Finite-time blow-up phenomena of Vlasov/Navier-Stokes equations and related systems

TL;DR

This paper investigates the finite-time blow-up of classical solutions in coupled kinetic-fluid systems like Vlasov/Navier-Stokes, demonstrating that initially smooth solutions can develop singularities in finite time, with new proof techniques and extensions to related models.

Contribution

It introduces a novel approach to proving finite-time blow-up for coupled kinetic-fluid systems and extends results to related fluid models, filling a gap in the existing literature.

Findings

Solutions can become singular in finite time despite initial smoothness.

New proof method for finite-time blow-up in coupled systems.

Results extend to isentropic compressible Navier-Stokes and two-phase fluids.

Abstract

This paper deals with the finite-time blow-up phenomena of classical solutions for Vlasov/Navier-Stokes equations under suitable assumptions on the initial configurations. We show that a solution to the coupled kinetic-fluid system may be initially smooth, however, it can become singular in a finite period of time. We provide a simple idea of showing the finite time blow up of classical solutions to the coupled system which has not been studied so far. We also obtain analogous results for related systems, such as isentropic compressible Navier-Stokes equations, two-phase fluid equations consisting of pressureless Euler equations and Navier-Stokes equations, and thick sprays model.