We consider the isentropic Navier-Stokes-Korteweg equations with a
non-decreasing pressure on the whole space $\mathbb{R}^d$ $(d \ge 2)$, 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 $L^p$-in-time and $L^q$-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 $H^{s + 1, q} (\mathbb{R}^d) \times H^{s, q}
(\mathbb{R}^d)^d$ with $s \ge [d/q] + 1$. 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. The proof relies on a
smoothing effect for the density due to the capillary terms and the maximal
regularity property for the negative of the Laplace operator.

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Keiichi Watanabe : Global existence of the Navier-Stokes-Korteweg equations with a non-decreasing pressure in $L^p$-framework https://t.co/xurF8PFC6a https://t.co/BrAiBppfJq