On the Boltzmann Equation withStochastic Kinetic Transport: Global Existence ofRenormalized Martingale Solutions

On the Boltzmann Equation with
Stochastic Kinetic Transport:
Global Existence of
Renormalized Martingale Solutions

Samuel Punshon-Smith
University of Maryland, College Park - punshs@math.umd.edu
Scott Smith Max Planck Institute for Mathematics in the Sciences, Leipzig - ssmith@mis.mpg.de
July 31, 2019
Abstract

This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions [17]) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. This study includes a criterion for renormalization, the weak closedness of the solution set, and tightness of velocity averages in .

1 Introduction

The Boltzmann equation

(2)

on is a nonlinear integro-differential equation describing the evolution of a rarefied gas, dominated by binary collisions, and in the presence of a external force field . The function describes the density of particles at time , position , with velocity , starting at from an initial density . The nonlinear functional , known as the collision operator, acts on the velocity variable only, and accounts for the effect of collisions between pairs of particles; it will be described in more detail below.

Several studies have been conducted regarding the well-posedness of the Cauchy problem for the Boltzmann equation (2) with a fixed (deterministic) external force, for instance [4, 7, 19, 45]. In general, the external force field may depend on . Such external forces may arise when considering the influence of gravity such as in the treatment of the Rayleigh-Benard problem in the kinetic regime [20, 3]. In fact, many external forces are not fixed, and are instead coupled with the density in a self consistent way. This is the case, for example, with the Vlasov-Poisson-Boltzmann and Vlasov-Maxwell-Boltzmann equations (see [12, 34] and references therein for more details on these systems).

This article focuses instead on the Cauchy problem for the Boltzmann equation with random external forcing. In particular, we are interested in the following SPDE

(SB)

where are one-dimensional Brownian motions and are a family of vector fields with . An implicit summation is taken over , and the expression denotes a transport type multiplicative noise, white in time and colored in , where the product is interpreted in the Stratonovich sense.

Physically, we view the quantity

(3)

as an environmental noise acting on the gas. In the absence of collisions, all particles evolve according to the stochastic differential equation

(4)

and are only distinguished from one another according to their initial location in the phase space. Let be the stochastic flow associated with the SDE (4), that is, solves (4) and satisfies . The Stratonovich form of the noise and the fact that ensures that the flow is volume preserving (with probability one). The density of the collision-less gas is then given by and evolves according to the free stochastic kinetic transport equation

(5)

The presence of collisions interrupts the stochastic transport process. In the low volume density regime, binary collisions are dominant and can be described by the Boltzmann collision operator . The stochastic Boltzmann equation (SB) accounts for both stochastic transport and binary collisions. In fact, formally (SB) can be written in mild form,

(6)

The stochastic Boltzmann equation (SB) can be interpreted as the so-called Boltzmann-Grad limiting description of interacting particles subject to the same environmental noise. In the deterministic setting, the Boltzmann-Grad problem has been studied extensively in the literature (see [25] for a recent review). In the stochastic setting, the Boltzmann-Grad problem has (to our knowledge) not yet been studied. However, a mean field limit to the Vlasov equation with stochastic kinetic transport has been shown recently by Coghi and Flandoli [14].

To our knowledge, this is the first study to obtain mathematically rigorous results on the Boltzmann equation with a random external force. However, a number of results on the fluctuating Boltzmann equation are available in the Math and Physics literature [8, 24, 37, 43, 41, 42, 44]. In particular, the articles of Bixon/Zwanzig [8] and Fox/Uhlenbeck [24] outline a formal derivation of Landau and Lifshitz’s equations of fluctuating hydrodynamics [33], from the fluctuating linear Boltzmann equation. The connection with macroscopic fluid equations arises from studying the correlation structure of the fluctuations at the level of the kinetic description. A more rigorous treatment of the fluctuation theory for the Boltzmann equation and its connection to the Boltzmann-Grad limit is given by Spohn [43, 41, 42].

Although our perspective differs from that of [24] and [8], we do expect to obtain various stochastic hydrodynamic equations (with colored noise) in different asymptotic regimes, using a Chapman-Enskog expansion and the moments method of Bardos/Golse/Levermore [6]. In fact, one of the original motivations for this article was to understand which of the common forms of noise in the stochastic fluids literature can be obtained by considering fluctuations of the stochastic kinetic description relative to an equilibrium state. This will be addressed in detail in future works.

