On uniqueness and stability for the Boltzmann–Enskog equation

The time-evolution of a moderately dense gas in a vacuum is described in classical mechanics by a particle density function obtained from the Boltzmann–Enskog equation. Based on a McKean–Vlasov equation with jumps, the associated stochastic process was recently constructed by modified Picard iterations with the mean-field interactions, and more generally, by a system of interacting particles. By the introduction of a shifted distance that exactly compensates for the free transport term that accrues in the spatially inhomogeneous setting, we prove in this work an inequality on the Wasserstein distance for any two measure-valued solutions to the Boltzmann–Enskog equation. As a particular consequence, we find sufficient conditions for the uniqueness and continuous-dependence on initial data for solutions to the Boltzmann–Enskog equation applicable to hard and soft potentials without angular cut-off.


The Boltzmann-Enskog model
In the classical description of a moderately dense gas in a vacuum, each particle is completely described by its position r ∈ R d and its velocity v ∈ R d , where d ≥ 3. Moreover, the particles are assumed to be indistinguishable and with equal mass. Any particle (r, v) moves with constant speed v until it performs a collision with another particle (q, u). Denote by v , u the resulting velocities after collision. We suppose that collisions are elastic, as a consequence conservation of momentum and kinetic energy hold, i.e. 25 Page 2 of 25 Friesen, Rüdiger, and Sundar NoDEA u + v = u + v |u| 2 + |v| 2 = |u | 2 + |v | 2 .
A commonly used parameterization of the deflected velocities v , u is given by the unit vector where (·, ·) denotes the euclidean product in R d . Note that, for fixed n ∈ S d−1 , the change of variables (v, u) −→ (v , u ) is an involutive transformation with Jacobian equal to 1. Let f 0 (r, v) ≥ 0 be the particle density function of the gas at initial time t = 0. The time evolution f t = f t (r, v) is then obtained from the Boltzmann-Enskog equation Here Q is a non-local, nonlinear collision integral operator given by where dn denotes the Lebesgue surface measure on the sphere S d−1 and B(|v − u|, n) ≥ 0 the collision kernel so that B(|v − u|, n)dn includes the effect of velocity cross-section. The particular form of B(|v − u|, n) depends on the particular microscopic model one has in mind, while β(r − q) ≥ 0 describes the rate at which a particle at position r performs a collision with another particle at position q. For instance, β(r − q) = δ 0 (r − q) describes the case of local collisions governed by the classical Boltzmann equation, while the particular choice β(r − q) = δ ρ (|r − q|) describes the case where particles behave like billiard balls of radius ρ > 0 and was studied by Rezakhanlou [21]. Following [1,16] we study in this work the case where β is a symmetric and smooth function. Applications, additional physical background and classical mathematical results are collected in the books of Cercignani [10] and Cercignani, Illner, Pulvirenti [11]. For recent review articles on this topic we refer to Villani [24] and Alexandre [2]. where θ ∈ (0, π] is the angle between u − v and u − v and φ ∈ (0, 2π] is the longitude angle, see Tanaka [22] or Horowitz and Karandikar [17]. This is summarized in Fig. 1. Note that Boltzmann's original model (1.4) satisfies Grad's angular cutoff assumption, i.e.

Examples in dimension
A mathematically more challenging class of models which does not satisfy Grad's angular cut-off assumption is provided by long-range interactions given by (1.5) where b is at least locally bounded on (0, π] and The parameters γ and ν are related by For long-range interactions one distinguishes between the following cases: (i) Very soft potentials s ∈ (2,3], γ ∈ (−3, −1] and ν ∈ [1, 2).

