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Low-regularity global well-posedness for the Boltzmann equation near vacuum

Xinfeng Hu, Shuangqian Liu, Haoran Peng, Yi Zhou

Abstract

We study the Boltzmann equation near vacuum in anisotropic low-regularity Besov spaces. We establish the global existence and uniqueness of strong solutions with the critical regularity index $2/p$ for $p\in[1,\infty)$ in $\mathbb{R}^3$. The proof relies on a new bilinear estimate for the nonlinear collision operator. Combined with a div-curl type lemma we develop, this allows us to close the a priori estimates and thereby obtain global well-posedness.

Low-regularity global well-posedness for the Boltzmann equation near vacuum

Abstract

We study the Boltzmann equation near vacuum in anisotropic low-regularity Besov spaces. We establish the global existence and uniqueness of strong solutions with the critical regularity index for in . The proof relies on a new bilinear estimate for the nonlinear collision operator. Combined with a div-curl type lemma we develop, this allows us to close the a priori estimates and thereby obtain global well-posedness.

Paper Structure

This paper contains 11 sections, 11 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. Main results
  4. Related works
  5. Strategies and ideas
  6. Structure of the paper
  7. Notations and Function Spaces
  8. Nonlinear Estimates
  9. Bilinear dissipation and the a priori estimate
  10. The global existence
  11. Appendix

Key Result

Theorem 1.1

Suppose that $p> 1$ and $\frac{3}{p}-2 < \gamma <\frac{2}{p}-1$, with the endpoint case $p=1$ corresponding to $\gamma=1$. Let $l>\max\left\{ 1-\dfrac{1}{p},3-\dfrac{4}{p}-\gamma\right\}.$ Assume there exists $\varepsilon>0$ such that where $w_0(x,v)=\langle x-v\rangle^{l}$. Then there exists a unique global solution $f(t,x,v)$ to Change variable BE satisfying $\blacktriangleleft$$\blacktriangle

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Theorem 4.1
  • proof
  • ...and 15 more