Small data solutions of the Vlasov-Poisson system and the vector field method
The aim of this article is to demonstrate how the vector field method of Klainerman can be adapted to the study of transport equations. After an illustration of the method for the free transport operator, we apply the vector field method to the Vlasov-Poisson system in dimension or greater. The main results are optimal decay estimates and the propagation of global bounds for commuted fields associated with the conservation laws of the free transport operators, under some smallness assumption. Similar decay estimates had been obtained previously by Hwang, Rendall and Velázquez using the method of characteristics, but the results presented here are the first to contain the global bounds for commuted fields and the optimal spatial decay estimates. In dimension or greater, it suffices to use the standard vector fields commuting with the free transport operator while in dimension , the rate of decay is such that these vector fields would generate a logarithmic loss. Instead, we construct modified vector fields where the modification depends on the solution itself.
The methods of this paper, being based on commutation vector fields and conservation laws, are applicable in principle to a wide range of systems, including the Einstein-Vlasov and the Vlasov-Nordström system.
- 1 Introduction
- 2 Preliminaries
- 3 Decay of velocity averages for the free transport operator via the vector field method
- 4 Small data solutions of the Vlasov-Poisson system
- 5 Proof when
- 6 The three dimensional case
A standard approach to the study of asymptotic stability of stationary solutions of non-linear evolution equations consists in an appropriate linearization of the system together222A third ingredient not needed in the present case is that of modulation theory, see for instance  for an application of modulation theory in the context of the Vlasov-Poisson system. with
a robust method for proving decay of solutions to the linearized equations,
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 , several methods for proving decay of solutions to the linear wave equation where is the wave operator of the flat Minkowski space333In case of perbutations around a non-flat solution with metric , the operator would naturally be replaced by , the wave operator of the metric . are a priori available. One of the classical methods to derive decay estimates is to use an explicit representation of the solutions, such as the Fourier representation, together with specific estimates for singular or oscillatory integrals. While this method provides very precise estimates on the solutions, it does not seem sufficiently robust to be applicable to quasilinear system of wave equations such as the Einstein equations, and the method of choice444More 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  for an application of these tools. for proving decay in view of such applications is the commutation vector field method of Klainerman  and its extensions using multiplier vector fields, see for instance [13, 14, 3]. The method of Klainerman is based on
A coercive conservation law: the standard energy estimate in the case of the wave equation.
Commutation vector fields: these are typically associated with the symmetries of the equations. In the case of the wave equation, these are the Killing and conformal Killing fields of the Minkowski space.
Weighted vector field idendities and weighted Sobolev inequalities: the usual vector fields are rewritten in terms of the commutation vector fields. The coefficients involved in these decompositions contain weights in and 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 systems of transport equations such as the Vlasov-Nordström system is the method of characteristics. This is an explicit representation of the solutions and thus, in our opinion, should be compared with the Fourier representation for solutions of the wave equation. What would then be the analogue of the vector field method for transport equations? The aim of this article is twofold. First, we will provide a vector field method for the free transport operator. In fact, in a joint work with J. Joudioux and D. Fajman, we have developped a vector field approach to decay of averages not only for the free (non-relativistic) transport operator but also for the massive and massless relativistic transport operators, see . In this paper, we will give two different proofs of Klainerman-Sobolev inequalities. The easier proof will give us a decay estimate for velocity averages of sufficiently regular distribution functions, i.e. quantities such as . However, this proof fails in the case of velocity averages of absolute values of distribution functions, i.e. quantities such as , because higher derivatives of will typically not lie in even if is in some high regularity Sobolev space. On the other hand, the decay estimate obtained via the method of characteristics can be applied equally well to and . We shall therefore give a second proof of Klainerman-Sobolev inequalities for velocity averages which will be applicable to absolute values of regular distribution functions. The first approach, which is closer to the standard proof of the Klainerman-Sobolev inequality for wave equations, consists essentially of two steps, a weighted Sobolev type inequality for functions in and an application of this inequality to velocity averages, exploiting the commutation vector fields. The improvement in the second approach comes from mixing the two steps together.
In the second part of this paper, we will apply our method to the Vlasov-Poisson system in dimension
where , , with , , is a sufficiently regular function of and is given by
Our main result can be summarized as follows (a more precise version is given in Section 4.2).
