A vector field method for relativistic transport equations with applications
We adapt the vector field method of Klainerman to the study of relativistic transport equations. First, we prove robust decay estimates for velocity averages of solutions to the relativistic massive and massless transport equations, without any compact support requirements (in or ) for the distribution functions. In the second part of this article, we apply our method to the study of the massive and massless Vlasov-Nordström systems. In the massive case, we prove global existence and (almost) optimal decay estimates for solutions in dimensions under some smallness assumptions. In the massless case, the system decouples and we prove optimal decay estimates for the solutions in dimensions for arbitrarily large data, and in dimension under some smallness assumptions, exploiting a certain form of the null condition satisfied by the equations. The -dimensional massive case requires an extension of our method and will be treated in future work.††University of Vienna preprint ID: UWThPh-2015-27
- 1 Introduction
- 2.1 Basic notations
- 2.2 The relativistic transport operators
- 2.3 The foliations
- 2.4 Geometry of the hyperboloids
- 2.5 Regular distribution functions
- 2.6 The linear equations
- 2.7 The commutation vector fields
- 2.8 Weights preserved by the flow
- 2.9 Multi-index notations
- 2.10 Vector field identities
- 2.11 The particle vector field and the stress energy tensor of Vlasov fields
- 2.12 Commutation vector fields and energy densities
- 2.13 (Approximate) conservation laws for Vlasov fields
- 3 The vector field method for Vlasov fields
- 4 Applications to the Vlasov-Nordström system
- A Distribution functions for massive particles with compact support in
- B Integral estimate
- C Geometry of Vlasov fields
The vector field method of Klainerman [Kla85] provides powerful tools which are at the core of many fundamental results in the study of non-linear wave equations, such as the famous proof of the stability of the Minkowski space [CK93]. In essence, the method takes advantage of the symmetries of a linear evolution equation to derive in a robust way boundedness and decay estimates of solutions. The robustness is crucial, as the final aim is typically to prove the non-linear stability of some stationary solution, so that the method should be stable when perturbed by the non-linearities of the equations.
In this paper, we are interested in the massive and massless relativistic transport equations444We will be using in the whole article the Einstein summation convention. For instance, in (1) stands for .
where is the mass555In the remainder of this article, we will often normalize the mass to be either or and thus consider mostly and . We will however sometimes keep the mass so that the reader can see how some of the estimates would degenerate as . of the particles and is a function of defined on if and otherwise, with being the dimension of the physical space.
Decay estimates via the method of characteristics for relativistic transport equations
For transport equations, the standard method to prove decay estimates is the method of characteristics. The origin of these decay estimates goes back in the non-relativistic case to the work of Bardos-Degond [BD85] on the Vlasov-Poisson system. Recall that if is a regular solution to say then, for all ,
and assuming that has initially compact support in , one can easily infer the velocity average estimate, for all and all ,
where is an upper bound on the size of the support in of at the initial time and as .
These estimates, while being relatively easy to derive, suffer from two important drawbacks when applied to a non-linear system.
They require compact support in of the solutions. In a non-linear setting, one therefore needs to bound an extra quantity, the size of this support at . In particular, we are enlarging the number of variables of the system. Moreover, there are many interesting models where the correct assumptions from the point of view of physics666See for instance the end of the introduction in [Vil10]. is to allow arbitrary large .
They require a strong control of the characteristics of the system.
These two inconveniences are in fact related and the first can be somehow mitigated by an even finer analysis of the characteristics, see for instance [Sch04]. Concerning the second problem, we note that there are many evolution problems for which the characteristics in a neighbourhood of a stationary solution will eventually diverge from the original ones, introducing extra difficulties in the analysis. A famous example of that is the stability of Minkowski space, where there is a logarithmic divergence, see [CK93, LR10]. Moreover, to prove decay estimates such as (2), one needs to control the Jacobian associated with the differential of the characteristic flow777In the context of the Vlasov equation on a curved Lorentzian manifold, this means that one needs estimates on the differential of the exponential map, or at least on its restriction to certain sub-manifolds. and in order to obtain improved decay estimates for derivatives, one also needs estimates on the derivatives of the Jacobian. See for instance [HRV11] where such a program is carried out for the Vlasov-Poisson system. In other words, one needs strong control on the characteristics to be able to prove sharp decay estimates via this method in a non-linear setting.
