Tanaka Theorem for Inelastic Maxwell Models

One-dimensional inelastic Boltzmann equation: Regularity \& uniqueness of self-similar profiles for moderately hard potentials

We prove uniqueness of self-similar profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, that is with collision kernel of the form | $\bullet$ | $γ$ for $γ$ > 0 small enough (explicitly quantified). Our result provides the first uniqueness statement for self-similar profiles of inelastic Boltzmann models allowing for strong inelasticity besides the explicitly solvable case of Maxwell interactions (corresponding to $γ$ = 0). Our approach relies on a perturbation argument from the corresponding Maxwell model through a careful study of the associated line…

Exponential trend to equilibrium for the inelastic Boltzmann equationdriven by a particle bath

We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient $α\in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which $α$ is close to $1$, and the rate of convergence is expli…

Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data

The Cauchy problem for the inelastic Boltzmann equation is studied for small data. Existence and uniqueness of mild and weak solutions is obtained for sufficiently small data that lies in the space of functions bounded by Maxwellians. The technique used to derive the result is the well known iteration process of Kaniel and Shinbrot.

On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity

We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ide…

非弹性与多组分混合 Boltzmann 方程的高速尾

High-velocity tails of the inelastic and the multi-species mixture Boltzmann equations

研究了非弹性与多组分混合 Boltzmann 方程的高速尾，通过扩展 Cancellation lemma 等工具证明了非截断非弹性情形下的指数下界，并证明了混合型方程的 Maxwellian 尾。

Propagation of $L^{1}$ and $L^{\infty}$ Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann Equation

We consider the $n$-dimensional space homogeneous Boltzmann equation for elastic collisions for variable hard potentials with Grad (angular) cutoff. We prove sharp moment inequalities, the propagation of $L^1$-Maxwellian weighted estimates, and consequently, the propagation $L^\infty$-Maxwellian weighted estimates to all derivatives of the initial value problem associated to the afore mentioned problem. More specifically, we extend to all derivatives of the initial value problem associated to this class of Boltzmann equations corresponding sharp moment (Povzner) inequalities and time propagati…

On the Measure Valued Solution to the Inelastic Boltzmann Equation with Soft Potentials

The goal of this paper is to extend the existence result of measure-valued solution to the Boltzmann equation in elastic interaction, given by Morimoto-Wang-Yang, to the inelastic Boltzmann equation with moderately soft potentials, also as an extensive work of our preceding result in the inelastic Maxwellian molecules case. We prove the existence and uniqueness of measure-valued solution under Grad's angular cutoff assumption, as well as the existence of non-cutoff solution, for both finite and infinite energy initial datum, by a delicate compactness argument. In addition, the moments propagat…

Inelastic Boltzmann Equation Driven by a Particle Thermal Bath

We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient $α\in(0,1)$, under the thermalization induced by a host medium with a fixed Maxwellian distribution and any fixed $e\in(0,1]$. When the restitution coefficient $α$ is close to 1 we prove existence and uniqueness of global solutions considering the close-to-equilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.

Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres

We consider the spatially homogeneous Boltzmann equation for {\em inelastic hard spheres}, in the framework of so-called {\em constant normal restitution coefficients} $α\in [0,1]$. In the physical regime of a small inelasticity (that is $α\in [α_*,1)$ for some constructive $α_*>0$) we prove uniqueness of the self-similar profile for given values of the restitution coefficient $α\in [α_*,1)$, the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation). Moreover, if the initial datum lies in $L^1_3$, and under some smallness condition on…

非弹性Boltzmann方程冷却过程的逐点定量估计

Quantitative pointwise estimates of the cooling process for inelastic Boltzmann equation

本文研究硬球齐次非弹性Boltzmann方程的冷却过程，证明了解的全局及零速度外的逐点含时上界、冷却时间无穷大，给出依赖法向恢复系数的上界并还原弹性碰撞情形的已知结果，最终得到各时刻的Maxwellian上界。

具有非弹性长程相互作用的空间齐次 Boltzmann 方程的测度值解

Measure Valued Solution to the Spatially Homogeneous Boltzmann Equation with Inelastic Long-Range Interactions

本文在无 Grad 角截断的非弹性 Boltzmann 方程中建立概率测度值解的适定性，并构造推广至非弹性情形的无限能量自相似渐近稳态解。