ( ¯ ϕ ) can be written asP α , jd β ( ¯ ϕ ) = P ( ¯ ϕ ) + X ϕ ∈ M n X i = C i , ϕ Y i Y ρ i , ϕ ( ϕ ) + X ϕ ∈ M n X i = P i ϕ ( ¯ ϕ ) ∂ x i Y η i , ϕ , ϕ ′ ( ϕ ),˜ P α , jd β ( ¯ ϕ ) = ˜ P ( ¯ ϕ ) + X ϕ ∈ M n X i = ˜ C i , ϕ Y i Y ˜ ρ i , ϕ ( ϕ ) + X ϕ ∈ M n X i = f P i ϕ ( ¯ ϕ ) ∂ x i Y ˜ η i , ϕ , ϕ ′ ( ϕ ), where1. P and ˜ P are multilinear forms of degree d and signature less than k satisfyingk ≤ | α | − k + | β | ≤ | α | ,2. Y i are the modified uniform motions defined at the beginning of Section 6.2, i.e.Y i = t ∂ x i + ∂ v i − P nk = Φ ki ∂ x k ,3. | ρ i , ϕ | = | α | − , ˜ ρ i , ϕ = | α | − , | η i , ϕ , ϕ ′ | ≤ | α | − , | ˜ η i , ϕ , ϕ ′ | ≤ | α | − ,4. C i , ϕ , ˜ C i , ϕ are constants and P i ϕ ( ¯ ϕ ) , f P i ϕ ( ¯ ϕ ) are polynomial of degree at most | α | in the ϕ ′ ∈ M .Proof. This is an easy proof by induction. The case | α | = α and let Z ∈ Γ . As before, it is sufficient to consider only Z ¡ ρ ( Z α ) ¢ . Wewill only be interested in the top order terms, that is to say terms containing Y η ( ϕ )with η = | α | +
1. Note that they can only be generated by applying a vector field toterms containing Y η ′ ( ϕ ) with η ′ = | α | .Using the induction hypothesis, the top order terms coming from the P multi-linear forms will give terms of the form X ϕ ∈ M n X i = C i , ϕ Z £ Y i Y ρ i , ϕ ( ϕ ) ¤ (35)and X ϕ , ϕ ′ ∈ M n X i = D i , ϕ , ϕ ′ Z £ ϕ ′ ∂ x i Y η i , ϕ , ϕ ′ ( ϕ ) ¤ . (36)For the first type of terms, we have Z £ Y i Y ρ i , ϕ ( ϕ ) ¤ = ( Y + ϕ ′′ ∂ x ) £ Y i Y ρ i , ϕ ( ϕ ) ¤ = Y Y i Y ρ i , ϕ ( ϕ ) + ϕ ′′ ∂ x Y i Y ρ i , ϕ ( ϕ ).Now the second term on the right-hand side has the right structure. The first termalso has the right structure, since [ Y , Y i ] can be written as a linear combinations ofterms of the form ∂ x , the modified uniform motions Y j and ϕ ′′ ∂ x and Y ′ ( ϕ ) ∂ x .The terms of the type (36) can be treated similarly, the only dangerous termsbeing of the form ϕ ′ Y ∂ x i Y η i , ϕ , ϕ ′ ( ϕ ), where the Y and ∂ x i can be commuted up tolower order terms.The top order terms coming from the Q multi-linear forms will give terms of theform ρ ³ Z [ Y η ( ∂ x ϕ ) Y β ( g )] ´ = ρ ³¡ Y + ϕ ′′ ∂ x ¢ h Y η ( ∂ x ϕ ) Y β ( g ) i´ = ρ ³ Y h Y η ( ∂ x ϕ ) Y β ( g ) i´ + ρ ³ ϕ ′′ ∂ x h Y η ( ∂ x ϕ ) Y β ( g ) i´ ,31here | η | = | α |−
1. Now the first term on the right-hand side will contribute only the Q forms so it can be ignored here. While the second term, repeating the argumentof the previous lemma, gives the following contribution to the P forms1 t ρ ³ Y ( ϕ ) ′′ Y η ( ∂ x ϕ ) Y β ( g ) ´ which, since | η | ≤ | α | − t ρ ³ ϕ ′′ Y Y η ( ∂ x ϕ ) Y β ( g ) ´ as well as1 t ρ ³ ϕ ′ ϕ ′′ ∂ x h Y η ( ∂ x ϕ ) Y β ( g ) i´ = t ρ ³ ϕ ′ ϕ ′′ ∂ x £ Y η ( ∂ x ϕ ) ¤ Y β ( g ) ´ + t ρ ³ ϕ ′ ϕ ′′ Y η ( ∂ x ϕ ) ∂ x h Y β ( g ) i´ (37)which are all of top order. Again, we use that ∂ x essentially commutes with anymodified vector field up to lower order terms, to put these last terms in the rightform. (cid:3) The following commutation property will also be usefull later.
