Low Mach number limit of the compressible Euler--Vlasov--Fokker--Planck system in the whole space

Abstract

Although there are many important contributions on compressible and incompressible fluid-particle interaction models respectively, how to connect the two-type fluid-particle models via the low Mach number limit remains a challenging open problem. In this paper, we resolve it for the compressible isentropic fluid-particle model (Euler--Vlasov--Fokker--Planck (Euler--VFP) system) in the whole space ${\mathbb{R}}^3$. First, we establish the global-in-time {\it a priori estimates} of strong solutions that are uniform with respect to the Mach number $\epsilon$ near the global Maxwellian. The proof relies on a refined energy method that combines the relaxation structure $b^{\epsilon }-u^{\epsilon }$ induced by the fluid-particle interaction and the symmetrized acoustic structure of the compressible Euler part in the model. Under the assumption of well-prepared initial data, we derive a {\it global-in-time} uniform error estimate in the $H^2$ framework between the solution of the compressible Euler--VFP system and that of the limiting incompressible Euler--VFP system. A key point is to introduce the corrected acoustic variable $q^{\epsilon }-\epsilon [P^{\prime }(1){]}^{-1}\pi$, which captures the pressure corrector in the low Mach number limit. This also allows us to exploit the exact cancellation of the singular acoustic terms and to close the {\it global-in-time} error estimate. The damping term $b^{\epsilon }-u^{\epsilon }$, which is absent in the pure Euler equations, plays an essential role in recovering the relative velocity dissipation and in controlling the coupled fluid-particle dynamics. As a consequence, we prove the low Mach number limit of the compressible Euler--VFP system with the convergence rate $\mathcal{O}(\epsilon )$ in the time-continuous $H^2$ topology.

34 pages. All comments are welcome