Literature on the Cauchy problem (1.2)
The analysis of the Cauchy problem (1.2) strongly depends on the particular choice of β, that is, on the distance of colliding particles. The classical Boltzmann equation formally corresponds to the particular choice β(x − y) ≡ δ 0 (|x−y|) (dirac distribution at zero) where colliding particles have to be at the same position. Hence the collision integral (1.3) is local (but highly singular) in the spatial variables. Classical results on the space-inhomogeneous Boltzmann equation can be found in [2,24]. More recently, there has been some interesting progress on global solutions close to equilibrium as studied in the works of Alexandre, Morimoto, Ukai, Xu, Yang [3][4][5][6]. Note that in contrast to our work, the solutions studied in these references are, in general, not probability distributions on R 2d .
Letting β(x − y) ≡ δ ρ (|x − y|) provides the description of particles which can only perform elastic collisions at a fixed radius ρ > 0. In this case the collision integral (1.3) is less singular than in the classical Boltzmann equation. The corresponding Cauchy problem was studied e.g. by Toscani, Bellomo [23], Arkeryd [7] and Arkeryd, Cercignani [8]. Based on an interacting particle system of binary collisions, the Boltzmann-Grad limit was established for true hard spheres by Rezakhanlou [21]. Most of the results obtained in this direction are mainly applicable under Grad's angular cut-off assumption.
Finally, taking 0 ≤ β ∈ C 1 c (R d ) to be a symmetric function removes the spatial singularity in the collision integral (1.3) which allows us to use stochastic methods. This can be seen as a mollified version of either δ 0 (|x − y|) or δ ρ (|x−y|). The analysis of the corresponding Cauchy problem was initiated by Povzner [20] while propagation of chaos was studied by Cercignani [9] performing the Boltzmann-Grad limit for the corresponding BBGKY-hierarchy. First results applicable without cut-off (including the case of Maxwellian molecules) have been recently obtained in [1,16], where the construction of the corresponding stochastic process (the so-called Enskog process) was studied. (1.5) it is also feasible to parameterize v , u in terms of the angle θ, i.e. n = n(v, u, θ, φ), and hence study continuity properties of (u−v, n)n in u, v for fixed θ, φ. It was already pointed out by Tanaka that in d = 3, (u, v) −→ (u − v, n)n cannot be smooth. To overcome this problem he introduced in [22] another transformation of parameters which is bijective, has Jacobian 1 and hence can be used on the right side of (1.1). Such ideas have been extended to arbitrary dimension d ≥ 3 and are briefly summarized in this section, see [15,18]. For this purpose set S d−2 = {ξ ∈ R d−1 | |ξ| = 1} and define The following is due to [15,22], see also [16] for this formulation.

Some notation
Here and below we let v := (1 + |v| 2 ) 1/2 and frequently use the elementary inequalities We denote by K, C > 0 generic constants which may vary from line to line. Finally, for k ∈ N we use the following function spaces • C k (R 2d ) the space of all continuous functions on R 2d which are k-times continuously differentiable. Friesen, Rüdiger, and Sundar ) such that f and its first k derivatives are bounded.
) such that f has compact support. • Lip(R 2d ) the space of all globally Lipschitz continuous functions.
Denote by P(R d ) the space of probability measures and let be the pairing between μ ∈ P(R 2d ) and an integrable function ψ.

Weak formulation for measure-solutions of the Boltzmann-Enskog equation
In this work we take any dimension d ≥ 3 and assume that the collision kernel B is given by the velocity cross-section σ ≥ 0 and a measure Q such that Without loss of generality we assume that β is bounded by 1.

NoDEA
On uniqueness and stability Page 7 of 25 25 and, for any Analogously we define a global weak solution to the Boltzmann-Enskog equation.
Note that one has A1 = 0 where 1 denotes the constant function equal to one. Hence total mass is conserved and we may restrict our study of the Boltzmann-Enskog equation without loss of generality to the case of probability distributions. A construction of such solutions was recently studied in [1,16].

Stability estimates for the Boltzmann-Enskog equation
In this work we prove stability estimates for weak solutions to the Boltzmann-Enskog equation in the Wasserstein distance. In contrast to the spacehomogeneous case studied in [15] the additional free transport term v · ∇ r prevents us from directly applying their methods. In order to take this transport of particles into account we introduce the shifted Wasserstein distance where S(t) is a one-parameter group of transformations defined by S(t) * denotes the adjoint operator to S(t) acting on measures μ ∈ P(R 2d ) and and W 1 denotes the classical Wasserstein distance, i.e.
where Lip(R 2d ) = {ψ | ψ Lip < ∞} is the space of all globally Lipschitz continuous functions and In the case of hard potentials we obtain the following.
Friesen, Rüdiger, and Sundar For soft potentials we obtain the following.
Condition (2.13) stems from the necessity to compensate the singularity of σ(|v − u|) at zero appearing in the case of soft potentials, see (1.5).
Above estimates are sufficient to imply uniqueness and stability (with respect to initial data) of weak solutions to the Boltzmann-Enskog equation. Indeed, by using a generalization of the Gronwall inequality as stated in the "Appendix" (see e.g. [12, Lemma 5.2.1, p. 89]) we obtain the following. • γ ∈ (−1, 0) and there exists δ > 0 with • γ ∈ (−1, 0) and there exists δ > 0 with Our proofs are partially inspired by the work of Fournier and Mouhot [15] where similar estimates for solutions to the space-homogeneous Boltzmann equation have been established. Other uniqueness results for the spacehomogeneous Boltzmann equation are based on additional regularity assumptions for the solution, see e.g. [13,26]. However, since we work in the spaceinhomogeneous setting we have to replace the classical Wasserstein distance W 1 to by a shifted distance W t 1 which compensates the free transport operator v · ∇ r appearing in the definition of A, see (2.6). For hard-potentials the authors have used in [15] Povzner inequalities to prove creation of exponential moments for solutions to the (space-homogeneous) Boltzmann equation, see also [19] and the references therein. Their proofs implicitly use the fact that any two particles may perform a collision. In contrast to that, in the spaceinhomogeneous setting studied here β is compactly supported and hence only particles being close enough may perform a collision. This prevents us from proving similar results on the creation of moments for the Boltzmann-Enskog equation with hard potentials.
While this is one of the few results where uniqueness and stability estimates are obtained for the space-inhomogeneous setting, future research may be devoted to the study of suitable moment estimates in the spirit of Povzner or to the regularity of solutions to the Enskog equation. Any progress in this

