UNIFORM L 1 STABILITY OF THE INELASTIC BOLTZMANN EQUATION WITH LARGE EXTERNAL FORCE FOR HARD POTENTIALS

. In this paper, we will study the uniform L 1 stability of the inelastic Boltzmann equation. More precisely, according to the existence result on the inelastic Boltzmann equation with external force near vacuum, we obtain the uniform L 1 stability estimates of mild solution for the hard potentials under the assumptions on the characteristic generated by force term which can be arbitrarily large. The proof is based on the exponentially decay estimate and Lu’s trick in [10].


1.
Introduction. The spatially inhomogeneous Boltzmann equation describes the phase space evolution of a distribution function f (t, x, v) of moderately dilute gas particles at time t ≥ 0 and position x ∈ R 3 with velocity v ∈ R 3 . In the presence of external forces F (t, x), it reads where Q(f, f ) is the inelastic collision operator: x, v) for simplicity. Here ω ∈ S 2 and the variable u = v − v * is the relative velocity between particles. v, v * denote the velocities of two particles before collision, v, v * represent the velocities after collision, and v = v − 1 + e 2 ( u · ω)ω, v * = v * + 1 + e 2 ( u · ω)ω.
The parameter e(0 < e < 1) is the restitution coefficient which is a function of |u·ω|.
In this paper, we consider the case of hard potentials 0 ≤ γ ≤ 1( including Maxwellian molecules model and hard sphere model), and we assume that b(θ) satisfy Grad s angular cut-off condition, i.e.: Under the case of space inhomogeneous, the global existence of a mild solution with the inelastic Boltzmann equation is obtained by Alonso [1], the uniform L 1 stability is gotten by Wu [15] using a Lyapunov functional. In the presence of some large external force, Wei and Zhang [11,12] obtained the existence of mild solutions for soft potentials to the inelastic Boltzmann equation and inelastic Enskog-Boltzmann equation, and they [11] got the existence of infinite energy solutions.
There are two different ways to study the stability estimate. One is the direct estimation method: Arkeryd [2] firstly proved the weighted L 1 stability for L 1 solution. Wennberg [14] got the stability in L p space for p > 1. DiPerna and Lions [13] obtained the weak stability for renormalized solutions. Yun [16] considered the L p stability of the Boltzmann equation near the local Maxwellian. Another method is to construct some Lyapunov functionals: Ha [7,8,9] obtained the L 1 stability estimates of Boltzmann equation near vacuum and the local Maxwellian.
Moreover, for the Boltzmann equation with small external forces, Duan et al. [5] obtained the L 1 and BV-type stability by some Lyapunov functionals. For the Boltzmann equation with some large external forces, Cheng [3] established the L 1 stability in the case of soft potentials by direct method. But for the non-integrability of |v − v * | γ f (t, x, v), both above methods can not obtain the L 1 stability of the Boltzmann equation with some large external for hard potentials.
So, the aim of this paper is to study the L 1 stability of mild solutions obtained by Wei and Zhang in [11] to the Cauchy problem for the inelastic Boltzmann equation with large external force for hard potentials (including hard sphere model). The main ideal come from Lu's trick in [10].
The rest of this paper is organized as follows: In Section 2, we give the main result and the assumptions about external force and restitution coefficient. In Section 3, we present some basic estimates and establish the uniform L 1 stability estimate for hard potentials .

2.
Assumptions and main results. The dynamic behavior of the particle trajectory is crucial in the existence and stability analysis of solutions, So for a given point (t, x, v) in R + × R 3 × R 3 , we denote by [X(s; t, x, v), V (s; t, x, v)] to be the particle trajectory which are the unique solutions of ODE system: We set Along the bicharacteristic, We integrate (1.1) to get a mild form The definition of mild solutions can be stated as follows.
if and only if f satisfies the integral equation (7) for all t ∈ [0, T ) and a.e (x, v) ∈ R 3 × R 3 .
In the following, we set the standard bounding functions decaying exponentially: Obviously, S(α, β, δ) is a Banach space with the norms: Throughout this paper, the assumptions on the external force F (t, x) can be summarized as follows, which come from reference [6].
(A1) The external force F (t, x) ensures the existence of global-in-time smooth solution to the system (2.1) and (2.2). Furthermore, there are two functions η 1 (s; t, x, v), η 2 (s; t, x, v) satisfy following conditions: for any ξ ∈ R 3 and    where η i (s; t, x, v)(i = 1, 2) is the derivative with respect to s, and η 0 ,η 0 are positive constants independent of s, t, x, v. where a is a positive constant.
Remark 2.2. Let s = t, (9) becomes For the coefficient of normal restitution we shall adopt the following assumptions, see [1]: (A2) e(r) is absolutely continuous and non-increasing from [0, +∞) to (0, 1], θ(r) = re(r) is strictly increasing. And for any c > 0, γ > −2, it holds that Under above assumptions, the global existence of solutions was obtained. For details, refer [11]. Precisely, one has the following proposition. The main objective of this paper is to consider the uniform L 1 stability of mild solutions of Proposition 2.1 for hard potentials. Hence, we need an extra assumption on η 1 (s, t, x, v), η 2 (s, t, x, v) appeared in (A1).

Remark 2.3. Obviously, the examples in Remark 2.2 satisfy assumption (A3).
Our main result about the stability is as follows.
3. Proof of Theorem 2.1. For L 1 stability estimate of mild solutions, we employ the Gronwall type inequality given in [10]. Next, we present several useful estimates.
Proof. Using the following inequality for any 0 ≤ γ ≤ 1 Then the proof of Lemma 3.2 is completed.
In order to obtain the L 1 norm estimates of the collision term Q(f, g), we must consider the following integral: where f ∈ S. Lemma 3.3. For I 1 , we have following estimate: for any R > 0.
Proof. By the definition of S(α, β, δ), it holds that and we take s = t and ξ = u in (9) or Remark 2.2 to obtain that Then,

SHAOFEI WU ET AL.
We now give two different estimates to above integrate based on decay properties in (x,v) space.
Case two. Letū = η 2 (s; s, x, v)u, we have Hence, This completes the proof of Lemma 3.3.