The goal of this article is to investigate global solutions to (SB) starting from general ‘large’ initial data . If the noise coefficients are identically zero, then this problem has already been addressed in the seminal work of DiPerna/Lions [17], where existence of renormalized solutions is proved. Our work is heavily inspired by [17], relying on a number of their insights together with various classical properties of the Boltzmann equation. Rather than give a detailed review, in the next subsection we will explain how these observations from the deterministic theory lead to the notion of renormalized martingale solution to (SB) in the present context. Finally, we should mention that our initial motivation for the choice of noise was heavily inspired by a number of interesting works on stochastic transport equations (see for instance [16, 22, 23, 21]). Finally, we should mention the work [11] on the 2-d stochastic Euler equations with a very similar noise to the one in this paper.

1.1 Notation and statement of the main result

We begin by giving a summary of the main notation used in the paper. To simplify the appearance of the function spaces, we will use a number of abbreviations. Since the functions we are dealing with are typically kinetic densities the variables and will be reserved for positions and velocity variables, while will be reserved for time. We will typically fix an arbitrary but finite time and consider evolution on the time interval and denote by the space of real valued functions on with norm

(7)

with the usual interpretation in terms of the essential supremum when or are . Naturally, is also short for the Lebesgue space . A similar subscript notation will hold for Sobolev spaces, namely will denote the space of functions whose -th order derivatives also belong to . Given a test function and a function in we will often denote the pairing by

(8)

This should, however, should not be confused with the velocity ‘average’

(9)

which integrates in only.

In general, given a topological space , with a particular topology, we denote by the space of continuous functions . Furthermore, given a Banach space , we denote the space endowed with its weak topology by , and therefore space of weakly continuous functions from to is . Also, given a Lebesgue space (or Sobolev Space), we denote the usual Frechet space of locally integrable functions endowed with it’s natural local topology.

We should also introduce some probablisitic notation. For a given probability space , we let be the expectation associated to the probability measure . Also, given a Banach space (not necessarily separable) with norm , we will denote by , the Lebesgue-Bochner space of equivalence classes of strongly measurable maps (random variables) such that is finite. We will often use the space

(10)

cosisting of random variables all moments finite (but not ). We will often suppress the dependence of a random variable on the probablistic variable , except when explicitly needed.

Let us now discuss the basics of the Boltzmann equation and introduce the analytical framework for the problem. We refer the reader to the books [12, 13] and the excellent set of notes [29] for a comprehensive introduction to the Boltzmann equation, as well as the review [47]. The collision operator describes the rate of change in particle density due to collisions. It contains all the information about collision rates between particles with different velocities. More precisely, it is defined through its action in as

(11)

where , , and are shorthand for , and , while denote pre-collisional velocities

(12)

Note that , parametrized by , are solutions to the equations describing pairwise conservation of momentum and energy,

(13)

The collision kernel is determined by details of the inter-molecular forces between particles and describes the rate at which particles with relative velocity collide with deflection angle . In this article, for technical reasons and simplicity of exposition, we restrict our attention to bounded, integrable kernels, though we intend to investigate (in a future work) the possibility of treating more singular kernels as in Alexandre/Villani [1] and other works. Our assumption on the collision kernel is the following:

Hypothesis 1.1.

The collision kernel depends solely on and only, and satisfies,

(14)

Since the nonlinear term is quadratic in , further properties of the operator must be exploited in order to obtain a priori bounds. A classical observation is that the symmetry assumptions on the collision kernel imposed in Hypothesis 1.1 and the definition of imply that for each smooth ,

(15)

Any quantity such that , is called a collision invariant. For any collision invariant , (15) implies that

(16)

As a result of the definition of , the quantities are collision invariants. Therefore, multiplying both sides of (SB) by a collision invariant and integrating in , the collision operator vanishes

(17)

In the case that in (17), one can close on estimate on , provided we have the following coloring hypothesis on :

Hypothesis 1.2.

For each , the noise coefficient satisfies . In addition, the sequence obeys:

(H1)
Remark 1.1.

