Global existence of the Navier-Stokes-Korteweg equations with a non-decreasing pressure in ${\mathrm{L}}^p$-framework

Abstract

We consider the isentropic Navier-Stokes-Korteweg equations with a non-decreasing pressure on the whole space ${\mathbb{R}}^n$ $(n\ge 2)$, where the system describes the motion of compressible fluids such as liquid-vapor mixtures with phase transitions including a variable internal capillarity effect. We prove the existence of a unique global strong solution to the system in the ${\mathrm{L}}^p$-in-time and ${\mathrm{L}}^q$-in-space framework, especially in the maximal regularity class, by assuming $(p,q)\in (1,2)\times (1,\infty )$ or $(p,q)\in \{2\}\times (1,2]$. We show that the system is globally well-posed for small initial data belonging to ${\mathrm{H}}^{s+1,q}({\mathbb{R}}^n)\times {\mathrm{H}}^{s,q}({\mathbb{R}}^n{)}^n$ provided $s>n/q$ if $q\le n$ and $s\ge 1$ if $q>n$. Our results allow the case when the derivative of the pressure is zero at a given constant state, that is, the critical states that the fluid changes a phase from vapor to liquid or from liquid to vapor. The arguments in this paper do not require any exact expression or a priori assumption on the pressure.

21 pages, no figure