[论文解读] Hydrodynamic limit of rarefaction wave for the Vlasov-Maxwell-Landau system with Coulomb potential

Guanghui Wang, Lingda Xu+2 more|arXiv (Cornell University)|Feb 25, 2026

Gas Dynamics and Kinetic Theory被引用 0

一句话总结

直接回答：证明带库仑势的两种粒子Vlasov-Maxwell-Landau系统的解在努塞数趋于零时收敛到稀疏波，通过能量方法（带速度权重与带放缩的时空分析）获得流体极限。

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.

研究动机与目标

• Motivate and rigorously justify the hydrodynamic limit to a rarefaction wave for the VML system with Coulomb potential.
• Develop an energy framework that overcomes weak dissipation from electromagnetic coupling and Lorentz force.
• Construct an approximate rarefaction wave and quantify convergence in a near-Maxwellian setting.
• Handle the micro-macro decomposition and weighted dissipation to obtain uniform-in-epsilon estimates.

提出的方法

• Macro-micro decomposition around a local Maxwellian for F1 and a macroscopic-microscopic split for F2.
• Velocity-weighted energy estimates with epsilon-dependent scalings and space-time rescaling.
• Construction of a smooth approximate 3-rarefaction wave via Burgers-type data and Riemann invariants.
• Derivation of epsilon-dependent a priori estimates and higher-order weighted estimates.
• Development of weighted energy inequalities that manage weak electromagnetic dissipation and Lorentz-force induced terms.
• Proving convergence of the scaled VML solution to the rarefaction wave on any finite time interval as epsilon -> 0.

实验结果

研究问题

• RQ1 Can the two-species Vlasov-Maxwell-Landau system with Coulomb potential be approximated by a rarefaction wave in the hydrodynamic limit (epsilon -> 0)?
• RQ2 What energy framework and dissipation structure are sufficient to control the coupling between microscopic components and electromagnetic fields?
• RQ3 How does one construct a smooth approximate rarefaction wave to enable rigorous convergence analysis in this kinetic-fluid setting?
• RQ4 What are the precise convergence rates and regularity requirements for the macro-variables and fields as epsilon vanishes?

主要发现

• The solution of the VML system converges to a rarefaction wave on any fixed time interval as epsilon -> 0.
• An epsilon-dependent energy method with velocity weights yields uniform-in-epsilon estimates despite weak dissipation from Lorentz force and field coupling.
• A refined a priori framework using macro-micro decomposition and weighted norms controls both macroscopic and microscopic components and the electromagnetic field.
• Convergence results are established under smallness assumptions on the rarefaction strength and far-field data, with explicit conditions on initial energy and parameters.
• The authors also develop convergence rates and extended results via a second theorem that optimizes the choice of spatial-temporal scaling to relate the kinetic solution to the rarefaction profile.