Note that Hypothesis 1.2 states that the coefficients are bounded in both and , but Lipschitz in only.

More generally, in Section 2 we show that Hypothesis 1.2 implies that a solution to (SB) satisfies the following formal a priori bound

(18)

for all and some positive constant (depending on ). In addition, a further estimate on is available due to the entropy structure of (SB). To obtain this, let be a sufficiently smooth function, which we will refer to as a renormalization. Since we use Stratonovich noise and , if is a solution of (SB), then formally should satisfy:

(RSB)

In particular, taking in (RSB) and integrating in yields

(19)

where

(20)

Equation (19) describes the local dissipation of the entropy density . The quantity is referred to as the entropy dissipation, and inherits non-negativity from . Since is unsigned, we cannot immediately use (19) to obtain an bound. However, combining this with (18), in Section 2 we show that for all

(21)

Although the a priori bounds (18) and (21) provide a useful starting point, they are unfortunately insufficient to give a meaning to in the sense of distributions. For bounded kernels, one can obtain an estimate on ,

(22)

However, since acts pointwise in , the operator sends to (a measurable function in ). A key observation of DiPerna and Lions [17] is that the renormalized collision operator is better behaved. More precisely, the following inequality holds:

(23)

Thus, if satisfies the a priori bounds (18) and (21), the quantity is well defined in . Hence, it becomes feasible to search for solutions satisfying (RSB) in the sense of distributions for a suitable class of renormalizations. Towards this end, we make the following definition:

Definition 1.1.

Define the set of renormalizations to consist of functions such that the mapping belongs to .

It is important to keep in mind that this class of renormalizations excludes the possibility of choosing or and therefore extra care must be taken to obtain the a priori estimates (18) and (21) above.

We note that for analytical purposes, relating to martingale techniques, it is often more convenient to work with (RSB) in Itô form. Thus, we introduce the matrix

(24)

and define the operator

(25)

Using the divergence free assumption for each , the random transport term in (RSB) can be converted to Itô form via the relation

(26)

We are now ready to define our notion of solution for (RSB).

Definition 1.2.

A density is defined to be a renormalized martingale solution to (SB) provided there exists a stochastic basis such that the following hold:

  1. For all , the quantity is a non-negative element of .

  2. The mapping defines an adapted process with continuous sample paths.

  3. For all renormalizations , test functions , and times ; the following equality holds almost surely:

  4. For all there exists a positive constant such that:

    (27)
Remark 1.2.

In light of the estimate (23), the estimates in condition 4 of Definition 1.2 ensure that the weak form (3) is well defined and the stochastic integral is a continuous-time martingale.

At present, we require a further technical hypothesis on and . This is related to the regularity needed on to renormalize a linear, stochastic kinetic transport equation, a crucial procedure in our analysis. This is discussed in more detail in Section 1.3 below.

Hypothesis 1.3.

There exists an such that for each compact set

The main result of this article is the following global existence theorem:

Theorem 1.1.

Let be a collection of noise coefficients satisfying Hypotheses 1.2 and 1.3 and assume that the collision kernel satisfies Hypothesis 1.1. For any initial data satisfying

(28)

there exists a renormalized martingale solution to (SB), starting from with noise coefficients .

Moreover satisfies

  • almost sure local conservation of mass

    (29)
  • average global balance of momentum

    (30)
  • average global energy inequality

    (31)
  • almost sure global entropy inequality

    (32)

The almost sure local conservation of mass holds almost surely in distribution, the average global momentum and energy balances hold for every , and the global entropy inequality holds almost surely for every .

1.2 Outline of the deterministic theory

In order to motivate many of the steps of the proof of theorem 1.1, and make more explicit the similarities and differences to the deterministic theory of renormalized solutions for the Boltzmann equation. We will outline a rough sketch of the strategy of proof in the the determinisitic setting and emphasize several key steps. For a more detailed exposition of the deterministic theory, we direct the reader to the original work of DiPerna/Lions [17] as well as some later improvements [34] with regard to dealing with passing limits in the collision operator. A detailed exposition of modern theory can be found in the notes [29] and a very nice sketch of the deterministic theory can be found in the book by Saint-Raymond [40].