Decay estimates for the wave equation
In the context of the wave equation
several methods exist to prove decay estimates of solutions. For instance, one standard way is to use the Fourier representation of the solution together with estimates for oscillatory integrals. In the fundamental article [Kla85], Klainerman introduced what is now referred to as the vector field method888Let us also mention that, complementary to the method of Klainerman, which uses vector fields as commutators, one can also use vector fields as multipliers, in the style of the work of Morawetz, see for instance [Mor62, Mor68].. Instead of relying on an explicit integral representation of the solutions, it uses
a coercive conservation law. In the case of the wave equation, this is simply the conservation of the energy.
a set of vector fields which commute with the equations. In the case of the wave equation, these are the Killing and conformal Killing fields of the Minkowski space.
weighted Sobolev inequalities: the standard derivatives , are rewritten in terms of the commutator vector fields before applying the usual Sobolev inequalities. The weights in these decompositions together with those arising from the conservation laws are then translated into decay rates.
This leads to the decay estimate
for solutions of the wave equation , where is an energy norm obtained by integrating and derivatives of (with weights) at time .
These types of estimates, being based on conservation laws and commutators, are quite robust, and as a consequence, are applicable in strongly non-linear settings, such as the Einstein equations or the Euler equations (see for instance [Chr07] for such an application).
A vector field method for transport equations
In our opinion, the decay estimate (2), being based on an explicit representation of the solutions, that given by the method of characteristics, should be compared to the decay estimates for the wave equation obtained via the Fourier or other integral representations. In this paper, we derive an analogue of the vector field method for the massive and massless relativistic transport equations (1). The coercive conservation law is given by the conservation of the norm of the solution, while the vector fields commuting with the operators are essentially obtained by taking the complete lifts of the Killing and conformal Killing fields, a classical operation in differential geometry which takes a vector field on a manifold to a vector field on the tangent bundle . The weighted Sobolev inequalities are slightly more technical. One of the main ingredients is that averages in possess good commutation properties with the Killing vector fields and their complete lifts. Our decay estimates can then be stated as
Theorem 1 (Decay estimates for velocity averages of massless distribution functions).
For any regular distribution function , solution to and any , we have
where the are multi-indices of length and the are differential operators of order obtained as a composition of vector fields of the algebra .
The detailed list of the vector fields and their complete lifts used here is given in Section 2.7.1. For the massive transport equation, we prove
Theorem 2 (Decay estimates for velocity averages of massive distribution functions).
For any regular distribution function , solution to , any and any , we have
where denotes the unit hyperboloid , is the restriction to of , is the contraction of the -velocity with the unit normal to and where the are differential operators obtained as a composition of vector fields of the algebra .
No compact support assumptions in or in are required for the statements of Theorems 1 or 2. Of course, for the norms on the right-hand sides of (4) or (5) to be finite, some amount of decay in and is needed. Note that from the point of view of non-linear applications, it is sufficient to propagate bounds for the norms appearing on the right-hand sides of (4) or (5), without any need to control pointwise the decay in or of the solutions, to get the desired decay estimates for the velocity averages.
As in the case of the Klein-Gordon equation, one can easily prove that for regular solutions to with data given at and decaying sufficiently fast as (in particular, solutions arising from data with compact support in ) the norm on the right-hand side of (5) is finite, so that the decay estimate applies999See also [Geo92], where decay estimates for the Klein-Gordon operator were obtained starting from non-compactly supported data at using (mostly) vector field type methods.. Thus, the use of hyperboloids is merely a technical issue. The restriction “” simply means in the future of the unit hyperboloid. We provide a classical construction in Appendix A, which explains how Theorem 2 can be applied to solutions arising from initial data with compact -support given at to obtain a decay of velocity averages in the whole future of the hypersurface.
The reader might wonder whether the same type of techniques could be applied for the classical transport operator . This question was addressed in [Smu15] where decay estimates for velocity averages of solutions to the classical transport operator were obtained. As an application, [Smu15] considered the study of small data solutions of the Vlasov-Poisson system and provided an alternative proof (with some additional information on the asymptotic behaviour of the solutions, concerning in particular the decay in and uniform bounds on some global norms) of the optimal time decay for derivatives of velocity averages obtained first in [HRV11]. One of the nice features of the vector field method is that improved decay estimates for derivatives follow typically easily from the main estimates, and [Smu15] was no exception. In the relativistic case, our vector field method also provides improved decay for derivatives. See Propositions 28 and 30 in Sections 3.2 and 3.3, respectively.
