Global regular motions for compressible barotropic viscous fluids: Stability

Abstract

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.