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

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.