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Hydrodynamic limit of rarefaction wave for the Vlasov-Maxwell-Landau system with Coulomb potential

Guanghui Wang, Lingda Xu, Tong Yang, Mingying Zhong

Abstract

In this paper, we investigate the hydrodynamic limit of rarefaction wave for the two-species Vlasov-Maxwell-Landau(VML) system with Coulomb potential. We prove that for any given time interval, the solution of the Vlasov-Maxwell-Landau system with appropriate initial data converges to a rarefaction wave as the Knudsen number $ε$ approaches zero. The main difficulty in the analysis lies in the loss of dissipation in the interaction between the electromagnetic field and the microscopic component, and the weak dissipation induced by the Lorentz force and the scaling with small parameter $ε$. For this, we introduce a velocity weight function and a space-time scaling parameter together with suitable $ε$-dependent energy estimates.

Hydrodynamic limit of rarefaction wave for the Vlasov-Maxwell-Landau system with Coulomb potential

Abstract

In this paper, we investigate the hydrodynamic limit of rarefaction wave for the two-species Vlasov-Maxwell-Landau(VML) system with Coulomb potential. We prove that for any given time interval, the solution of the Vlasov-Maxwell-Landau system with appropriate initial data converges to a rarefaction wave as the Knudsen number approaches zero. The main difficulty in the analysis lies in the loss of dissipation in the interaction between the electromagnetic field and the microscopic component, and the weak dissipation induced by the Lorentz force and the scaling with small parameter . For this, we introduce a velocity weight function and a space-time scaling parameter together with suitable -dependent energy estimates.
Paper Structure (28 sections, 3 theorems, 371 equations)

This paper contains 28 sections, 3 theorems, 371 equations.

Table of Contents

  1. Introduction
  2. The model and previous results
  3. Macro-micro equations and dissipation
  4. Rarefaction wave and approximation
  5. Notations and main results
  6. Reformulation and a priori estimate
  7. Scaling and reformulation
  8. A Priori estimates
  9. Lower order energy estimates
  10. Estimates of the macro-components and electromagnetic field
  11. Low order energy estimates on $n$
  12. Estimates on $\partial_y\phi$ and $\partial_{\tau}(\phi,\psi,\zeta,n)$
  13. Estimate on $(\mathbf{g}_1,\mathbf{g}_2)$
  14. High order energy estimates
  15. Estimate on derivatives of $(\phi,\psi,\zeta)$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $(\bar{\rho},\bar{u},\bar{\theta})(t,x)$ be the smooth rarefaction wave defined in smooth and $M_{[\bar{\rho}, \bar{u}, \bar{\theta}]}(t, x,v)$ denotes the corresponding local Maxwellian. Assume that $\eta_0:=\sup\limits_{t\ge 0,x\in\mathbb{R}}\{|\bar{\rho}(t,x)-1|+|\bar{u}(t,x)|+|\bar{\theta}(t admits a unique solution $F_1(t,x,v),F_2(t,x,v), E(t,x), B(t,x)$ for all $t\in[0,T]$ and it satisfi

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 16 more