Lemma 6.5.
For any multi-index α and any ≤ i ≤ , we have [ ∂ x i , Y α ] = | α | X d = X | β |≤| α |− ≤ j ≤ P α , ji , d β ( ∂ x ¯ ϕ ) Y β ∂ x j , where the P α , ji , d β ( ∂ x ¯ ϕ ) are multilinear forms of degree d of the form (33) with a signa-ture less than k such that k + | β | ≤ | α | − .Proof. We prove the | α | = Y be modified vector field. We will write schematically Y = Z + ϕ∂ x ,where Z is a non-modified vector field and ϕ∂ x stands for a linear combination ofproducts of some ϕ ∈ M and some ∂ x k . For simplicity, assume that Z commuteswith ∂ x i (the cases where Z do not commute with ∂ x i can be treated similarly, sincethe resulting error terms do not involve any terms depending on the coefficients in M .) An easy computation then shows that[ ∂ x i , Y ] = [ ∂ x i , ϕ∂ x ] = ∂ x i ( ϕ ) ∂ x ,which is of the desired form. (cid:3) Finally, let us also remark that
Lemma 6.6.
For any multi-index α ,Y α ∇ φ = Z α ∇ φ + t | α | X d = P α d β ( ¯ ϕ ) Z β ∇ φ , where P α d β ( ¯ ϕ ) are multilinear forms of degree d and signature less than k such thatk ≤ | α | − and k + | β | ≤ | α | . roof. We only do the | α | = Y ′ be a modified vector field and write schematically Y ′ = Z ′ + ϕ∂ x . We have Y ′ ∇ φ = ¡ Z ′ + ϕ∂ x ¢ ∇ φ = Z ′ ∇ φ + ϕ t [ t ∂ x ] ∇ φ ,which is of the correct form since t ∂ x ∈ Γ . (cid:3) Let now f be a solution to the Vlasov-Poisson system in dimension n = T ≥ t ∈ [0, T ] and all x ∈ R ,1. E N , δ [ f ( t )] ≤ ǫ , (38)2. For all 0 < δ ′ ≤ δ , there exists a C δ ′ > α with | α | ≤ N , ||∇ Z α φ ( t ) || L + δ ′ ≤ C δ ′ ǫ . (39)3. For all multi-index α with | α | ≤ N − (9/2 + |∇ Z α φ ( t , x ) | . ǫ + t (40)and | Y α ∇ φ ( t , x ) | . ǫ + t (41)4. For all multi-index α with | α | ≤ N − (9/2 + ϕ ∈ M , ¯¯ Y α ( ϕ )( t , x , v ) ¯¯ . ǫ ¯¯ + log(1 + t ) ¯¯ , (42)5. For all ϕ ∈ M , all 1 ≤ i ≤ | α | ≤ N − (9/2 +
3) , ¯¯ ∂ x i Y α ( ϕ )( t , x , v ) ¯¯ . ǫ , (43)It follows from the initial data assumption and standard arguments that T >
0. Inthe rest of the proof, we will try to improve each of the above assumptions, whichwould show that T = +∞ .Note that in view of Lemma 6.5, (43) is equivalent to ¯¯ Y α ∂ x i ( ϕ )( t , x , v ) ¯¯ . ǫ and we will switch freely between the two in the rest of the article. Note moreoverthan in view of the Gagliardo- Niremberg inequality, assumption (39) immediatelyimplies that ||∇ Z α φ || L q ≤ C q ǫ , (44)with q = < + δ ′ )2 − δ ′ <
2, in view of the definition of δ . In particular, q can be takenas close to 3/2 as wanted.Finally, the notation A . B used in (40), (41), (42), (43) stand for A ≤ C N , δ , n B where C N , δ , n is a constant depending only on N , n , δ (with here n = ǫ on the right-handsides of (40), (41), (42), (43) by ǫ , choosing ǫ sufficiently small to absorb the C N , δ , n .33 .4 Klainerman-Sobolev inequalities with modified vector fields Using the bootstrap assumptions above, we have
Proposition 6.1.
