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

TL;DR

This work analyzes the three-dimensional Navier–Stokes–Vlasov–Fokker–Planck (NSVFP) system, which models a viscous incompressible fluid coupled to dispersed particles via a friction force. The authors construct global suitable weak solutions by first solving a regularized problem and then passing to the limit using Tao’s Littlewood–Paley decomposition, DiPerna–Lions compactness, and Aubin–Lions compactness for the velocity. They establish new global and local energy inequalities, prove convergence of particle density and its moments, and derive the Hausdorff-dimension result for the singular set, along with α-Hölder continuity of f at regular points. The results provide a rigorous partial regularity framework for a fully coupled kinetic-fluid model in 3D, illuminating the interaction between fluid dynamics and kinetic transport in two-phase flows. These findings lay groundwork for further analysis of two-phase flow stability and regularity in kinetic-fluid systems under broad initial data and moment conditions.

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 $α$-Hölder continuity of $f$ at the regular points of $u$.

Theorems & Definitions (71)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Lemma 2.1: Lemma 1 in carpioLongtimeBehaviourSolutions1998
• Lemma 2.2: Lemma 4.2 in carpioWellPosednessIntegrodifferential2016
• proof : Proof of \ref{['5']}