Asymptotic stability of steady states for the compressible Navier-Stokes-Riesz system in the presence of vacuum

TL;DR

This paper proves the asymptotic stability of steady states for a one-dimensional compressible Navier-Stokes-Riesz system with vacuum, establishing global solutions and convergence rates despite degeneracy and non-local effects.

Contribution

It introduces a novel analysis of the stability and convergence of solutions to a vacuum free boundary problem with Riesz potential, handling degeneracy and non-locality.

Findings

Existence of unique global-in-time strong solutions.

Lyapunov-type stability of steady states.

Convergence rates in weighted Sobolev spaces.

Abstract

We consider a one-dimensional physical vacuum free boundary problem on the compressible Navier-Stokes-Riesz system for an attractive Riesz potential $|x|^{2s-1}/(2s-1)$ with $0<s<1/2$. It is proved that for the adiabatic constant $\gamma$ satisfying $2(1-s)<\gamma<1+2s/3$ under the additional condition that $3/8<s<1/2$, there exists a unique global-in-time strong solution. Specifically, we establish the Lyapunov-type stability of the compactly supported steady states in the Lagrangian coordinates and we also obtain the time rate of convergence for the strong solution to steady states with the same mass in weighted Sobolev spaces where the weights indicate the behavior of solutions near the vacuum free boundary. The difficulties and challenges in the proof are caused not only by the degeneracy due to the vacuum free boundary but also by the non-local feature of the Riesz potential.