For all multi-index | α | ≤ N − , ρ ¡¯¯ Y α f ( t ) ¯¯¢ . + t + | x | ) X | β |≤| α |+ || Y β f ( t ) || L ( R x × R v ) , . + t + | x | ) X | β |≤ N || Y β f ( t ) || L ( R x × R v ) . Proof.
Similarly to the proof of Proposition 3.3, let us fix ( t , x ) ∈ R × R and let e ψ bedefined by e ψ : B (0,1/2) ∋ y → ρ ¡¯¯ Y α f ¯¯¢ ¡ t , x + ( t + | x | ) y ¢ . We fix δ ′ = and apply a1 d Sobolev inequality ρ ( | Y α f | )( t , x ) . Z | y |≤ δ ′ ¡¯¯ ∂ y e ψ ¯¯ + | e ψ | ¢ ( y ,0,0) d y ,where as before ∂ y e ψ ( y ) = ( t + | x | ) ∂ x ρ ( | Y α f | ) ¡ t , x + ( t + | x | ) y ¢ = t Z v ∂ x ¡ | Y α f | ¢ ¡ t , x + ( t + | x | ) y , v ¢ d v + | x | Z v ∂ x ¡ | Y α f | ¡ t , x + ( t + | x | ) y , v ¢¢ d v .Now, t Z v ∂ x ¡ | Y α f | ¢ d v = Z v "à t ∂ x + ∂ v − n X i = Φ j ∂ x j + n X j = Φ j ∂ x j ! ¡ | Y α f | ¢ d v = Z v Y ¡ | Y α f | ¢ d v + Z v n X j = Φ j ∂ x j ¡ | Y α f | ¢ d v .The first term on the right-hand side is simply estimated by ¯¯¯¯Z v Y ¡ | Y α f | ¢ d v ¯¯¯¯ ≤ Z v | Y Y α f | d v .For the second term, we again try to force the apparition of our modified vectorfields Z v Φ j ∂ x j ¡ | Y α f | ¢ d v = Z v Φ j t à t ∂ x j + ∂ v j − n X k = Φ kj ∂ x k ! ¡ | Y α f | ¢ d v − Z v Φ j t à ∂ v j − n X k = Φ kj ∂ x k !¡ | Y α f | ¢ d v = Z v Φ j t Y j ¡ | Y α f | ¢ d v − Z v Φ j t à ∂ v j − n X k = Φ kj ∂ x k !¡ | Y α f | ¢ d v .The first term on the right-hand side can then be estimated as above, using that | Φ j t | is uniformly bounded from the bootstrap assumptions (42). For the remainder34erms, we first note than in view of the bootstrap assumptions (42), the terms of theform Z v Φ j t Φ kj ∂ x k ¡ | Y α f | ¢ can be estimated by Z v ¯¯ ∂ x k ¡ | Y α f | ¢¯¯ d v .For the last type of terms, we integrate by parts in v Z v Φ j t ∂ v j ¡ | Y α f | ¢ d v = − Z v ∂ v j Φ j t ¡ | Y α f | ¢ d v .We now rewrite ∂ v j Φ j as ∂ v j Φ j = à t ∂ x j + ∂ v j − n X k = Φ kj ∂ x k ! Φ j − à t ∂ x j − n X k = Φ kj ∂ x k ! Φ j = Y j ( Φ j ) − à t ∂ x j − n X k = Φ kj ∂ x k ! Φ j .The first term only grow like | + log(1 + t ) | according to (42) and this growth can beabsorbed thanks to the 1/ t factor. For the second term, using (43) and (42) ¯¯¯ t ∂ x j ( Φ j ) ¯¯¯ + ¯¯¯ Φ kj ∂ x k ( Φ j ) ¯¯¯ ≤ ǫ t + ǫ | + log(1 + t ) | ,and again we can absorb the growth using the 1/ t factor.Putting everything together we have obtained that ρ ( | Y α ( f ) | )( t , x ) . Z | y |≤ δ ′ Z v ¡ | Y Y α ( f ) | + | Y α ( f ) | ¢ ¡ t , x + ( t + | x | )( y ,0,0) ¢ d vd y .The remaining of the proof follows as in the proof of (3.3), repeating the previousarguments for each of the variables and applying the usual change of coordinates. (cid:3) Y α ( ϕ ) Y β ( f ) Due to the form of the commutators of Section 6.2.2, we will need to estimate termsof the form Y α ( ϕ ) Y β ( f ). When α is sufficiently small, we will have access to point-wise estimates on Y α ( ϕ ) so there is no difficulty. When α is large, say α = N , wehave for the moment no estimate on Y α ( ϕ ) and we can certainly not hope to provepointwise estimates for these quantities, because, in view of the transport equationssatisfied by the coefficients ϕ , these estimates would in turn require pointwise esti-mates on ∇ Y α ( φ ), and these estimates do not hold at the top order. Instead, we willprove directly estimates on the products Y α ( ϕ ) Y β ( f ), taking advantage of the factthat Y β ( f ) are integrable in v . More precisely, Proposition 6.2.
