2024/04/11 by Huang, Yucong, Hashimoto, Itsuko, Nishibata, Shinya
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2404.07469
We investigate an inflow problem for the multi-dimensional isentropic compressible Navier-Stokes equations. The fluid under consideration occupies the exterior domain of unit ball, Ω=\x∈ℝn \vert |x|≥ 1\, and a constant stream of mass is flowing into the domain from the boundary ∂Ω=\|x|=1\. It is shown in Hashimoto-Matsumura(2021) that if the fluid velocity at the far-field is assumed to be zero, then there exists a unique spherically symmetric stationary solution, denoted as (ρ,u)(r) with r≡ |x|. In this paper, we show that either ρ is monotone increasing or ρ attains a unique global minimum, and this is classified by the boundary condition of density. In addition, we also derive a set of spatial decay rates for (ρ,u) which allows us to prove the time-asymptotic stability of (ρ,u) using the energy method. More specifically, we prove this under small initial perturbation on (ρ,u), provided that the density at the far-field is supposed to be strictly positive but suitably small, in other words, the far-field state of the fluid is not vacuum but suitably rarefied. The main difficulty for the proof is the boundary terms that appears in the a-priori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.