Existence and partial regularity of suitable weak solutions to the 3D Navier-Stokes-Vlasov-Fokker-Planck equations

TL;DR

This paper establishes the existence of suitable weak solutions for the 3D Navier-Stokes-Vlasov-Fokker-Planck system, analyzing their regularity and singularity structure using advanced convergence and energy inequality techniques.

Contribution

It introduces a new class of suitable weak solutions satisfying energy inequalities, and characterizes their singular set and regularity properties.

Findings

Existence of suitable weak solutions with energy estimates.

Characterization of the singular set's Hausdorff dimension.

Proven regularity and Hölder continuity of the particle density function.

Abstract

In this paper, we investigate the incompressible Navier-Stokes equations coupled with the Vlasov-Fokker-Planck equation, which describes a two-phase mixture of the viscous incompressible fluid with particles or bubbles through a frictional force term. In the three-dimensional whole space, we construct a new class of suitable weak solutions to the Navier-Stokes-Vlasov-Fokker-Planck system satisfying energy estimates and three local or global energy inequalities of different forms. These obtained local energy inequalities play an important role in characterizing the measure of the singularity set of weak solutions. The main difficulties in deriving these inequalities lie in establishing the convergence of the density function $f$ in bounded or unbounded domains and dealing with the convergence of the non-local frictional force term. The strong convergence of both $f$ and $f \log f$ weighted by $|v|^k$ is proved by exploring some new a priori quantities of the velocity with the help of Tao's $L^p$ decomposition and the DiPerna-Lions compactness method. Moreover, as an immediate consequence of the existence result, we are able to describe the Hausdorff dimension of set of singular points of the fluid velocity $u$ and also establish the $\alpha$-H\"{o}lder continuity of $f$ at the regular points of $u$.