To illustrate the existence theory for renormalized solutions, it suffices to prove a stability result. That is, suppose we have a sequence of renormalized solutions to the deterministic Boltzmann equation

(33)

with compact initial data and satisfying the uniform apriori estimates

(34)

Then we would like to show that any limit point of the sequence is also a renormalized solution to the Boltzmann equation and satisfies the same apriori estimates. Indeed, this was the approach initially take by DiPerna/Lions [17].

The proof of the deterministic stability result can conceptually be broken into three main steps: a weak compactness step using the apriori estimates, a strong compactness step using the equation and velocity averaging Lemmas, and a limit passage step using both the weak and strong compactness to pass the limit in the renormalized equation.

The first conceptual step is to obtain several weak compactness results using the apriori estimates (34). Using the classical Dunford Pettis criterion, one can easily obtain from the first uniform estimate in (34) that the sequence is weakly relatively compact in . Furthermore, using the uniform entropy dissipation bound in (34) and a bound on the gain part of the collision operator, one can show that the sequence

(35)

The next step is to use the equation to deduce various strong compactness results using the equation. Here one chooses a sequence of renormalizations

(36)

which approximate the identity as . Since is a suitable renormalizer then solves

(37)

Using the weak compactness of deduced from the apriori estimates, we may deduce that is relatively compact in . Furthermore, we are now in a position to use the subtle regularizing effects present in the kinetic equation (37) known as velocity averaging. Here, one can deduce that for every test function the velocity averages

(38)

where the angle brackets denote integration in velocity. Using the weak relative compactness of one can show that approximates as uniformly in and strongly in , namely

(39)

This allows one to deduce compactness results on from concluding that is compact in and the velocity averages are relatively compact in .

At this stage one has enough compactness to pass certain limits in the collision operators, suitably renormalized. Using the product limit Lemma B.1, one can deduce the following that, up to a subsequence, for each ,

(40)

The next step is to choose a bounded sequence of renormalizers,

(41)

and using the weak compactness already deduced, conclude that

(42)

and

(43)

Again, using the uniform estimate (39), with replacing , one can show the following strong convergence

(44)

The idea is then to pass the limit in the equation

(45)

The strong convergence (44) is enough to pass the limit on the left side in the sense of distribution. It remains to pass the limit on the right-hand side as . This is the most technical part of the analysis and requires limit (40) as well as all of the compactness and apriori estimates deduced on . The main result is that

(46)

which allows one to pass the limit on the right-hand side of (45) and conclude

(47)

It remains to show the the the equation also holds for any admissible renormalization .

(48)

This can be done by renormalizing equation (47) with a truncation of

(49)

and using the bound on to pass the limit in the truncation.

1.3 Overview of the article

Now we proceed to an overview of the article, addressing the main difficulties overcome and the relation of our work to the existing literature on kinetic equations and stochastic PDE’s.

Our analysis begins with formal a priori estimates which point to the natural functional framework for (SB). Namely, in Section 2 we show that under the coloring Hypothesis (H1), solutions to (SB) formally satisfy

With these formal a priori bounds at hand, the remainder of the paper splits roughly into two parts. In Sections 3 and 4, we analyze linear stochastic kinetic equations, while Sections are devoted to the proof of Theorem 1.1.

In Sections 3 and 4 we move to a detailed discussion of stochastic kinetic equations of the form

(50)

Here is a deterministic initial density, while is a certain random variable with values in . We will focus on so-called weak martingale solutions to (50). Roughly speaking (see Definition 3.1 of Section 3.1 for the precise meaning), these are valued stochastic processes satisfying (50) weakly in both the PDE and the probabilistic sense. In this context, probabilistically weak means that the filtered probability space and the Brownian motions are not fixed in advance, but found as solutions to the problem, along with the process solving (50) in the sense of distribution.

For convenience we introduce the following language to refer to solutions of (50), we say that: is a solution to the stochastic kinetic equation driven by and starting from , relative to the noise coefficients and the stochastic basis . In the case that the coefficients , the filtration , and the Brownian motions are implicitly known or irrelevant, we may omit them from the statement, saying instead: is a solution to the stochastic kinetic equation driven by and starting from .