Let σ > . Then, there exists a C σ > such that for any multi-indices α , β , | α | ≤ N − , | β | ≤ + and any ϕ ∈ M , we have for all t ∈ [0, T ] and all ≤ i ≤ , | Y α ( ϕ )( t ) Y β ( f )( t ) || L ( R x × R v ) ≤ C σ (1 + t ) σ ǫ , || Y i Y α ( ϕ )( t ) Y β ( f )( t ) || L ( R x × R v ) ≤ C σ (1 + t ) σ ǫ , || ∂ x i Y α ( ϕ )( t ) Y β ( f )( t ) || L ( R x × R v ) ≤ C σ (1 + t ) σ ǫ , Proof.
Let σ > α be a multi-index satisfying | α | ≤ N and such that if | α | = N then Y α = Y j Y α ′ or Y α = ∂ x j Y α ′ with | α ′ | = N − T φ ³ Y α ( ϕ ) Y β ( f ) ´ = T φ ¡ Y α ( ϕ ) ¢ Y β ( f ) + Y α ( ϕ ) T φ ³ Y β ( f ) ´ = I + I ,where I = T φ ¡ Y α ( ϕ ) ¢ Y β ( f ) and I = Y α ( ϕ ) T φ ¡ Y β ( f ) ¢ . In view of Lemma 2.9, itsuffices to show that || I , I || L x , v ≤ C σ (1 + t ) − σ .1. Estimates on I We have I = [ T φ , Y α ]( ϕ ) Y β ( f ) + Y α [ T φ ( ϕ )] Y β ( f ) = I + I ,with I = [ T φ , Y α ]( ϕ ) Y β ( f ) and I = Y α [ T φ ( ϕ )] Y β ( f ). For I , we use thecommutation formula (30) I = | α |+ X d = n X i = X | γ |≤| α | , | η |≤| α | P α , id γη ( ¯ ϕ ) ∂ x i Z γ ( φ ) Y η ( ϕ ) Y β ( f ),where P α , id γη satisfies the requirement of Lemma 6.2, in particular, it has signa-ture less than k such that k ≤ | α | − k + | γ | + | η | ≤ | α | + | γ | ≤ N − (9/2 + ¯¯ ∂ x i Z γ ( φ ) ¯¯ ≤ ǫ + t .Moreover, since k + | η | ≤ | α | + ≤ N +
1, either k ≤ N − (9/2 + ¯¯¯ P α , id γη ( ¯ ϕ ) ¯¯¯ ≤ ǫ d /2 ¡ + log(1 + t ) ¢ d or k > N − (9/2 + | η | ≤ N + − k ≤ N − (9/2 + N ≥
14, so that we now have access to pointwise estimates on Y η ( ϕ ). Note alsothat since k ≤ N +
1, there is at most one factor in each of products of Y ρ j ( ¯ ϕ )in the decomposition of P α , id γη ( ¯ ϕ ) for which we do not have access to pointwiseestimates. In conclusion, it follows that we have an estimate of the form ¯¯¯ P α , id γη ( ¯ ϕ ) ∂ x i Z γ ( φ ) Y η ( ϕ ) ¯¯¯ . ǫ ¡ + log(1 + t ) ¢ d + t X | ρ |≤| α |− | Y ρ ( ϕ ) | .36ase 2: γ > N − (9/2 + k +| η | ≤ N + −| γ | ≤ N − (9/2 +
2) since N ≥
14 and we can bound P α , id γη ( ¯ ϕ )and Y η ( ϕ ) pointwise. Since | β | ≤ +
3, we have access to pointwise boundon ρ ( | Z β ( f ) | ), so that we can estimate || P α , id γη ( ¯ ϕ ) ∂ x i Z γ ( φ ) Y η ( ϕ ) Y β ( f ) || L x , v . ǫ ¡ + log(1 + t ) ¢ d + || ∂ x i Z γ ( φ ) || L qx || ρ ³ | Y β ( f ) | ´ || L px ,where 1/ q + p =
1. Taking q as in (44) with δ ′ as small as needed (dependingonly on σ ), we obtain, using the pointwise estimates of Proposition 6.1, that || ρ ³ | Y β ( f ) | ´ || L px . C σ ǫ (1 + t ) − σ ,so that || P α , id γη ( ¯ ϕ ) ∂ x i Z γ ( φ ) Y η ( ϕ ) Y β ( f ) || L x , v . ǫ ¡ + log(1 + t ) ¢ d + (1 + t ) − σ ,which, assuming σ <
