On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum

TL;DR

This paper investigates the existence, uniqueness, and potential finite-time blowup of regular solutions for the coupled Vlasov and compressible Navier-Stokes equations with degenerate viscosities and vacuum, providing rigorous results on singularity formation.

Contribution

It establishes the first rigorous conditions for finite-time singularity formation in the Vlasov/Navier-Stokes system with degenerate viscosities and vacuum.

Findings

Existence and uniqueness of local-in-time regular solutions with large initial data and vacuum.

Sufficient conditions for finite-time blowup of solutions.

Rigorous validation of previous conjectures on singularity formation.

Abstract

We analyze the Vlasov equation coupled with the compressible Navier--Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative velocity between particle and fluid. We first establish the existence and uniqueness of local-in-time regular solutions with arbitrarily large initial data and a vacuum. We then present sufficient conditions on the initial data leading to the finite-time blowup of regular solutions. In particular, our study makes the result on the finite-time singularity formation for Vlasov/Navier-Stokes equations discussed by Choi [J. Math. Pures Appl., 108, (2017), 991-1021] completely rigorous.