Compressible fluid limit for smooth solutions to the Landau equation

Abstract

Although the compressible fluid limit of the Boltzmann equation with cutoff has been extensively investigated in Caflisch [Comm. Pure Appl. Math. 33 (1980), 651–666] and Guo, Jang, and Jiang [Comm. Pure Appl. Math. 63 (2010), 337–361], obtaining analogous results in the case of the angular non-cutoff or even in the grazing limit which gives the Landau equation, still remains largely open, essentially due to the velocity diffusion effect of the collision operator such that $L^{\infty }$ estimates are hard to obtain without using Sobolev embeddings. In this 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 $\epsilon >0$, yielding the $O(\epsilon )$ 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 $\epsilon$-dependent energy functional to capture the dissipation in the compressible fluid limit with the feature that only the highest-order derivatives are most singular.