Global existence of the Navier-Stokes-Korteweg equations with a non-decreasing pressure in $L^p$-framework
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 (add twitter)
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Mathematics - Analysis of PDEs

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07/18/19 06:05PM
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mathAPb: 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
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