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

Abstract

We are concerned with a system of equations in ${\mathbb{R}}^d(d\ge 2)$ 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 $P^{\prime }({\rho }^{\ast })=0$, 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 ${\bar{\nu }}^2\ge 4\bar{\kappa }$, 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 $1\le p\le d$ if $d\ge 2$. The proof mainly relies on new nonlinear Besov (-Gevrey) estimates for product and composition of functions.