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.
European Union's Seventh Framework Programme FP7/2007-2013/, Grant/Award Number: 319012; Funds for International Co-operation under Polish Ministry of Science and Higher Education, Grant/Award Number: 2853/7.PR/2013/2The 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.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. The authors thank to the referee for very important comments.
