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Smoothing Estimates of the Vlasov-Poisson-Landau System

Dingqun Deng

Abstract

In this work, we consider the smoothing effect of Vlasov-Poisson-Landau system for both hard and soft potential. In particular, we prove that any classical solutions becomes immediately smooth with respect to all variables. We also give a proof on the global existence to Vlasov-Poisson-Landau system with optimal large time decay. These results give the regularity to Vlasov-Poisson-Landau system. The proof is based on the time-weighted energy method building upon the pseudo-differential calculus.

Smoothing Estimates of the Vlasov-Poisson-Landau System

Abstract

In this work, we consider the smoothing effect of Vlasov-Poisson-Landau system for both hard and soft potential. In particular, we prove that any classical solutions becomes immediately smooth with respect to all variables. We also give a proof on the global existence to Vlasov-Poisson-Landau system with optimal large time decay. These results give the regularity to Vlasov-Poisson-Landau system. The proof is based on the time-weighted energy method building upon the pseudo-differential calculus.

Paper Structure

This paper contains 12 sections, 22 theorems, 325 equations.

Table of Contents

  1. Introduction
  2. Model and Equation.
  3. Notations.
  4. Main results.
  5. Preliminaries
  6. Macroscopic Estimate
  7. Global Existence
  8. Step 4.
  9. Regularity
  10. Appendix
  11. Pseudo-differential calculus
  12. Acknowledgments:

Key Result

Theorem 1.1

Let $\gamma+2\ge -1$, $K\ge 3$, $\psi=1$. Assume $l_0\ge K$ satisfies that Define and where $E_0(x)=E(0,x)$, $l_1=\frac{5(\gamma+2)}{4(1-p)\gamma}$, $l_2=\frac{5(\gamma+2)}{4\gamma}$ for soft potential $-1\le\gamma+2< 0$ and $l_1=l_2=0$ for hard potential $\gamma+2\ge 0$. Let $f_0(x,v)=(f_{0,+}(x,v),f_{0,-}(x,v))$ satisfying $F_\pm(0,x,v)=\mu(v)+(\mu(v))^{1/2}f_{0,\pm}(x,v)\ge 0$. If for any $t

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • ...and 30 more