Global regular motions for compressible barotropic viscous fluids. Stability

TL;DR

This paper proves the stability of special spherically symmetric solutions in viscous compressible barotropic fluids within a bounded domain, establishing existence and regularity of solutions close to these special solutions.

Contribution

It demonstrates the stability of special solutions and constructs solutions close to them with detailed regularity properties in a bounded domain.

Findings

Existence of solutions close to special solutions with specified regularity.

Stability of the special solutions $v_s$, $ ho_s$ established.

Solutions exist with controlled norms in Sobolev spaces.

Abstract

We consider viscous compressible barotropic motions in a bounded domain $\Omega \subset \mathbb{R}^3$ with the Dirichlet boundary conditions for velocity. We assume the existence of some special sufficiently regular solutions $v_s$ (velocity), $\varrho_s$ (density) of the problem. By the special solutions we can choose spherically symmetric solutions. Let $v$, $\varrho$ be a~solution to our problem. Then we are looking for differences $u=v-v_s$, $\eta=\varrho-\varrho_s$. We prove existence of $u$, $\eta$ such that $u,\eta\in L_\infty(kT,(k+1)T;H^2(\Omega))$, $u_t,\eta_t\in L_\infty(kT,(k+1)T;H^1(\Omega))$, $u\in L_2(kT,(k+1)T;H^3(\Omega))$, $u_t\in L_2(kT,(k+1)T;H^2(\Omega))$, where $T>0$ is fixed and $k \in \mathbb{N} \cup \{0 \}$. Moreover, $u$, $\eta$ are sufficiently small in the above norms. This also means that stability of the special solutions $v_s$, $\varrho_s$ is proved. Finally, we proved existence of solutions such that $v=v_s+u$, $\varrho=\varrho_s+\eta$.