Let and if and if . Let . Then, there exists such that for all , if , where is a norm555See Section 4.1 for a precise definition of the norms. The encodes some additional integrability properties of the solutions. containing up to derivatives of , then the classical solution of (1)-(3) exists globally in time666Under 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, are the propagation of the global bounds and the optimal space and time decay estimates for the solutions. and satisfies the estimates, and ,
Space and time pointwise decay of averages of
for any multi-index
where is a differential operator of order obtained as a combination of commuting vector fields and is a constant depending only on .
Improved decay estimates for derivatives of
for any multi-index
Boundedness of the norms of and
Space and time decay of the gradient of the potential and its derivatives
for any multi-index with
as well as the improved decay estimates
Stronger bounds can be propagated by the equations provided the data enjoy additional integrability conditions. More precisley, the improved decay estimates for derivatives of can be improved to
and the improved decay estimates for derivatives of the gradient of can be improved to
the point being that additional derivatives now bring additional decay in and . These stronger estimates hold provided the initial data have stronger decay in than what is needed in the proof of Theorem 1.1. Similarly, one can propagates norms with for and provided additional decay of the initial data is assumed.
Similar time decay estimates have been obtained in  for derivatives of and using the method of characteristics under different assumptions on the initial data. On the other hand, the optimal decay rates in space and the propagation of the global bounds (4) were, as far as we know, not known prior to our work.
As is clear from the proof below and is typical of strategies based on commutation 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 energy for the system (1)-(3). An illustration of this robustness will be given in  where we will 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 introduction in  for a good introduction to the subject and only quote here the most important results from the point of view of this article. In the pioneered work , small data global existence in dimension for the Vlasov-Poisson system was established together with optimal time decay rates for and but no decay was obtained for their derivatives. The optimal time (but not spatial) decay rates for derivatives of and has been only much later obtained in , covering at the same time all dimensions . Both these works use decay estimates obtained via the method of characteristics. In fact, in  and even more in , precise estimates on the deviation of the characteristics from the characteristics of the free transport operator are needed in order to obtain the desired decay estimates. Parallely 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 dimension for the solutions of (1)-(3), see [17, 11]. The strongest results concerning the stability of non-trivial stationnary solutions of (1)-(3) with have been obtained in . They are not based on decay estimates but on a variational characterisation of the stationary solutions. On the other hand, this type of method does not provide asymptotic stability of the solutions but orbital stability. It is likely that any result addressing the question of asymptotic stability will need to go back to an appropriate linearization of the equations combined with robust decay estimates777See for instance  for some stability results using the linearization approach in the case of the spherically-symmetric King model.. We believe that, once again, the vector field method would be totally appropriate for the derivation of such decay estimates. Finally, let us mention the celebrated work  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 to revisit this question using vector field methods.
Outline of the paper
In Section 2, we introduce the vector fields commuting with the free transport operator and the notations that we will use throughout the paper. In Section 3, we present and prove decay estimates for velocity averages of solutions to the transport equation. In the following section, we present our results on the Vlasov-Poisson system. The remaining last two sections are devoted to the proof of these results, first in dimension and then in dimension using modified vector fields.
This project was motivated by my joint work with David Fajman and Jérémie Joudioux on relativistic transport equations. I would like to thank both of them, as well as Christophe Pallard and Frédéric Rousset for many interesting discussions on these topics. I would also like to thank Pierre Raphaël for an extremely stimulating conversation which took place during the conference “Asymptotic analysis of dispersive partial differential equations” held in October 2014 at Pienza, Italy. Some of this research was done during this conference. Finally, I would like to acknowledge partial funding from the Agence Nationale de la Recherche ANR-12-BS01-012-01 (AARG) and ANR SIMI-1-003-01.
Throughout this article, will denote a sufficiently regular function of with and . By sufficiently regular, we essentially mean that is such that all the terms appearing in the equations make sense as distributions and that all the norms appearing in the estimates are finite. For simplicity, the reader might just assume that is smooth with compact support in (but any sufficient fall-off will be enough).
We will denote by the free transport operator i.e.
where and for all , . Similarly, for any sufficiently regular scalar function , will denote the perturbed transport operator
where , corresponding to an attractive or repulsive force. Since we are dealing only with small data solutions, the sign of will play no role in the rest of this article.
