Compressible fluid limit for smooth solutions to the Landau equation

TL;DR

This paper proves the compressible Euler and acoustic limits of the Landau equation with Coulomb potentials, demonstrating uniform estimates and convergence to fluid solutions in a high-order Sobolev space, addressing challenges posed by velocity diffusion effects.

Contribution

It establishes the compressible Euler and acoustic limits for the Landau equation with Coulomb potentials, providing uniform estimates and convergence results in a high-order Sobolev space.

Findings

Uniform estimates independent of Knudsen number 

Convergence of Landau solutions to Euler solutions at rate O()

Establishment of the acoustic limit in optimal scaling

Abstract

Although the compressible fluid limit of the Boltzmann equation with cutoff has been well investigated in [6] and [13], it still remains largely open to obtain analogous results in case of the angular non-cutoff or even in the grazing limit which gives the Landau equation, essentially due to the velocity diffusion effect of collision operator such that $L^\infty$ estimates are hard to obtain without using Sobolev embeddings. In the paper, we are concerned with the compressible Euler and acoustic limits of the Landau equation for Coulomb potentials in the whole space. Specifically, over any finite time interval where the full compressible Euler system admits a smooth solution around constant states, we construct a unique solution in a high-order weighted Sobolev space for the Landau equation with suitable initial data and also show the uniform estimates independent of the small Knudsen number $\varepsilon>0$, yielding the $O(\varepsilon)$ convergence of the Landau solution to the local Maxwellian whose fluid quantities are the given Euler solution. Moreover, the acoustic limit for smooth solutions to the Landau equation in an optimal scaling is also established. For the proof, by using the macro-micro decomposition around local Maxwellians together with techniques for viscous compressible fluid and properties of Burnett functions, we design an $\varepsilon$-dependent energy functional to capture the dissipation in the compressible fluid limit with feature that only the highest order derivatives are most singular.