1, is integrable in t .We now turn to the estimates on I . Recall that we have T φ ( ϕ ) = t ∂ x i Z ( φ ) forsome Z and some x i , unless Z is the spatial scaling vector field, in which case T φ ( ϕ ) = t ∂ x i ¡ Z ( φ ) − φ ¢ . Since the extra term can be handled similarly, wewill only treat the case of the non spatial scaling vector fields below. ApplyingLemma 6.6, we have Y α ¡ T φ ( ϕ ) ¢ = tY α ∂ x i Z ( φ ) = t Z α ∂ x i Z ( φ ) + t t | α | X d = P α d η ( ¯ ϕ ) Z η ∂ x i Z ( φ ), = t Z α ∂ x i Z ( φ ) + | α | X d = P α d η ( ¯ ϕ ) Z η ∂ x i Z ( φ ), (45)where P α d η ( ¯ ϕ ) is a multi-linear form of degree d and signature less than k with k ≤ | α | − k + | η | ≤ | α | .Assume first that | α | ≤ N −
1. For the first term on the right-hand side of (45),we have | t Z α ∂ x i Z ( φ ) | . t X | η |≤| α | | ∂ x i Z η Z ( φ ) | .Since | α | ≤ N −
1, we have | η |+ ≤ N in the above sum. Thus, | t Z α ∂ x i Z ( φ ) || Z β ( f ) | can be estimated as before using the Hölder inequality, pointwise estimateson ρ ( | Z β ( f ) | ) and the estimate (44).If now | α | = N , then by assumption, Y α = Y j Y α ′ or Y α = ∂ x j Y α ′ with | α ′ | = N −
1, so that Z α = t ∂ x i Z α ′ or Z α = ∂ x i Z α ′ . Thus, | t Z α ∂ x i Z ( φ ) | . t (1 + t ) X | η |≤ N − | ∂ x ∂ x i Z η Z ( φ ) | , . t (1 + t ) X | η |≤ N | ∂ x Z η ( φ ) | .37e can now estimate | t Z α ∂ x i Z ( φ ) | ρ ( | Z β ( f ) | ) using Hölder inequality with p = − σ = + σ ′ with σ ′ = σ − σ > σ < q = σ . Recall that || ∂ x Z η ( φ ) || L p is bounded thanks to the boostrap assumption (39) provided σ is sufficiently small. Moreover, using the pointwise estimate on ρ ( | Z β ( f ) | ), wehave || ρ ( | Z β ( f ) | ) || L q ( R x ) . ǫ (1 + t ) − σ .Whether | α | ≤ N − | α | = N , the above estimates gives Z x , v | t Z α ∇ Z ( φ ) || Z β ( f ) | d xd v ≤ C σ ǫ (1 + t ) − σ which, after integration, gives rise to the t σ growth in the statement of theproposition.For the second term on the right-hand side of (45), we have either | η | + ≤ N − (9/2 + | ∂ x i Z η Z ( φ ) | . ǫ (1 + t ) and thus, P α d η ( ¯ ϕ ) | ∂ x i Z η Z ( φ ) | Z β ( f ) | . ǫ ¡ + log(1 + t ) ¢ d (1 + t ) X | ρ |≤ N − | Y ρ ( ϕ ) || Z β ( f ) | ,where we have used the fact that there is at most one term in P α d η ( ¯ ϕ ) for whichwe do not have access to pointwise estimates, or we have | η |+ > N − (9/2 + k ≤ N − (9/2 +
2) and we can bound pointwise P d η ( ¯ ϕ ) as | P d η ( ¯ ϕ ) | . ¡ + log(1 + t ) ¢ d . P α d η ( ¯ ϕ ) | ∂ x i Z η Z ( φ ) | Z β ( f ) | can then be estimated as before, using Hölder in-equality, pointwise estimates on ρ ( | Z β ( f ) | ) and the estimate (44).2. Estimates on I . Using (30) again, we have | I | . | β |+ X d = n X i = X | γ |≤| β | , | η |≤| β | | P β , id γη ( ¯ ϕ ) || ∂ x i Z γ ( φ ) || Y η ( f ) || Y α ( ϕ ) | ,where P β , id γη ( ¯ ϕ ) are multilinear forms of degree d and signature k ≤ | β | − k + | γ | + | η | ≤ | β | +
1. Since | β | ≤ +
3, we have | β | − ≤ + ≤ N − (9/2 + P β , id γη ( ¯ ϕ ) can be bounded pointwise. Thus, wehave | I | . ¡ + log(1 + t ) ¢ | β |+ n X i = X | γ |≤| β | , | η |≤| β | , | γ |+| η |≤| β |+ | ∂ x i Z γ ( φ ) || Y η ( f ) || Y α ( ϕ ) | .38ow since | γ | ≤ | β | ≤ + | γ | ≤ N − (9/2 + N ≥