Applications to the massive and massless Vlasov-Nordström systems
In the second part of this paper, we will apply our vector field method to the massive and massless Vlasov-Nordström systems
where in the massless case and in the massive case, is the standard wave operator of Minkowski space, is a scalar function of and is as before a function of with , if , if . A good introduction to this system can be found in [Cal03]. See also the classical works [Cal06, Pal06].
Roughly speaking, the Vlasov-Nordström system can be derived from the Einstein-Vlasov system by considering only a special class of solutions (that of metrics conformal to the Minkowski metric) and by neglecting some of the non-linear interactions in the (Einstein part of) the equations. Since most of the simplifications concern difficulties which we already know how to handle (in the style of [CK93] or [LR10]) and since the method that we are using here is of the same type as the one used to study the Einstein vacuum equations, we believe it is a good model problem before addressing the full Einstein-Vlasov system via vector field methods.
Before presenting our main results for the massive and massless Vlasov-Nordström systems, let us explain the main differences between the and cases. First, as easily seen from (6)-(7), when , the system degenerates to a partially decoupled system101010In fact, using as an unknown, we can obtain an even simpler form of the equations where the right-hand side of (9) is put to . See (51).
Because of the decoupling, the first equation is simply the wave equation on Minkowski space and the second can be viewed as a linear transport equation, where the transport operator is the massless relativistic transport operator plus some perturbations. In particular, all solutions are necessarily global as long as the initial data is sufficiently regular so that the linear equations can be solved. Thus, our objective is solely to derive sharp asymptotics for the solutions of the transport equation. Moreover, since we have in mind future applications to more non-linear problems, the only estimates that we will use on will be those compatible with what can be derived via a standard application of the vector field method.
Apart from the decoupling just explained, let us mention also two important pieces of structure present in the above equations. First, another great simplification comes from the existence of an extra scaling symmetry present only in the massless case: the vector field commutes with the massless transport operator and it is precisely this combination of derivatives in which appears in the equation (9). This fact will make all the error terms obtained after commutations much better than if a random set of derivatives in was present in (9). Another property of (8)-(9) is the existence of a null structure, similar to the null structure of Klainerman for wave equations. More precisely, we show that has roughly the structure
where denotes derivatives tangential to the outgoing cone, denotes arbitrary derivatives of and are weights which are bounded along the characteristics of the linear massless transport operator. Since has better decay properties than a random derivative , we see that products of the form , where is a solution to have better decay properties than expected111111It is interesting to compare this form of the null condition to the one uncovered in [Daf06] for the massless Einstein-Vlasov system in spherical symmetry. In fact, two null conditions were used there. The obvious one consists essentially in understanding why null components of the energy momentum tensor of decay better than expected. A more secret null condition is used in the analysis of the differential equation satisfied by the part of the velocity vector tangent to the outgoing cone. Our null condition is closely related to this one, even though we exploit it in a different manner since we are not using directly the characteristic system of ordinary differential equations associated with the transport equations.. Similar to the study of dimensional wave equations with non-linearities satisfying the null condition, the extra decay obtained means that in dimension (or greater), all the error terms in the (approximate) conservation laws are now integrable.
We now state our main results for the massless Vlasov-Nordström system.
Theorem 3 (Asymptotics in the massless case for dimension ).
Let and . Let be a solution of (8) satisfying for some sufficiently regular functions . Then, if , where is an energy norm containing up to derivatives of and if , where is a norm containing up to derivatives of , then the unique classical solution to (9) satisfying verifies
Global bounds: for all ,
where is a constant depending only on .
Pointwise estimates for velocity averages: for all and all multi-indices satisfying ,
In dimension , similar results can be obtained provided an extra smallness assumption on the initial data for the wave function as well as stronger decay for the initial data of hold.
Theorem 4 (Asymptotics in the massless case for dimension ).
