arXiv 1907.077522019-07math.AP

$\mathrm{L}^p$框架下压力不减的Navier-Stokes-Korteweg方程解的整体存在性

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

Keiichi Watanabe

备注：: 21 pages, no figure

摘要：

本文考虑全空间$\mathbb{R}^n$ $(n \ge 2)$上压力不减的等熵Navier-Stokes-Korteweg方程组，该方程组描述了可压缩流体如具有相变的汽液混合物的运动，包括可变的内部毛细效应。通过假设$(p, q) \in (1, 2) \times (1, \infty)$或$(p, q) \in \{2\} \times (1, 2]$，在$\mathrm{L}^p$ -in-time和$\mathrm{L}^q$ -in-space框架下，特别是在极大正则类下，证明了该系统存在唯一的全局强解.我们表明，该系统是全局适定的小初始数据属于$\mathrm{H}^{s + 1, q} (\mathbb{R}^n) \times \mathrm{H}^{s, q} (\mathbb{R}^n)^n$提供$s > n/q$，如果$q \le n$和$s \ge 1$，如果$q > n$。我们的结果允许的情况下，当压力的导数为零，在一个给定的恒定状态，也就是说，流体改变相从蒸汽到液体或从液体到蒸汽的临界状态。本文的论证不需要对压力有任何精确的表达式或先验的假设。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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