Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global Regularity of the Vlasov-Poisson-Boltzmann System Near Maxwellian Without Angular Cutoff for Soft Potential

Dingqun Deng

Abstract

We consider the non-cutoff Vlasov-Poisson-Boltzmann (VPB) system of two species with soft potential in the whole space $\mathbb{R}^3$ when an initial data is near Maxwellian. Continuing the work Deng [Comm. Math. Phys. 387, 1603-1654 (2021)] for hard potential case, we prove the global regularity of the Cauchy problem to VPB system for the case of soft potential in the whole space for the whole range $0<s<1$. This completes the smoothing effect to the Vlasov-Poisson-Boltzmann system, which shows that any classical solutions are smooth with respect to $(t,x,v)$ for any positive time $t>0$. The proof is based on the time-weighted energy method building upon the pseudo-differential calculus.

Global Regularity of the Vlasov-Poisson-Boltzmann System Near Maxwellian Without Angular Cutoff for Soft Potential

Abstract

We consider the non-cutoff Vlasov-Poisson-Boltzmann (VPB) system of two species with soft potential in the whole space when an initial data is near Maxwellian. Continuing the work Deng [Comm. Math. Phys. 387, 1603-1654 (2021)] for hard potential case, we prove the global regularity of the Cauchy problem to VPB system for the case of soft potential in the whole space for the whole range . This completes the smoothing effect to the Vlasov-Poisson-Boltzmann system, which shows that any classical solutions are smooth with respect to for any positive time . The proof is based on the time-weighted energy method building upon the pseudo-differential calculus.
Paper Structure (8 sections, 16 theorems, 215 equations)

This paper contains 8 sections, 16 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Equations
  3. Collision Kernel
  4. Reformulation
  5. Notations
  6. Main results
  7. Preliminaries
  8. Regularity

Key Result

Theorem 1.1

Let $-\frac{3}{2}-2s<\gamma\le -2s$, $0<s<1$, $0<\tau<T\le \infty$ and $l\ge 0$. For any $K\ge 4$ and multi-indices $|\alpha|+|\beta|\le K$, assume $\psi=t^N$ with $N>0$ large when $|\alpha|\le 4$ and $N=N(\alpha)>0$ defined by 106a when $|\alpha|>4$. Let $(f,E)$ be the solution to 7, 8 and 9 satisf Then the followings hold true. (1) If is sufficiently small, then for $|\alpha|+|\beta|\le K$, $T<

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 2.1: Duan2013, Theorem 1.1
  • proof
  • Lemma 2.2: Deng2020a, Lemma 2.3
  • Lemma 2.3: Deng2020a, Lemma 2.4
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • ...and 17 more