Unconditional stability of stationary flows of compressible heat-conducting fluids driven by large external forces
type: Item
instance of: scholarly article
title: Unconditional stability of stationary flows of compressible heat-conducting fluids driven by large external forces
zbMATH Open document ID: 0931.35123
author: Eduard Feireisl
author: Hana Petzeltová
published in: Journal of Mathematical Fluid Mechanics
publication date: 2000-02-23T00:00:00Z
review text: The authors discuss the long-time behaviour of a one-dimensional version of the Navier Stokes equation which involves five scalars \(\rho,u,p,e,q\) with \(\rho\) the fluid density, \(p\) the pressure, \(e\) the specific internal energy and \(q\) the heat flux. Set \(D=\rho (e+{1\over 2} u^2)\). The system in question is: \[ \rho_t+ (\rho u)_x=0, \quad (\rho u)_t +(\rho u^2)_x-\nu u_{xx}+ p_x=\rho f, \quad D_t+ (uD+up)_x-\nu (uu_x)_x+ q_x=\rho fu. \tag{1} \] Here \(\nu\) is the viscosity and \(f=f(x)\) an outer force. There are two constitutive laws: \[ p(\rho,e)= (\gamma-1) \rho e,\quad q(e,e_x)= -k(e)e_x,\tag{2} \] with \(k(z)\) a given function and \(\gamma>1\) the adiabatic constant. One assumes: \[ u(t,0)=u(t,1)= q(t,0)= q(t,1)=0,\;t>0.\tag{3} \] There is an equilibrium solution \(\rho_s(x)\), \(u_s\equiv 0\) whose explicit form is omitted here for simplicity. The aim of the paper is the proof of Theorem 1.1, whose simplified form is as follows: Theorem. Let \(f\) be bounded and Hölder continuous on \((0,1)\). Let \(k(z)\geq k_0>0\) for \(z>0\) and \(\lim k(z)=\infty\) as \(z\to\infty\). Let \(\rho>0\), \(e>0\), \(u\) be classical solutions of (1)--(3). Then there exist constants \(\rho_0,\rho_1\) such that \(0<\rho_0\leq \rho(t,x)\leq\rho_1\) for \(t>0\), \(x\in (0,1)\). In addition we have: (a) \(\rho\to\rho_s\) strongly in \(L^1 (0,1)\) as \(t\to\infty\), (b) \(u\to 0\) strongly in \(L^2(0,1)\) as \(t\to \infty\). The proof is based on a series of delicate estimates which are obtained by exploiting the specific structure of the system.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 76E19
zbMATH DE Number: 1339301
zbMATH Keywords: Navier-Stokes equation
zbMATH Keywords: heat conducting fluids
zbMATH Keywords: long-time behaviour
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
full work available at URL: s000210050008
OpenAlex ID: W2013467209
DOI: 10.1007/S000210050008
description: scientific article; zbMATH DE number 1339301