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The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian

Chuqi Cao, Dingqun Deng, Xingyu Li

Abstract

In this work, we consider the Vlasov-Poisson-Boltzmann system without angular cutoff and the Vlasov-Poisson-Landau system with Coulomb potential near a global Maxwellian $μ$. We establish the global existence, uniqueness and large time behavior for solutions in a polynomial-weighted Sobolev space $H^2_{x, v}( \langle v \rangle^k)$ for some constant $k >0$. The proof is based on extra dissipation generated from semigroup method and energy estimates on electrostatic field.

The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian

Abstract

In this work, we consider the Vlasov-Poisson-Boltzmann system without angular cutoff and the Vlasov-Poisson-Landau system with Coulomb potential near a global Maxwellian . We establish the global existence, uniqueness and large time behavior for solutions in a polynomial-weighted Sobolev space for some constant . The proof is based on extra dissipation generated from semigroup method and energy estimates on electrostatic field.

Paper Structure

This paper contains 24 sections, 28 theorems, 366 equations.

Table of Contents

  1. Introduction
  2. Models and equations
  3. Reformulation
  4. Domains
  5. Torus
  6. Union of cubes
  7. Notations
  8. Main results
  9. Strategies and ideas of the proof
  10. Comments on the weight \ref{['eA']}.
  11. Difference between the polynomial case and the exponential weight case.
  12. Organization of the paper
  13. Basic estimates
  14. Preliminaries on the Boltzmann/Landau equation
  15. Upper and lower bound for the non-cutoff Boltzmann operator.
  16. ...and 9 more sections

Key Result

Theorem 1.2

Consider the Cauchy problem vplocal1 for Vlasov-Poisson-Boltzmann/Landau system. Suppose $\gamma \in (-3, 1]$, $s \in [\frac{1}{2}, 1)$, $\gamma+2s>-1$ for the Boltzmann case and $\gamma \in [-3, 1]$ for the Landau case. Let $l=0$ for the hard potential case and $l>\frac{|\gamma|}{2}$ for the soft p Remind $\mathcal{E}_k(t)$ is given in DefE. If the initial data $f_0$ satisfies $F_0(x,v)=\mu+f_0(x

Theorems & Definitions (50)

  • Remark 1.1
  • Theorem 1.2: Global existence, uniqueness and large-time decay
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: H, Theorem 1.1
  • Corollary 2.5
  • ...and 40 more