Let , and . Let be a solution of (8) satisfying for some sufficiently regular functions . Then, if , where is an energy norm containing up to derivatives of and if , where is a norm121212The index refers to powers of certain weights. See (4.2.4) for a precise definition of the norms. containing up to derivatives of , then the unique classical solution to (9) satisfying verifies
Global bounds with loss: for all ,
where depends only on .
Improved global bounds for lower orders: for any and any ,
Pointwise estimates for velocity averages: for all and all multi-indices ,
Perhaps counter-intuitively, the massive case turned out to be harder to treat. While it is true that in the massive case, the pointwise decay of velocity averages is not weaker along the null cone, there are two important extra difficulties, namely
The equations are now fully coupled. In particular, one cannot close an energy estimate for (6) unless we have some decay for the right-hand side. On the other hand, our decay estimates, being based on commutators, necessarily lose some derivatives. In turns, this would imply commuting (9) more, but we would then fail to close the estimates at the top order. We resolve this issue by another decay estimate for inhomogeneous transport equations with rough source terms satisfying certain product structures. This other type of decay estimates only provides time decay of the velocity averages, which is precisely what is required to close the energy estimate for (6). The proof of this decay estimate itself can be reduced to our estimates, so that it can also be obtained using purely vector field type methods.
The vector field does not commute with the massive transport operator. This implies that commuting with (some of) the standard vector fields will lose a power of of decay compared to the massless case.
Because of the last issue, the results that we will present here are restricted to dimension . One way to treat the -dimensional case would be to improve upon the commutation formulae to eliminate the most dangerous terms. For instance, one could try to use modified vector fields in the spirit of [Smu15]. We plan to address the dimensional case in future work.
A slightly more technical consequence of this last issue is that it introduces weights in the estimates, which are not constant on the leaves of the hyperboloidal foliation that we wish to use. Together with the fact that the energy estimates are weaker on hyperboloids, this implies that the error terms arising in the top-order approximate conservation laws can be shown to be space-time integrable only in dimension . To address the dimension, instead of estimating directly , where is the unknown distribution functions and is a combination of vector fields, we estimate instead a renormalized quantity of the form where the is a (small) non-linear term constructed from the solution. The extra terms appearing in the equation when the transport operator hits will then cancel some of the worst terms in the equations.
Our main result in the massive case can then be stated as follows.
Let and . Let be sufficiently large depending only on . For any , denote by the hyperboloid
For any sufficiently regular function defined on , denote by its restriction to . Similarly, for any sufficiently regular function defined on , denote by its restriction to . Then, there exists an such that for any , if , where and are norms depending on respectively derivatives of and derivatives of , then there exists a unique classical solution to (6)-(7) satisfying the initial conditions
such that exists globally131313Here, globally means at every point lying in the future of the initial hyperboloid . In , this, of course, would already follow from the work [Cal06] for regular initial data of compact support given on a constant time slice. and satisfies the estimates
Global bounds, for all ,
where when and when .
Pointwise decay for : for all multi-indices such that and all with , we have
Pointwise decay for : for all multi-indices and such that , and all with , we have
where when and when .
Finally, the following -estimates on hold: for all multi-indices with , and all with , we have
where when and when .
As for the linear decay estimates of Theorem 2, it is not essential to start on an initial hyperboloid for the conclusions of Theorem 5 to hold. In particular, an easy argument based on finite speed of propagation, similar to that given in Appendix A, shows that our method and results apply to the case of sufficiently small initial data with compact -support given at .
In [Fri04], solutions of the massive Vlasov-Nordström system in dimension arising from small, regular, compactly supported (in and ) data given at were studied and the asymptotics of velocity averages of the Vlasov field and up to two derivatives of the wave function were obtained. However, no estimates were obtained for derivatives of the Vlasov field or for higher derivatives of the wave function. Thus, [Fri04] is the analogue of [BD85] for the Vlasov-Nordström system while we obtained here (in dimension and greater) results more in the spirit of [HRV11, Smu15].
A posteriori, it is straightforward to propagate higher moments of the solutions in any of the situations of Theorems 3, 4, 5, provided that these moments are finite initially. Moreover, we recall that improved decay for derivatives of and follows from the statements of Theorems 3, 4 and 5. See for instance Propositions 28 and 30 below.