The notation will be used to specify that there exists a universal constant such that , where typically will depends only on the number of dimensions and a few other fixed constants, such as the maximum number of commutations.
2.1 Macroscopic and microscopic vector fields
Consider first the following set of vector fields
Translations in space and time , ,
Uniform motion in one spatial direction888Recall that these vector fields are the generators of the Gallilean transformations of the form , where . ,
Scaling in space ,
Scaling in space and time .
The above set of vector fields is associated with the Gallilean invariance of macroscopic fields and equations. We will denote by the set of all such vector fields
One easily check that while the translations commute with , uniform motions, rotations or the scaling in space do not. The correct replacement for these vector fields is most easily explained using the language of differential geometry; the interested reader may consult [18, 4] for detailed constructions (in the case of the relativistic transport operator). We shall here only present the resulting objects which are the vector fields
Uniform motions in one direction in microscopic form ,
Rotations in microscopic form ,
Scaling in space in microscopic form .
One can then easily check
Lemma 2.1 (Commutation with the transport operator).
If is any of the translations, microscopic uniform motions or microscopic rotations, then .
If is the microscopic scaling in space, then .
If is the scaling in space and time, then .
From the two scaling commuting vector fields, it follows automatically that also commutes with in the sense that
This vector field will be used to obtain improved decay for derivatives of velocity averages.
To ease the notation, we will denote by the rotation vector fields in and by the rotation vector fields in . The full microscopic rotation vector fields are thus of the form . Similarly, we will denote by the scaling in space in microscopic form, with and .
Let now be the set of all the above microscopic vector fields including the translations and the space and time scaling i.e.
2.2 Multi-index notations
Let be an ordering of . For any multi-index , we will denote by the differential operator of order given by the composition .
In view of the above discussion, to any vector field of , we can associate a unique vector field of . More precisely,
Thus, to any ordering of we can associate an ordering of . We will by a small abuse of notation, denote again by the elements of such an ordering since it will be clear that, if is applied to a macroscopic quantity, such as a velocity average, then and if is applied to a microscopic quantity, i.e. any function depending on (and possibly ), then . Similarly, for any multi-index , we will also denote by the differential operator of order given by the composition obtained from the vector fields of .
For some of the estimates below, it will be sufficient to only consider a subset of all the vector fields of and . Let us thus denote by the set of all the macroscopic vector fields apart from and , which are the only vector fields containing time derivatives and by the corresponding set of microscopic vector fields, i.e.
The notation (respectively ) will be used to denote a generic differential operator of order obtained as a composition of vector fields in (respectively in ). The standard notation will also be used to denote a differential operator of order obtained as a composition of translations among the vector fields.
The following lemmae can easily be checked.
Lemma 2.2 (Commutation within ).
For any , , where are multindices, we have
for some constant coefficients . Moreover, if , then all the of the right-hand side belongs to .
Lemma 2.3 (Commutation of and weights in ).
Let . For any sufficienly regular function of and for any where is a multi-index, we have
Moreover, if then all the belong to in the above inequality.
2.3 Velocity averages and commutators
For any integrable function of , we will denote by the quantity
We have the following lemma.
For any sufficiently regular function of we have
for all ,
for all and all ,
for all and ,
where and are the spatial scaling vector fields in macroscopic and microscopic forms.
The proof is straigtforward and consist in identifying total derivatives in . For instance, we have
where we have integrated by parts in each of the . Similarly, in the case of rotations, it suffices to note that for any , is an angular derivative in and thefore, ∎
In the remainder of this paper, we shall write the preceding lemma as
where we are using, by a small abuse of notation, the letter to denote a generic macroscopic vector field and its corresponding microscopic version and where unless is the spatial scaling vector field, in which case .
2.4 Vector field identities
The following well-known identity will be used later
For any , we have
and thus, at any ,
where the coefficients are all homogeneous of degree and therefore uniformly bounded.
The following higher order version will be used often in the derivation of the Klainerman-Sobolev inequalities of the next section.
For any multi-index ,
where the coefficients are all uniformly bounded.
