Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Gevrey Gelfand-Shilov regularizing effect of the Landau equation with soft potential

Xiao-Dong Cao, Chao-Jiang Xu, Yan Xu

Abstract

This paper studies the Cauchy problem for the spatially inhomogeneous Landau equation with soft potential in the perturbative framework around the Maxwellian distribution. Under a smallness assumption on the initial datum with exponential decay in the velocity variable, we establish the optimal Gevrey Gelfand-Shilov regularizing effect for the solution to the Cauchy problem.

The Gevrey Gelfand-Shilov regularizing effect of the Landau equation with soft potential

Abstract

This paper studies the Cauchy problem for the spatially inhomogeneous Landau equation with soft potential in the perturbative framework around the Maxwellian distribution. Under a smallness assumption on the initial datum with exponential decay in the velocity variable, we establish the optimal Gevrey Gelfand-Shilov regularizing effect for the solution to the Cauchy problem.

Paper Structure

This paper contains 6 sections, 14 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Methodology and preliminary results
  3. Commutators between weights and Landau operators with vector fields
  4. Energy estimates for one directional derivations
  5. Energy estimates for multi-directional derivations
  6. Proofs of technical lemmas

Key Result

Theorem 1.1

Assume that the initial datum $\|f_{0}\|_{H^{3}_{x}L^{2}_{v}(\omega_{0})}$ small enough, then the Cauchy problem 1-2 admits a unique solution satisfying $\omega_t f(t) \in G^{\sigma}\left(\mathbb{T}^3_x; S^{\sigma}_{\sigma}\left(\mathbb{R}^3_{v}\right)\right)$ for $t > 0$ with $\sigma = \max\left\{1

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Proposition 3.2
  • Corollary 3.3
  • ...and 12 more