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Landau damping, collisionless limit, and stability threshold for the Vlasov-Poisson equation with nonlinear Fokker-Planck collisions

Jacob Bedrossian, Weiren Zhao, Ruizhao Zi

Abstract

In this paper, we study the Vlasov-Poisson-Fokker-Planck (VPFP) equation with a small collision frequency $0 < ν\ll 1$, exploring the interplay between the regularity and size of perturbations in the context of the asymptotic stability of the global Maxwellian. Our main result establishes the Landau damping and enhanced dissipation phenomena under the condition that the perturbation of the global Maxwellian falls within the Gevrey-$\frac{1}{s}$ class and obtain that the stability threshold for the Gevrey-$\frac{1}{s}$ class with $s>s_{\mathrm{k}}$ can not be larger than $γ=\frac{1-3s_{\mathrm{k}}}{3-3s_{\mathrm{k}}}$ for $s_{\mathrm{k}}\in [0,\frac{1}{3}]$. Moreover, we show that for Gevrey-$\frac{1}{s}$ with $s>3$, and for $t\ll ν^{\frac13}$, the solution to VPFP converges to the solution to Vlasov-Poisson equation without collision.

Landau damping, collisionless limit, and stability threshold for the Vlasov-Poisson equation with nonlinear Fokker-Planck collisions

Abstract

In this paper, we study the Vlasov-Poisson-Fokker-Planck (VPFP) equation with a small collision frequency , exploring the interplay between the regularity and size of perturbations in the context of the asymptotic stability of the global Maxwellian. Our main result establishes the Landau damping and enhanced dissipation phenomena under the condition that the perturbation of the global Maxwellian falls within the Gevrey- class and obtain that the stability threshold for the Gevrey- class with can not be larger than for . Moreover, we show that for Gevrey- with , and for , the solution to VPFP converges to the solution to Vlasov-Poisson equation without collision.
Paper Structure (34 sections, 17 theorems, 427 equations, 1 table)

This paper contains 34 sections, 17 theorems, 427 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main result
  3. Main ideas
  4. The time-dependent Gaussian weight
  5. The time-dependent Fourier multiplier
  6. Three levels of energy functionals
  7. Further discussion
  8. The working system
  9. Reformulation (I): the distribution function
  10. Reformulation (II): the density
  11. Relationship between unknowns $f$ and $f^w$
  12. Proof of the main theorems
  13. Bootstrap
  14. Proof of Theorem \ref{['Thm:main2']}
  15. Some estimates on $M_\theta$
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $s_{\mathrm{k}}\in [0,\frac{1}{3}]$, $\lambda_{\mathrm{in}}>c_0>0$, $\sigma_0 \geq n+20$, and $m\geq n+10$. For any $1\geq s>s_{\mathrm{k}}$, we define the following weighted Gevrey norm Then the stability threshold $\gamma(\mathcal{G}^{s,\lambda_{\mathrm{in}},\sigma}_{m+2,2\nu})$ can not be larger than $\frac{1-3s_{\mathrm{k}}}{3-3s_{\mathrm{k}}}$, namely, if then there exists $\epsilon_0>0

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.2
  • Remark 1.7
  • Lemma 2.1: from $f^w$ to $f$
  • ...and 22 more