2018/10/11 by Wojciech M. Zajączkowski, Zajaczkowski, Wojciech M.
Engineering · Mathematics · #35A01 #35B10 #76N10 #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1810.04926
openalex publication_date 2018/10/11 · openalex created_date 2018/10/26 · openalex updated_date 2026/07/28
We consider barotropic motions described by the compressible Navier-Stokes equations in a box with periodic boundary conditions. We are looking for density \varrho in the form \varrho=a+η, where a is a constant and η|t=0 is sufficiently small in H2-norm. We assume existence of potentials φ and ψ such that v=∇φ+rotψ. Next we assume that ∇φ|t=0 is sufficiently small in H2-norm too. Finally, we assume that the second viscosity coefficient ν is sufficiently large. Then we prove long time existence of solutions such that v∈ L_∞(0,T;H2(Ω))∩ L2(0,T;H3(Ω)), v,t∈ L_∞(0,T;H1(Ω))∩ L2(0,T;H2(Ω)), where the existence time T is proportional to ν. Next for T sufficiently large we obtain that v(T) is correspondingly small so global existence is proved using the methods appropriate for problems with small data.