14 and thus, we can bound | ∂ x i Z γ ( φ ) | pointwise using the boostrapassumption (40). We have thus obtained || I ( t ) || L ( R nv ) × L ( R nx ) . ǫ (1 + t ) + δ ′ X | η |≤| β | || Y α ( ϕ )( t ) Y η ( f )( t ) || L ( R x × R v ) ,for some δ ′ > F ( t ) = X | η |≤| β | X | α |≤ N − || Y α ( ϕ )( t ) Y η ( f )( t ) || L ( R x × R v ) , F ( t ) = X | η |≤| β | X i = X | α |= N − || Y i Y α ( ϕ )( t ) Y η ( f )( t ) || L ( R x × R v ) , F ( t ) = X | η |≤| β | X i = X | α |= N − || ∂ x i Y α ( ϕ )( t ) Y η ( f )( t ) || L ( R x × R v ) , F = F + F + F .Combining all the ingredients above, We have obtained that, there exists some δ ′ > F ( t ) . Z t ǫ (1 + s ) + δ ′ F ( s ) d s + C σ ǫ t σ .Applying Gronwall inequality and using the smallness of the initial data thenfinishes the proof. (cid:3) Using (8), we have similarly
Proposition 6.3.
Let σ > . Then, there exists a C σ > such that for any multi-indices α , β , | α | ≤ N − , | β | ≤ + and any ϕ ∈ M , we have for all t ∈ [0, T ] , and all ≤ i ≤ , || (1 + v ) δ ( δ + + δ ) Y α ( ϕ )( t ) Y β ( f )( t ) || L + δ ( R x × R v ) ≤ C σ (1 + t ) σ ǫ , (46) || (1 + v ) δ ( δ + + δ ) Y i Y α ( ϕ )( t ) Y β ( f )( t ) || L + δ ( R x × R v ) ≤ C σ (1 + t ) σ ǫ , (47) || (1 + v ) δ ( δ + + δ ) ∂ x i Y α ( ϕ )( t ) Y β ( f )( t ) || L + δ ( R x × R v ) ≤ C σ (1 + t ) σ ǫ . (48) Proof.
The proof is almost identical to the proof of the previous proposition andtherefore left to the reader. (cid:3)
Finally, we can get rid of the small growth provided with look at a product of theform Y α ∂ x ( ϕ ) Y β ( f ). Proposition 6.4.
For any multi-indices α , β , | α | ≤ N − , | β | ≤ + and any ϕ ∈ M ,we have for all t ∈ [0, T ] and all ≤ j ≤ , | Y α ( ∂ x j ϕ )( t ) Y β ( f )( t ) || L ( R x × R v ) . ǫ , as well as || (1 + v ) δ ( δ + + δ ) Y α ( ∂ x j ϕ )( t ) Y β ( f )( t ) || L + δ ( R x × R v ) . ǫ . Proof.
First note that the previous arguments used in the proof of Proposition 6.2still apply and that we may only focus on the terms leading to the t σ growth in theproof of Proposition 6.2, which were contained in the error term I . More precisely,the term leading to the t σ growth is the first one on the right-hand side of (45), i.e.the term t Z α ∂ x i Z ( φ ). In our case, this term should be replaced by t Z α ∂ x i ∂ x j Z ( φ ),with | α | ≤ N −
1. Commuting the ∂ x and Z α , we have | t Z α ∂ x i ∂ x j Z ( φ ) | ≤ t X | η |≤ N | ∂ x Z η φ | .We can then repeat the previous arguments (i.e. use Hölder inequality and thebootstrap assumption (39)), except that we have gained a 1/ t factor. This gain nowmeans that the resulting error will decay like 1/ t − σ , which is integrable in t andtherefore does not lead to any growth. The L + δ x , v weighted estimates can be treatedsimilarly. (cid:3) We are now in a position to improve each of the boostrap assumptions. Z α ( φ )Similarly to Lemma 5.2, we can improve assumption (39) to Lemma 6.7.