Aside: the Einstein-Vlasov system
As explained above, the Vlasov-Nordström system is a model problem for the more physically relevant Einstein-Vlasov system. We refer to the recent book141414Apart from a general introduction to the Einstein-Vlasov system, the main purpose of this book is to present a proof of stability of exponentially expanding spacetimes for the Einstein-Vlasov system, see [Rin13]. [Rin13] for a thorough introduction to this system. The small data theory around the Minkowski space is still incomplete for the Einstein-Vlasov system. The spherically symmetric cases in dimension have been treated in [RR92, RR96] for the massive case and in [Daf06] for the massless case with compactly supported initial data. A proof of stability for the massless case without spherical symmetry but with compact support in both and has recently been announced in [Tay15]. As in [Daf06], the compact support assumptions and the fact that the particles are massless are important as they allow to reduce the proof to that of the vacuum case outside from a strip going to null infinity. Interestingly, the argument in [Tay15] is quite geometric, relying for instance on the double null foliation, in the spirit of [KN03], as well as several structures associated with the tangent bundle of the tangent bundle of the base manifold.
We hope to address the stability of the Minkowski space for the Einstein-Vlasov system in the massive and massless case (without the compact support assumptions) using the method developed in this paper in future works.
Structure of the article
Section 2 contains preliminary materials, such as basic properties of the transport operators, the definition and properties of the foliation by hyperboloids used for the analysis of the massive distribution function, the commutation vector fields and elementary properties of these vector fields. In Section 3, we introduce the vector field method for relativistic Vlasov fields and prove Theorems 1 and 2. In Section 4, we apply our method first to the massless case in dimension (Section 4.2.3) and (Section 4.2.4) and then to the massive case in dimension (Section 4.3). In Appendix A, we provide a classical construction which explains how our decay estimates in the massive case can be applied to data of compact support in given at . Some integral estimates useful in the course of the paper are proven in Appendix B. Finally, Appendix C contains a general geometrical framework for the analysis of the Vlasov equation on a Lorentzian manifold.
We would like to thank Martin Taylor for several interesting discussions on his work on the massless Einstein-Vlasov system. We would also like to thank Olivier Sarbach for his geometric introduction to Vlasov fields. The second and third authors are partially funded by ANR-12-BS01-012-01 (AARG). The third author is also partially funded by ANR SIMI-1-003-01. The first author gratefully acknowledges the travel support by ANR-12-BS01-012-01 (AARG).
2.1 Basic notations
Throughout this paper we work on the -dimensional Minkowski space , where the standard Minkowski metric is globally defined in Cartesian coordinates by . We denote spacetime indices by Greek letters and spatial indices by Latin letters . We will sometimes use to denote the partial derivatives ,
Since we will be interested in either massive particles with or massless particles , the velocity vector will be parametrized by and in the massless case, in the massive case.
The indices and will be used to denote objects corresponding to the massless or massive case, such as the massless transport operator and the massive one and should not be confused with spatial or spacetime indices for tensor components (we use bold letters on the transport operators to avoid this confusion).
The notation will be used to denote an inequality of the form , for some constant independent of the solutions (typically will depend on the number of dimensions, the maximal order of commutations , the value of the mass ).
2.2 The relativistic transport operators
For any and any , let us define the massive relativistic transport operator by
Similarly, we define for any , the massless transport operator by
For the sake of comparison, let us recall that the classical transport operator is given by
In the remainder of this work, we will normalize the mass to be either or , so that the massive transport operators we will study are
2.3 The foliations
We will consider two distinct foliations of (some subsets of) the Minkowski space.
Let us fix global Cartesian coordinates , on and denote by the hypersurface of constant . The hypersurfaces , then give a complete foliation of . The second foliation is defined as follows. For any , define by
For any , is thus only one sheet of a two sheeted hyperboloid.
The above subset of will be referred to as the future of the unit hyperboloid. On this set, we will use as an alternative to the Cartesian coordinates the following two other sets of coordinates.
We first consider spherical coordinates on , where denotes spherical coordinates on the dimensional spheres and . then defines a coordinate system on the future of the unit hyperboloid. These new coordinates are defined globally on the future of the unit hyperboloid apart from the usual degeneration of spherical coordinates and at .
These are the coordinates . These new coordinates are also defined globally on the future of the unit hyperboloid.
For any function defined on (some part of) the future of the unit hyperboloid, we will move freely between these three sets of coordinates.
