On the compressible Navier-Stokes-Korteweg equations

Abstract

In this paper, we consider compressible Navier-Stokes-Korteweg (N-S-K) equations with more general pressure laws, that is the pressure $P$ is non-monotone. We prove the stability of weak solutions in the periodic domain $\Omega=\mathbb{T}^{N}$, when $N = 2,3$. Utilizing an interesting Sobolev inequality to tackle the complicated Korteweg term, we obtain the global existence of weak solutions in one dimensional case. Moreover, when the initial data is compactly supported in the whole space $\mathbb{R}$, we prove the compressible N-S-K equations will blow-up in finite time.