Viscous compressible barotropic motions (described by v-velocity and ϱ-density) in a bounded domain Ω ⊂ R3 with v=0 on the boundary are considered. Assuming existence of some special global sufficiently regular solutions (vs velocity and ϱs-density), we prove their stability by assuming that initial differences of u=v-vs and η=ϱ−ϱs are sufficiently small in some norms. Then we prove existence of u, η such that u,η∈L∞(kT,(k+1)T;H2(Ω)), ut,ηt∈L∞(kT,(k+1)T;H1(Ω)), u∈L2(kT,(k+1)T;H3(Ω)), ut∈L2(kT,(k+1)T;H2(Ω)), where T>0 and k ∈ N ∪ {0}. Moreover, u, η are sufficiently small in the above norms. Finally, the existence of global regular solutions such that v = vs +u,𝜚ϱ = ϱs +η is proved.
The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7/2007-2013/ under REA grant agreement 319012 and from the Funds for International Co-operation under Polish Ministry of Science and Higher Education grant agreement 2853/7.PR/2013/2. The authors thank professor Y. Shibata for discussions, which helped to improve the proof of Lemma 4.8. The authors thank to the referee for very important comments.
