2015/08/25 by Hyeong‐Ohk Bae, Wojciech M. Zajączkowski, Bae, H-O. +1
Mathematics · #35A01 #35Q30 #76N10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1508.06127
openalex publication_date 2015/08/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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), \varrhos (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-vs, η=\varrho-\varrhos. 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, \varrhos is proved. Finally, we proved existence of solutions such that v=vs+u, \varrho=\varrhos+η.