2.4 Geometry of the hyperboloids
The Minkowski metric is given in coordinates by
where is the standard round metric on the dimensional unit sphere, so that for instance
in standard spherical coordinates for the -sphere. The dimensional volume form is thus given by
where is the standard volume form of the dimensional unit sphere.
The Minkowski metric induces on each of the a Riemannian metric given by
A normal differential form to is given by while is a normal vector field. Since
the future unit normal vector field to is given by the vector field
Finally, the induced volume form on , denoted , is given by
2.5 Regular distribution functions
For the massive transport operator, we will consider distribution functions as functions of or defined on
i.e. we are looking at the future of the unit hyperboloid, or a subset of it, times .
For the massless transport operator, we need to exclude and we will only use the foliation so that we will consider distribution functions as functions of defined on , .
In the remainder of this article, we will denote by regular distribution function any such function that is sufficiently regular so that all the norms appearing on the right-hand sides of the estimates are finite. For simplicity, the reader might assume that is smooth and decays fast enough in and at infinity and in the massless case, that is integrable near and similarly for the distribution functions obtained after commutations.
In physics, distribution functions represent the number of particles and are therefore required to be non-negative. This will play no role in the present article so we simply assume that distribution functions are real valued.
2.6 The linear equations
In the first part of this paper, we will study, for any , the homogeneous transport equation
as well as the inhomogeneous transport equation
where in the massive case and in the massless case and where the source term is a regular distribution function, as explained in Section 2.5.
where for the massive case and in the massless case. In the inhomogeneous case, we obtain via the Duhamel formula that if solves (14) with initial data, then
2.7 The commutation vector fields
2.7.1 Complete lifts of isometries and conformal isometries
Let us recall that the set of generators of isometries of the Minkowski space, that is to say the set of Killing fields, denoted by , consists of the translations, the rotations and the hyperbolic rotations, i.e.
Mostly in the case of the massless transport operator, it will be useful, as in the study of the wave equation, to add the scaling vector field to our set of commutator vector fields. Let us thus define the set
The vector fields in or lie in the tangent bundle of the Minkowski space. To any vector field on a manifold, we can associate a complete lift, which is a vector field lying on the tangent bundle to the tangent bundle of the manifold. In Appendix C, we recall the general construction on a Lorentzian manifold. For the sake of simplicity, let us here give a working definition of the complete lifts only in coordinates.
Let be a vector field of the form , then let
where with in the massless case, in the massive case, be called the complete lift151515This is in fact a small abuse of notation, as, with the above definition, actually corresponds to the restriction of the complete lift of to the submanifold corresponding to in the massive case and in the massless case. See again Appendix C for a more precise definition of . of .
We will denote by
the sets of the complete lifts of and .
Finally, let us also define and as the sets composed respectively of and and a scaling vector field161616Here, by a small abuse of notation, we denote with the same letter , the vector field irrespectively of whether we consider it as a vector field on or a vector field on (some subsets of) . in only:
In Cartesian coordinates, the complete lifts of the elements of and are given by the following formulae:
2.7.2 Commutation properties of the complete lifts
Commutation rules for the massive transport operator
where is the usual scaling vector field.
Commutation rules for the massless transport operator
Note that from the expression of and the two commutation rules for and and for and , it follows that
Thus, we have in a certain sense two scaling symmetries, one in and one in .
It is interesting to note that while the Klein-Gordon operator () does not commute with the scaling vector field, the massive transport equation does commute in the form of equation (19). What does not commute is the second scaling vector field .
We also have the following commutation relation within and .
For any , there exist constant coefficients such that
Similarly, for any , there exist constant coefficients such that
2.8 Weights preserved by the flow
Recall that in a general Lorentzian manifold with metric , if is a geodesic with tangent vector and denotes a Killing field, then is preserved along . In this section, we explain how to transpose this fact to the transport operators on Minkowski space.
We define the set of weights
The following lemma can be easily checked.
For all , .
For all , .
The weights in or also have nice commutation properties with the vector fields in and .
For any and any ,
where the are constant coefficients.
Similarly for any and any ,
for some constant coefficients .
This follows from straightforward computations. ∎
2.9 Multi-index notations
Recall that a multi-index of length is an element of for some such that .
Let be an ordering of . For any multi-index , we will denote by