Global regular motions for compressible barotropic viscous fluids. Stability

We consider viscous compressible barotropic motions in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ 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 $u,\eta \in L_{\mathrm{\infty }}(kT,(k+1)T;H^2(\mathrm{\Omega }))$, $u_t,\eta_t\in L_\infty(kT,(k+1)T;H^1(\Omega))$,$u\in L_2(kT,(k+1)T;H^3(\Omega))$,$u_t\in L_2(kT,(k+1)T;H^2(\Omega))$,where$T>0$isfixedand$k \in $\mathbb{N}$ \cup {0 }$\mathrm{.}Moreover,$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 $v=v_s+u$, $\varrho ={\varrho }_s+\eta$.