Global regular motions for compressible barotropic viscous fluids. Stability

الأصل · EN

We consider viscous compressible barotropic motions in a bounded domain Ω⊂ R³ with the Dirichlet boundary conditions for velocity. We assume the existence of some special sufficiently regular solutions vₛ (velocity), ₛ (density) of the problem. By the special solutions we can choose spherically symmetric solutions. Let v, be a solution to our problem. Then we are looking for differences u=v-vₛ, η=-ₛ. 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 is fixed and k ∈ N ∪ {0 }. Moreover, u, η are sufficiently small in the above norms. This also means that stability of the special solutions vₛ, ₛ is proved. Finally, we proved existence of solutions such that v=vₛ+u, =ₛ+η.