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On the Boltzmann equation with soft potentials: Existence, uniqueness and smoothing effect of mild solutions

Ling-Bing He, Jie Ji, Wei-Xi Li

Abstract

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff for soft potentials. For any given initial datum such that the mass, energy and entropy densities are bounded and the mass is away from vacuum, we establish the local-in-time existence and uniqueness of mild solutions, and further provide the first result on sharp smoothing effect in analytic space or Gevrey space for soft potentials.

On the Boltzmann equation with soft potentials: Existence, uniqueness and smoothing effect of mild solutions

Abstract

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff for soft potentials. For any given initial datum such that the mass, energy and entropy densities are bounded and the mass is away from vacuum, we establish the local-in-time existence and uniqueness of mild solutions, and further provide the first result on sharp smoothing effect in analytic space or Gevrey space for soft potentials.

Paper Structure

This paper contains 21 sections, 21 theorems, 367 equations.

Table of Contents

  1. Introduction
  2. Function spaces and statement of the main results
  3. Brief review
  4. Difficulties and methodology
  5. Notations and quantitative estimate for the collision operator
  6. Dyadic decomposition and sharpness of Gevrey index
  7. Organization of the paper
  8. Preliminaries
  9. Regularity in velocity
  10. Commutator estimates and applicatons
  11. Estimates in the spatial and velocity spaces
  12. A priori estimate
  13. Proof of the a priori estimate: Analytic regularity
  14. Proof of the a priori estimate: smoothing effect in Gevrey space
  15. Linear Boltzmann equation
  16. ...and 6 more sections

Key Result

Theorem 1.2

Assume the non-cutoff collision kernel $B$ satisfies $\mathbf{(A1)}-\mathbf{(A3)}$ above with the numbers $s,\gamma$ therein satisfying $\mathbf{(A4)}$. Suppose that the initial datum $f_{in}$ in Bolt is non-negative, satisfying condition finite and that for some constant $a_0>0.$ Then the following assertions hold true.

Theorems & Definitions (53)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • ...and 43 more