Mild formulation for the Boltzmann-Enskog equation
In order to prove the desired stability estimates for the shifted distance W t 1 , it is reasonable to use another formulation of the Boltzmann-Enskog equation which involves the semigroup S(t). This is precisely the content of this section. Define for (r, v), (q, u) ∈ R 2d and ψ ∈ Lip(R 2d ) Then there exists a constant C > 0 such that for each ψ ∈ Lip(R 2d ) one has The next result is crucial for estimating weak solutions to the Boltzmann-Enskog equation. holds for each ψ ∈ Lip(R 2d ).

Since (μ t ) t≥0 satisfies the Boltzmann-Enskog equation and S(t−s)ψ
for a.a. s ∈ [0, t). Next we will prove that as h → 0. Combining then (3.5), (3.6) and using the definition of A and B proves (3.4). In order to prove (3.6) we let and denote the corresponding pointwise derivative with respect to h at h = 0 by Using the estimate Friesen, Rüdiger, and Sundar Combining these estimates yields (3.6) as h → 0 and letting R → ∞ thus completes the proof of Proposition 3.1.

The coupling inequality
Below we first provide another representation for the metric W t 1 (μ, ν) in terms of optimal couplings. Additional details on the classical Wasserstein distance and optimal transport are given in [25]. Let μ, ν ∈ P(R 2d ). A coupling H of (μ, ν) is a probability measure on R 4d such that its marginals are given by μ and ν, respectively. Let H(μ, ν) the space of all such couplings. Define a one-parameter family of norms on R 2d via Related to this family we define the Lipschitz norms Note that these Lipschitz norms are all equivalent. We will use the following simple observation.
Proof. The first equality in (4.1) follows by definition of W t 1 combined with (2.11). For the second equality in (4.1) observe that, for any ψ with ψ 0 ≤ 1, we get S(−t)ψ t ≤ 1. Conversely any ψ satisfying ψ t ≤ 1 can be written Then In order to prove this result we need continuity properties for the collision integral, i.e. to compare (2.3) for different values of u, v. It was already pointed out by Tanaka that (u, v) −→ α(v, u, θ, ξ) cannot be smooth for any choice of (θ, ξ). Using the parameterization (2.2) Tanaka [22,Lemma 3.1] has shown that if we allow to shift the angles ξ in a suitable way, then a weaker form of continuity holds. The latter estimate is sufficient for this work. Below we recall Tanaka's result for arbitrary dimension d ≥ 3 which is due to [15].

Lemma 4.3. [15, Lemma 3.1] There exists a measurable map ξ
With this parameterization we obtain from Lemma 4.3, for all u, v, u, v ∈ R d , all θ ∈ [0, π] and all ξ ∈ S d−2 , we have the inequality We are now prepared to prove our main coupling inequality of this section.

Proof of Theorem 2.4
Assume that γ ∈ [0, 2] and let δ be given by (2.12). Let us first show that under the conditions of Theorem 2.4, also (4.2) is satisfied. Indeed, using σ(|v − u|) ≤ C v 2 u 2 this follows from where we have used Jensen's inequality Thus we can apply the coupling inequality. In order to estimate Ψ as defined in Proposition 4.2, we need the following lemma.
Lemma 5.1. There exists a constant C = C(γ, σ) > 0 such that the following holds: Note that for γ = 0 the left-hand side is zero and hence the assertion is trivially satisfied.
Proof. (a) We use, for x, y ≥ 0 and a, b > 0, the elementary inequality with some constants c a,b , C a,b > 0 (see e.g. [14] for a similar application) to obtain The second term can be estimated in the same way.
From this lemma we deduce the following estimate on the function Ψ.
The coupling inequality combined with Lemma 5.2 gives for some con- A similar estimate to (5.2) was shown in [15, p.820], while (5.1) did not appear in the space-homogeneous setting studied there.
Proof. Here and below we denote by K > 0 some generic constant (independent of (μ t ) t∈[0,T ] and (ν t ) t∈[0,T ] ) which may vary from line to line. For b > 0, we obtain Estimating the second term in I(b) in the same way gives This gives the estimate Letting b 1+γ = 1+δ δ 2 δ |ln (W s 1 (μ s , ν s ))| yields (5.1). The inequality (5.2) can be shown in the same way.
Proof. (a) Following [15, p. 821] we obtain The other term can be estimated in the same way. and similarly for the second term.