For all < δ ′ ≤ δ , there exists a C δ ′ > such that for all multi-index α with | α | ≤ N , ||∇ Z α φ ( t ) || L + δ ′ ≤ C δ ′ ǫ . (49) Applying the Gagliardo-Nirenberg inequality, we deduce that for all t ∈ [0, T ] , ||∇ Z α φ ( t ) || L q ( R n ) . ǫ , for all < q < + δ )2 − δ .Proof. Using the commuted equation (32) for Z α φ , we have || ∆ Z α ( φ ) || L + δ ′ . J + J where J = | α | X j = | α |+ X d = X | β |≤| α | t j || ρ ³ P α , jd β ( ¯ ϕ ) Y β ( g ) ´ || L + δ ′ ,40here the P α , jd β ( ¯ ϕ ) are multilinear forms of degree d and signature less than k satis-fying k ≤ | α | , k + | β | ≤ | α | ,and, in view of Lemma 6.4, such that when | α | = N the only top order terms in P α , jd β ( ¯ ϕ ) are of the form Y i Y β ( ϕ ) or ∂ x i Y β ( ϕ ) with | β | = N −
1, so that we can ap-ply Propositions 6.2 and 6.3,and where J = | α | X d = X | β |≤| α | || ρ ³ Q α d β ( ∂ x ¯ ϕ ) Y β ( g ) ´ || L + δ ′ ,where the Q α d β ( ∂ x ¯ ϕ ) are multilinear forms of degree d of the form (33) and signatureless than k ′ satisfying k ′ ≤ | α | − k ′ + | β | ≤ | α | .Following the strategy of the proof of Lemma 5.2, we see that it is sufficient to prove L px bounds on ρ ( J ) and ρ ( J ). With this in mind, recall that, for i = || ρ ( J i ) || L px = ¯¯¯¯¯¯¯¯Z v J i d v ¯¯¯¯¯¯¯¯ L px . µZ v χ ( v ) − d v ¶ q ¯¯¯¯ χ ( v ) q J i ¯¯¯¯ L px , v ,for any weight function χ ( v ). Choosing χ ( v ), p and q as in the proof of 5.2, we onlyneed to prove the v -weighted L px , v bounds for J and J .1. Estimates on J . Since k ≤ | α | , there can be at most one term in the decom-position of each of the P α , jd β for which we do not have access to pointwise esti-mates. Thus, we have | P α , jd β ( ¯ ϕ ) Y β ( g ) | ≤ (1 + (log(1 + t )) d − X | η |+| β |≤| α | X ϕ ∈ γ m | Y η ( ϕ ) || Y β ( g ) | .Now in the above sum, either | η | > N − (9/2 + | β | ≤ + | η | ≤ N − (9/2 + | Y η ( ϕ ) | pointwise. In this case, we use the v -weighted bounds on Y β ( f ) contain in the norm E N , see (20). In both case,we can absorb the t -growth thanks to the t weights in the definition of J .2. Estimates on J . These are obtained similarly, using Proposition 6.4 instead of6.3, since there is no t weight in J to absorb any t growth. (cid:3) The preceding lemma improves the boostrap assumption (39).Similarly to Lemma 5.3 and Corollary 5.2, the pointwise estimates on ρ ¡ | Y β ( f ) | ¢ can be transformed into pointwise estimates for ∇ Z α φ . Lemma 6.8.
