On the time-decay with the diffusion wave phenomenon of the solution to the compressible Navier-Stokes-Korteweg system in critical spaces

TL;DR

This paper investigates the global behavior and decay properties of solutions to the compressible Navier-Stokes-Korteweg system in critical spaces, revealing diffusion wave phenomena and decay estimates in a rigorous mathematical framework.

Contribution

It establishes global-in-time solutions and $L^p$-$L^1$ decay estimates in critical Besov spaces, and analyzes the diffusion wave property with a novel approach using Fourier-Besov norms.

Findings

Proved global existence of solutions around equilibrium states.

Derived $L^p$-$L^1$ decay estimates for solutions.

Identified diffusion wave phenomena in the solution behavior.

Abstract

We consider the initial value problem of the compressible Navier-Stokes-Korteweg equations in the whole space $\mathbb{R}^d$ ($d \ge 2$). The purposes of this paper are to obtain the global-in-time solution around the constant equilibrium states $(\rho_*,0)$ and investigate the $L^p$-$L^1$ type time-decay estimates in a scaling critical framework, where $\rho_*>0$ is a constant. In addition, we study the diffusion wave property came from the wave equation with strong damping for the solution with the initial data belonging to the critical Besov space. The key idea of the proof is the derivation of the time-decay for the Fourier-Besov norm with higher derivatives by using $L^1$-maximal regularity for the perturbed equations around $(\rho_*,0)$.