Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp regularization effect for the non-cutoff Boltzmann equation with hard potentials

Jun-Ling Chen, Wei-Xi Li, Chao-Jiang Xu

Abstract

For the Maxwellian molecules or hard potentials case, we verify the smoothing effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. Given initial data with low regularity, we prove its solutions at any positive time are analytic for strong angular singularity, and in Gevrey class with optimal index for mild angular singularity. To overcome the degeneracy in the spatial variable, a family of well-chosen vector fields with time-dependent coefficients will play a crucial role, and the sharp regularization effect of weak solutions relies on a quantitative estimate on directional derivatives in these vector fields.

Sharp regularization effect for the non-cutoff Boltzmann equation with hard potentials

Abstract

For the Maxwellian molecules or hard potentials case, we verify the smoothing effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. Given initial data with low regularity, we prove its solutions at any positive time are analytic for strong angular singularity, and in Gevrey class with optimal index for mild angular singularity. To overcome the degeneracy in the spatial variable, a family of well-chosen vector fields with time-dependent coefficients will play a crucial role, and the sharp regularization effect of weak solutions relies on a quantitative estimate on directional derivatives in these vector fields.
Paper Structure (12 sections, 5 theorems, 174 equations)

This paper contains 12 sections, 5 theorems, 174 equations.

Table of Contents

  1. Introduction and main result
  2. Notations and function spaces
  3. Statement of the main result
  4. Sharpness of the Gevrey index
  5. Difficulties and Methodologies
  6. Arrangement of the paper
  7. Preliminaries
  8. Commutator estimates
  9. Analytic smoothing effect for strong angular singularity
  10. Quantitative estimate for directional derivatives
  11. Proof of Theorem \ref{['mainresult']}: analytic regularization effect for $\boldsymbol{\frac{1}{2} \leq s<1}$
  12. Optimal Gevrey smoothing effect for mild angular singularity

Key Result

Theorem 1.1

Let $G^{r}(\mathbb{T}_x^3 \times \mathbb{R}_v^3)$ be the Gevrey space defined above. Assume that the cross-section satisfies kern and angu with $\gamma \geq 0$ and $0< s <1$. There exists a sufficiently small constant $\epsilon>0$ such that if then the Boltzmann equation eqforper admits a global-in-time solution $f$ satisfying that $f \in G^{\tau}(\mathbb{T}_x^3 \times \mathbb{R}_v^3)$ for all $t

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 3.1
  • Remark 3.2
  • proof : Proof of Proposition \ref{['lemgamma']}
  • Proposition 3.3
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • ...and 2 more