The lemma is a consequence the previous decomposition, the fact is part of our algebra of commuted vector fields and that is homogeneous of degree . ∎
2.5 The commuted equations
We now turn to the study of the transport operator defined by (5). Many of the estimates below are only valid provided has sufficient regularity. In the applications to the Vlasov-Poisson system of this article, we will eventually control the regularity of via a bootstrap argument. For all the estimates below, we therefore assume that is a sufficiently regular999For instance, one can assume that is a smooth function on with compact support in . function of defined on , for some , which decays sufficiently fast as .
The following lemma can then easily be checked.
Let be a sufficiently regular function of (t,x,v) and let be a multi-index. Then, there exists constant coefficients such that,
Moreover if , then and in the above decomposition.
Similarly one has
Let be a sufficiently regular function of and let be the solution to the Poisson equation . Then, for any multi-index , is solution to an equation of the form
where are constants.
The lemma is an easy consequence of the fact that all macroscopic vector fields apart from the two scalings commute with while for the spatial scaling and the space-time scaling we have and .
2.6 Conservation laws
We shall use the following (approximate) conservation laws.
For any sufficiently regular function of , we have, for all ,
Similarly, we have for all , for all ,
and for all ,
Note in particular that the conclusions of the lemma hold true when (i.e. when ).
These are classical estimates so we only sketch their proofs.
One has (in the sense of distribution),
Using a standard procedure101010For instance, assume first that has compact support in with a uniform bound on the support of in for . For all , consider the function , where is a smooth cut-off function which is on the support of and vanishes for large and . Apply then the previous estimates to and take the limit . A standard density argument deals with the case of non-compact support., one can regularize the previous inequality. We shall therefore neglect regularity issues here. Integrating the previous line in leads to
On the other-hand, remembering that and integrating by parts in and , we obtain
3 Decay of velocity averages for the free transport operator via the vector field method
Since the main purpose of this article is to illustrate how the vector field method can lead to robust decay estimates for velocity averages, let us for the sake of comparison recall the Bardos-Degond decay estimate and its proof.
Proposition 3.1 ().
Let be a sufficiently regular solution of . Then, we have the estimate, for all and all ,
The proof of this classical estimate is based on the method of charateristics. More precisly, if is a regular solution to , then it follows that
for all , .
We then have
Applying now the change of coordinates for leads to
In the above proof, the two key ingredients are
the explicit representation obtained via the method of characteristics,
the change of variables .
Note that in the presence of a perturbation of the free transport operator, to exploit a similar change of variables would require estimates on the Jacobian associated with the differential of the characteristic flow, see [1, 6].
Let us now show how the vector field method can be used as an alternative to obtain similar decay estimates. As explained in the introduction, we will give two different proofs.
The first proof will give us decay estimates for quantities of the form . The starting point of this approach is the following Klainerman-Sobolev inequality using norms of commuted fields.
Lemma 3.1 ( Klainerman-Sobolev inequality).
For any sufficiently regular function defined on , we have
This is relatively standard material and we adapt here the presentation given in  Chap.2 to our setting.
Fix and let be the function
Applying a standard111111Recall that, while the general embedding requires , the special case only needs . Sobolev inequality, we have
where denote the ball in of radius . On the other hand, we have and thus, for ,
where we have used Lemma 2.6 in the last step and the fact that and are comparable for . Inserting the last line in the Sobolev inequality (13) and applying the change of variables conclude the proof of the lemma.
Using Lemma 2.4, we now note that for any vector field and any sufficiently regular function of , we have
Combined with the previous Klainerman-Sobolev inequality, we obtain
Proposition 3.2 (Global Klainerman-Sobolev inequality for velocity averages).
For any sufficiently regular function defined on , we have, for all and all ,
Note that the above inequality cannot be apply to even is is say a smooth, compactly supported function. Indeed, if is say in , then is in but, unless has some extra special properties, when . On the other hand, (10) clearly holds both for and for .
Two disctinct steps lead to the proof of (14), the Klainerman-Sobolev inequality of Lemma 3.1 and the special commutation properties of the velocity averaging operator as described in Lemma 2.4. To improve upon (14), the strategy is to try to 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 following improvement.
Proposition 3.3 (Global Klainerman-Sobolev inequality for velocity averages of absolute values).
For any sufficiently regular function defined on , we have, for all and all ,