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Global regular motions for compressible barotropic viscous fluids. Stability

We consider viscous compressible barotropic motions in a bounded domain $Ω\subset \mathbb{R}^3$ 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 $u=v-v_s$, $η=\varrho-\varrho_s$. We prove existence of $u$, $η$ such that $u,η\in L_\infty(kT,(k+1)T;H^2(Ω))$, $u_t,η_t\in L_\infty(kT,(k+1)T;H^1(Ω))$, $u\in L_2(kT,(k+1)T;H^3(Ω))$, $u_t\in L_2(kT,(k+1)T;H^2(Ω))$, where $T>0$ is fixed and $k \in \mathbb{N} \cup \{0 \}$. Moreover, $u$, $η$ 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 $v=v_s+u$, $\varrho=\varrho_s+η$.