Blow-up of Classical Solutions to the Isentropic Compressible Barotropic Navier-Stokes-Langevin-Korteweg Equations

Abstract

In this paper, we study the barotropic Navier-Stokes-Langevin-Korteweg system in $\mathbb{R}^{3}$. Assuming  the derivatives of the square root of the density and the velocity field decay to zero at infinity, we can prove the classical solutions blow up in finite time when the initial energy has a certain upper bound. We obtain this blow up result by a contradiction argument based on the conservation of the total mass and the total quasi momentum.