Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Kunlun Qi

Abstract

This paper is to study the inelastic Boltzmann equation without Grad's angular cutoff assumption, where the well-posedness theory of the solution to the initial value problem is established for the Maxwellian molecules in a space of probability measure defined by Cannone-Karch in [Comm. Pure. Appl. Math. 63 (2010), 747-778] via Fourier transform and the infinite energy solutions are not a priori excluded as well. Meanwhile, the geometric relation of the inelastic collision mechanism is introduced to handle the strong singularity of the non-cutoff collision kernel. Moreover, we extend the self-similar solution to the Boltzmann equation with infinite energy shown by Bobylev-Cercignani in [J. Stat. Phy. 106 (2002), 1039-1071] to the inelastic case by a constructive approach, which is also proved to be the large-time asymptotic steady solution with the help of asymptotic stability result in a certain sense.

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

Abstract

This paper is to study the inelastic Boltzmann equation without Grad's angular cutoff assumption, where the well-posedness theory of the solution to the initial value problem is established for the Maxwellian molecules in a space of probability measure defined by Cannone-Karch in [Comm. Pure. Appl. Math. 63 (2010), 747-778] via Fourier transform and the infinite energy solutions are not a priori excluded as well. Meanwhile, the geometric relation of the inelastic collision mechanism is introduced to handle the strong singularity of the non-cutoff collision kernel. Moreover, we extend the self-similar solution to the Boltzmann equation with infinite energy shown by Bobylev-Cercignani in [J. Stat. Phy. 106 (2002), 1039-1071] to the inelastic case by a constructive approach, which is also proved to be the large-time asymptotic steady solution with the help of asymptotic stability result in a certain sense.

Paper Structure

This paper contains 19 sections, 15 theorems, 179 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Motivation
  4. Main Theorems
  5. Plan of the paper
  6. Preliminary
  7. Some Properties of Characteristic Function
  8. Useful Estimates about Inelastic Variables and Collision Operator
  9. Well-posedness under the Cutoff Assumption
  10. Technical Lemma of the Cutoff Collision Operator
  11. Well-posedness under Cutoff Assumption
  12. Existence and Uniqueness with Non-Cutoff assumption
  13. Technical Lemma of the Non-Cutoff Collision Operator
  14. Proof of Theorem \ref{['noncutoffwellposedness']}
  15. Large-time Asymptotic Behavior to Self-similar Solutions for the Inelastic Boltzmann Equation
  16. ...and 4 more sections

Key Result

Theorem \oldthetheorem

(Well-posedness under non-cutoff assumption) Assume that $e\in (0,1]$ and the collision kernel $b$ satisfies the non-cutoff assumption noncutoffb for some $\alpha_{0}\in\left[0,2\right]$, then for each $\alpha\in\left[\alpha_{0},2\right]$ and initial condition $\varphi_{0}\in\mathcal{K}^{\alpha}$, t where the finite parameter $\lambda_{e,\alpha}$ is defined as,

Figures (1)

  • Figure 1: Illustration of the inelastic collision mechanism with $\cos\theta = \frac{\xi\cdot\sigma}{|\xi|}$ and $\eta_{e}^{+} = \xi_{e}^{+} - \zeta_{e}$.

Theorems & Definitions (36)

  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 26 more