Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Low Regularity Solutions for the Vlasov-Poisson-Landau/Boltzmann System

Dingqun Deng, Renjun Duan

Abstract

In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov-Poisson-Landau/Boltzmann system around global Maxwellians in torus or finite channel. The main goal is to establish the global existence and large time behavior of small amplitude solutions for a class of low regularity initial data. The molecular interaction type is restricted to the case of hard potentials for two classical collision operators because of the effect of the self-consistent forces. The result extends the one by Duan-Liu-Sakamoto-Strain [{\it Comm. Pure Appl. Math.} 74 (2021), no.~5, 932--1020] for the pure Landau/Boltzmann equation to the case of the VPL and VPB systems.

Low Regularity Solutions for the Vlasov-Poisson-Landau/Boltzmann System

Abstract

In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov-Poisson-Landau/Boltzmann system around global Maxwellians in torus or finite channel. The main goal is to establish the global existence and large time behavior of small amplitude solutions for a class of low regularity initial data. The molecular interaction type is restricted to the case of hard potentials for two classical collision operators because of the effect of the self-consistent forces. The result extends the one by Duan-Liu-Sakamoto-Strain [{\it Comm. Pure Appl. Math.} 74 (2021), no.~5, 932--1020] for the pure Landau/Boltzmann equation to the case of the VPL and VPB systems.

Paper Structure

This paper contains 19 sections, 10 theorems, 324 equations.

Table of Contents

  1. Introduction
  2. Equations
  3. Reformulation
  4. Spatial domain and boundary condition
  5. Case I: Torus
  6. Case II: Finite channel
  7. Notations
  8. Function space
  9. Case I: Torus
  10. Case II: Finite channel
  11. Main results
  12. Case of torus.
  13. Case of finite channel.
  14. Preliminary
  15. Macroscopic estimates on torus
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Omega=\mathbb{T}^3$ and $w$ be chosen by w2. Assume that $f_0(x,v)$ satisfies 4 with $t=0$. Then the followings hold. (1) For the VPL system with $-2\le\gamma< -1$, there exist $\varepsilon_0,\delta>0$ such that if $F_0(x,v)=\mu+\mu^{1/2}f_0(x,v)\ge 0$ and then there exists a unique global mild solution $f=f(t,x,v)$ to the problem 1 and VPL satisfying that $F(t,x,v)=\mu+\mu^{1/2}f(t,x,v)\ge

Theorems & Definitions (20)

  • Theorem 1.1: Existence and large-time behavior
  • Theorem 1.2: Propagation of spatial regularity
  • Theorem 1.3
  • Theorem 1.4: Propagation of spatial regularity in $\bar{x}$.
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • ...and 10 more