Smooth solution of the compressible Navier-Stokes equations in an unbounded domain with inflow boundary condition
type: Item
instance of: scholarly article
title: Smooth solution of the compressible Navier-Stokes equations in an unbounded domain with inflow boundary condition
zbMATH Open document ID: 0910.35093
author: Jae Ryong Kweon
author: R. Bruce Kellogg
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-03-14T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76N10
zbMATH DE Number: 1218586
zbMATH Keywords: unbounded two-dimensional domain
zbMATH Keywords: steady state flow
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5872
OpenAlex ID: W2020397521
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DOI: 10.1006/JMAA.1997.5872
description: scientific article; zbMATH DE number 1218586