Analyticity and asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg equations with the zero sound speed in scaling critical spaces

TL;DR

This paper studies the analyticity and long-term behavior of solutions to the compressible Navier-Stokes-Korteweg equations with zero sound speed, providing global existence, decay estimates, and asymptotic formulas in critical spaces.

Contribution

It establishes the analyticity of solutions, derives decay estimates in Fourier-Herz spaces, and presents the first order asymptotic formulas for solutions in the zero sound speed case.

Findings

Global-in-time solutions around equilibrium states

Decay estimates in scaling critical spaces

First order asymptotic formulas for solutions

Abstract

We consider the initial-value problem in the $d$-dimensional Euclidean space $\mathbb{R}^d$ $(d \ge 3)$ for the compressible Navier-Stokes-Korteweg equations under the zero sound speed case (namely, $P'(\rho_*)=0$, where $P=P(\rho)$ stands for the pressure). The system is well-known as the Diffuse Interface model describing the motion of a vaper-liquid mixture in a compressible viscous fluid. The purposes of this paper are to obtain the global-in-time solution around the constant equilibrium states $(\rho_*,0)$ $(\rho_*>0)$ satisfying the estimate on the analyticity as established by Foias-Temam (1989), and investigate the $L^p$-$L^1$ type time-decay estimates in scaling critical settings based on Fourier-Herz spaces. In addition, we also derive the first order asymptotic formula with higher derivatives for solutions as the application of the analyticity.