A key workhorse for our analysis is a stability result (Proposition 3.3) for weak martingale solutions to stochastic kinetic equations. In the deterministic setting, this simply corresponds to the observation that the space of solutions to linear, kinetic equations is closed with respect to convergence in distribution. More precisely, if there are functions , and satisfying

(51)

in the sense of distributions and converges to in the sense of distributions, then it readily follows from the linear structure of the equation that the limits also satisfy

(52)

in the sense of distributions. In the stochastic framework, an additional subtlety arises. Namely, one should distinguish between stability of stochastically strong solutions, where a stochastic basis has been fixed, and stability of stochastically weak solutions, where each solution comes equipped with its own stochastic basis. For a fixed stochastic basis and noise coefficients , one can use the linearity of together with a method of Pardoux [39] to make a direct passage to the limit on both sides of the equation. However, for stochastically weak solutions, the Brownian motions are not fixed, and the mapping is nonlinear, prohibiting the passage of weak limits. In this situation, a martingale method is used to overcome this difficulty and produce another weak martingale solution with a new stochastic basis. This result is detailed in Proposition 3.3.

Section 3.3 is devoted to renormalizing weak martingale solutions to stochastic kinetic equations. The technique of renormalization of deterministic transport equations originates from the now classical results of Di’Perna and Lions [18], where they were able to show uniqueness to certain linear transport equations when the drift has lower regularity that the classical theory of characteristics would allow. Formally, the strategy is as follows: if satisfies (50) and is a smooth renormalization, then satisfies

(53)

If one can justify such a computation, then upon integrating both sides of the equation (53) for certain non-negative choices of that vanish only at , for instance , then one can get explicit bounds on in terms of the initial data, which, by linearity, implies uniqueness. However, since we are working with analytically weak solutions to (50), this formal calculation may fail if the individual are too rough. In particular (to our knowledge), only requiring the coloring hypothesis (H1) is insufficient. The ability to renormalize stochastic kinetic transport equations will turn out to be a crucial property in the final stages of main existence proof. However, as in the case of the deterministic Boltzmann equation, it does not imply uniqueness of the equation, due to the nonlinear nature of the equation.

Our strategy in Section 3.3 uses the method of DiPerna and Lions reduces the renormalizability of stochastic kinetic equations to the vanishing of certain commutators between smoothing operators and the differential action of the rough vector fields. Specifically, given a smooth renormalization with bounded first and second derivatives, we begin by smoothing a solution to (50) in the variables with mollifier . The regularity improvement allows us to renormalize the equation by at the expense of a remainder comprised of commutators and double commutators of and convolution by ,

(54)

As is well known from the classical theory of renormalization by [18] that the single commutator

(55)

as long as and with . As it turns out, the double commutator also vanishes

(56)

provided that and . However one of the primary differences between the deterministic and stochastic theory is an interesting consequence of Itô’s formula. Specifically the remainder involves the square of the single commutator . Due to the limited integrability and regularity of , this imposes that and for this contribution to vanish in (see Proposition 3.4 for more details on this). Based on this method of proof, we are presently unable to treat the case . The main result of this section (Proposition 3.4) shows show that a weak martingale solution , to (50) is renormalizable provided we have the following regularity conditions on ,

(57)

We believe these results are consistent with the work of Lions/Le-Bris [10] on deterministic parabolic equations with rough diffusion coefficients. There should also be a connection with the more recent work of Bailleul/Gubinelli [5]. In the case that , the conditions (57) become precisely the assumptions (LABEL:eq:Noise-Assumption-3) and (LABEL:eq:Noise-Assumption-4) on the noise coefficients.

Section 4 concerns the subtle regularizing effects for stochastic kinetic equations. These are captured by studying the velocity averages of the solution, and have a long history in the deterministic literature [9, 27, 28, 30] as well as several more recent results in the SPDE literature [15, 26, 35]. Since equation (50) is of transport type, without more information on , one does not expect to obtain any further regularity on the solution than is present in the initial data . However, in view of the deterministic theory it is natural to expect a small gain in the regularity of velocity averages

(58)

where is a test function in velocity only. Using a method of Bouchut/Desvillette [9] based on the Fourier transform, we prove that if is a weak martingale solution to (50) and , then