Smooth solution of the compressible Navier-Stokes equations in an unbounded domain with inflow boundary condition
The authors consider the steady state flow of a barotropic compressible fluid in the 2D strip \(\Omega=(0,1)\times (-\infty,+\infty)\). The velocity \(v\), the pressure \(p\) and the density \(\rho\) satisfy the system \[ -\mu\Delta v-\nu\nabla\text{div}v+\rho(v\cdot\nabla)v+ \nabla p=0, \qquad \text{div}(\rho v)=0 \quad \text{in } \Omega, \] \[ v=v_0\quad\text{on }\{x=0,\;x=1\}, \qquad p(0,y)=p_0(y),\quad y\in \mathbb{R}. \] It is assumed that \(v_0\geq c_0= \text{const}>0\), \(\rho=\rho (p)\). The main result is the Theorem: Let \(k\geq 0\) be an integer. If \(\| \nabla v_0\| _{H^{k+2}}+ \| \nabla p_0\| _{H^{k+2}}\) is sufficiently small then the problem has a unique solution \((v,p)\in H^{k+3}\times H^{k+2}\) in the ball \[ \| v- v_0\|_{H^{k+3}}+\| p- p_0\|_{H^{k+2}}\leq c, \] where the constant \(c\) depends on the data of the problem. The proof of the solvability is based on Schauder's fixed point theory and on the analysis of a linearized problem.
scientific article; zbMATH DE number 1218586
scientific article