Compressible Euler-Maxwell limit for global smooth solutions to the Vlasov-Maxwell-Boltzmann system

TL;DR

This paper rigorously justifies the hydrodynamic limit from the Vlasov-Maxwell-Boltzmann system to the compressible Euler-Maxwell system for smooth solutions near equilibrium, providing explicit convergence rates over almost global times.

Contribution

It establishes the hydrodynamic limit for the Vlasov-Maxwell-Boltzmann system to the Euler-Maxwell system near equilibrium, with new energy estimates handling large velocities.

Findings

Explicit convergence rate in psilon for smooth solutions.

Validation of the hydrodynamic limit in the whole space.

Development of psilon-dependent energy estimates.

Abstract

Two fundamental models in plasma physics are given by the Vlasov-Maxwell-Boltzmann system and the compressible Euler-Maxwell system which both capture the complex dynamics of plasmas under the self-consistent electromagnetic interactions at the kinetic and fluid levels, respectively. It has remained a long-standing open problem to rigorously justify the hydrodynamic limit from the former to the latter as the Knudsen number $\varepsilon$ tends to zero. In this paper we give an affirmative answer to the problem for smooth solutions to both systems near constant equilibrium in the whole space in case when only the dynamics of electrons is taken into account. The explicit rate of convergence in $\varepsilon$ over an almost global time interval is also obtained for well-prepared data. For the proof, one of main difficulties occurs to the cubic growth of large velocities due to the action of the classical transport operator on local Maxwellians and we develop new $\varepsilon$-dependent energy estimates basing on the macro-micro decomposition to characterize the asymptotic limit in the compressible setting.