For all multi-index α such that | α | ≤ N − ||∇ Z α φ || L ( R n ) . ǫ t , and for any multi-index | α | ≤ N − (9/2 + and Z α ∈ Γ | α | s , we have for all t ∈ [0, T ], ¯¯ ∇ Z α φ ¯¯ . ǫ (1 + | x | + t ) t .41 roof. We use again the commutation formula (32). Apart from the top order terms,we can estimate all the quantities on the right-hand side using the pointwise esti-mates (42) on Y ρ ( ϕ ) for | ρ | ≤ N − (9/2 + Y ρ ( ∂ϕ ) for | ρ | ≤ N − (9/2 + Y ρ ( ϕ ). Those coming from the P multilinear forms are of the type1 t ρ ¡ Y ρ ( ϕ ) f ¢ with | ρ | the largest integer such that | ρ | ≤ N − (9/2 + | ρ | = N −
6. Now, since | ρ | ≤ N −
6, we can consider Y β ( Y ρ ( ϕ ) f ) for | β | ≤
3. Thus, we may apply Proposition6.1 directly to the product Y ρ ( ϕ ) f and we find ¯¯¯¯ t ρ ¡ Y ρ ( ϕ ) f ¢¯¯¯¯ . ǫ t − σ (1 + t + | x | ) ,where we have used Proposition 6.2 to bound the norms appearing on the right-hand side after applications of Proposition 6.1. Choosing σ <
1, these terms there-fore decay better than what is needed for the statement of the Lemma.The top order terms coming from the Q forms are of type ρ ¡ Y ρ ( ∂ x i ϕ ) f ¢ ,with | ρ | ≤ N − (9/2 + | ρ | ≤ N −
5. We can then proceed similarly. Since wehave no extra t decay in front of the Q forms, it is important not to lose any t decayhere and thus we use the improvements of Proposition 6.4. The rest of the proof isidentical to that of Lemma 5.3 and therefore omitted. (cid:3) The preceeding lemma improves (40). To improve (41) is then suffices to useLemma 6.6 as well as the pointwise bounds on Y α ( ϕ ) (42). Thus, it remains only toimprove (38), (42) and (43). We have
Lemma 6.9.
For all t ∈ [0, T ] , E N , δ [ f ( t )] ≤ ǫ , which improves (38) .Proof. Using the commutation formula (30), we have[ T φ , Y α ]( f ) = | α |+ X d = n X i = X | γ |≤| α | , | β |≤| α | P α , id γβ ( ¯ ϕ ) ∂ x i Z γ ( φ ) Y β ( f ),where the P α , id γβ ( ¯ ϕ ) are multilinear forms of degree d and signature less than k suchthat k ≤ | α | − ≤ N − k + | γ | + | β | ≤ | α | + ≤ N +
1. When k ≤ N − (9/2 + P α , id γβ ( ¯ ϕ ) by a polynomial power of 1 + log(1 + t ) and we can treat the otherterms as in Section 5.3.1. Otherwise k > N − (9/2 +
2) and thus | β | ≤ + L ¡ R t ; L ( R nx × R nv ) ¢ and the lemma follows from Lemma2.9. (cid:3) .6.3 Improving the pointwise bounds on Y α ( ϕ ) and Y α ( ∂ x i ϕ )Finally, we conclude the proof of the 3d case by improving (42) and (43). Lemma 6.10.
For all ϕ ∈ M and for all t ∈ [0, T ] ,1. for all multi-index α with | α | ≤ N − (9/2 + , ¯¯ Y α ( ϕ ) ¯¯ . ǫ ¡ + log(1 + t ) ¢ , (50)
2. for all multi-index α with | α | ≤ N − (9/2 + and all ≤ i ≤ n, ¯¯ Y α ( ∂ x i ϕ ) ¯¯ . ǫ . (51) Proof.
Using the commutation formula (30), we have[ T φ , Y α ]( ϕ ) = | α |+ X d = n X i = X | γ |≤| α | , | β |≤| α | P α , id γβ ( ¯ ϕ ) ∂ x i Z γ ( φ ) Y β ( ϕ ),where the P α , id γβ ( ¯ ϕ ) are multilinear forms of degree d and signature less than k suchthat k ≤ | α |− ≤ N − k +| γ |+| β | ≤ | α |+ ≤ N +
1. Given the range of the indices,we can estimates all terms on the right-hand side pointwise and find that ¯¯ [ T φ , Y α ]( ϕ ) ¯¯ . ¡ + log(1 + t ) ¢ | α |+ ǫ t ,which is integrable in t . On the other hand, we have Y α T φ ( ϕ ) = Y α t ∇ Z ( φ ). UsingLemma 6.6, it follows that Y α T φ ( ϕ ) = t Z α ∇ Z φ + | α | X d = P α d β ( ¯ ϕ ) Z β ∇ Z φ ,where the second term is integrable in t , while the first term satisfied only the weakbound, for all | t Z α ∇ Z φ | . ǫ + t .Since any solution to T φ ( g ) = h with 0 initial data satisfies, for all t >
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