Compressible fluid limit for smooth solutions to the Landau equation

Abstract

Although the compressible fluid limit of the Boltzmann equation with cutoff has been well investigated in [6] and [33], 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 ∞ 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 ε > 0, yielding the O(ε) 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 ε-dependent energy functional to capture the dissipation in the compressible fluid limit with feature that only the highest order derivatives are most singular.

The Vlasov-Poisson-Landau system near a local Maxwellian

“…In the incompressible regime, Guo [36] gave the rigorous proof of the limit to the incompressible Navier-Stokes system for smooth solutions to the Landau equation near global Maxwellians; see also two recent works [11] and [57]. In the compressible regime, the compressible Euler limit for smooth solutions to the Landau equation near global Maxwellians was recently studied by Duan-Yang-Yu [23] and Lei-Liu-Xiao-Zhao [51] via different energy methods. In the setting of one space dimension, regarding solutions with wave patterns, the only results are [24] for rarefaction waves and [68] for contact waves.…”

“…By using this decomposition, we can obtain the rescaled compressible Euler-Poisson equations with some source terms such that we can observe the existence of dispersive behavior, particularly the KdV limit under consideration. As mentioned in [23], on the one hand, the inverse of linearized operator L −1 M in the macroscopic equation is very complicated due to the velocity diffusion effect of collision operator. To this end, we make use of the Burnett functions and velocity-decay properties to handled the terms involving L −1 M so that the estimates can be obtained in a clear way, see the identities (4.21) and (4.22) for details.…”