Hilbert Expansion for the Relativistic Landau Equation

TL;DR

This paper proves the local-in-time validity of the Hilbert expansion for the relativistic Landau equation, showing solutions converge to relativistic Euler equations as the Knudsen number approaches zero, using novel weighted energy methods.

Contribution

It introduces the first Hilbert expansion analysis for the relativistic Landau equation, employing new time-dependent weights to control momentum growth.

Findings

Solutions converge to relativistic Euler equations as Knudsen number tends to zero

Develops new weighted energy methods to handle momentum growth

First rigorous Hilbert expansion result for Landau-type equations

Abstract

In this paper, we study the local-in-time validity of the Hilbert expansion for the relativistic Landau equation. We justify that solutions of the relativistic Landau equation converge to small classical solutions of the limiting relativistic Euler equations as the Knudsen number shrinks to zero in a weighted Sobolev space. The key difficulty comes from the temporal and spatial derivatives of the local Maxwellian, which produce momentum growth terms and are uncontrollable by the standard $L^2$-based energy and dissipation. We introduce novel time-dependent weight functions to generate additional dissipation terms to suppress the large momentum. The argument relies on a hierarchy of energy-dissipation structures with or without weights. As far as the authors are aware of, this is the first result of the Hilbert expansion for the Landau-type equation.