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.