Global existence and analyticity of $L^p$ solutions to the compressible fluid model of Korteweg type

TL;DR

This paper proves the global existence and analyticity of solutions to a compressible fluid model of Korteweg type in certain Besov spaces, especially when the sound speed is zero, using new nonlinear estimates.

Contribution

It establishes the global-in-time existence and Gevrey analyticity of solutions for the Korteweg fluid model with zero sound speed in $L^p$-based Besov spaces, extending previous $L^2$ results.

Findings

Global existence and Gevrey analyticity of solutions.

Improved $L^p$ bounds for density and velocity.

Validity under specific viscosity and capillary conditions.

Abstract

We are concerned with a system of equations in $\mathbb{R}^{d}(d\geq2)$ governing the evolution of isothermal, viscous and compressible fluids of Korteweg type, that can be used as a phase transition model. In the case of zero sound speed $P'(\rho^{\ast})=0$, it is found that the linearized system admits the \textit{purely} parabolic structure, which enables us to establish the global-in-time existence and Gevrey analyticity of strong solutions in hybrid Besov spaces of $L^p$-type. Precisely, if the full viscosity coefficient and capillary coefficient satisfy $\bar{\nu}^2\geq4\bar{\kappa}$, then the acoustic waves are not available in compressible fluids. Consequently, the prior $L^2$ bounds on the low frequencies of density and velocity could be improved to the general $L^p$ version with $1\leq p\leq d$ if $d\geq2$. The proof mainly relies on new nonlinear Besov (-Gevrey) estimates for product and composition of functions.