Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor
type: Item
instance of: scholarly article
title: Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor
zbMATH Open document ID: 1064.35140
published in: Journal of Evolution Equations
publication date: 2004-09-27T00:00:00Z
review text: The author investigates the compressible Navier-Stokes equations for isothermal fluids \[ \begin{gathered} (\rho u)_t+ \text{div}(\rho u\otimes u)+ \nabla\rho- \text{div\,}P(Du)=\rho f,\quad x\in\Omega,\\ \rho_t+ \text{div}(\rho u)= 0,\quad u(x,t)= 0\quad\text{for }x\in \Omega.\end{gathered}\tag{1} \] Here \(Du\) is the symmetric part of the velocity gradient \(\nabla u\), while \(P\) represents the nonlinear dependence of the stress tensor on the velocity field. Moreover, one is given initial data \(\rho(x,0)= \rho_0(x)\geq 0\), \((\rho u)(x, 0)= q_0(x)\), \(x\in \Omega\). The function spaces, in which weak solutions of (1) are found, are Orlicz spaces \(L_\psi(\Omega)\) defined in terms of a Young function \(\psi\) and its complementary function \(\overline\psi\), and provided with a norm \(\|\;\|_\psi\). The notion of Orlicz space is then extended to vector fields, giving rise to a space \(E_\psi\), with respect to equation (1). Based on this definitions and a series of basic assumptions (A) involving the stress tensor \(P\) among others, a Definition 4.4 of weak solutions to (1) is given. The basic aim of the paper is to prove Theorem 1.1 which states the following. Let \(\rho\), \(u\) be a weak solution of Definition 4.4, let \(f=\nabla g\), \(g\in W^{2\infty}(\Omega)\). Under assumption (A) there is a unique \(\rho_\infty\in L_{\phi_1}(\Omega)\) such that \[ \lim\|\rho(t)- \rho_\infty(t)\|_{\phi_1}= 0,\quad t\to\infty, \] where \(\nabla\rho_\infty= \rho_\infty f\) a.e. in \(\Omega\), and \(\int_\Omega\rho_\infty dx= \int_\Omega \rho_0 dx\), \(\rho_\infty\geq 0\); moreover for any \(a>0\) we have \[ \lim\int^{t+a}_{t-a}\| Du(s)\|^2_{\psi_2} \,ds= 0,\quad \lim\int^{t+a}_{t-a} \int_\Omega\rho|u|^2 \,dx\,ds= 0,\quad t\to\infty. \] Here, \(\phi_1\), \(\psi_2\) are certain Young functions. The proof is based on a large number of lemmas, some of them are really interesting.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2103745
zbMATH Keywords: compressible Navier-Stokes equations for isothermal fluids
zbMATH Keywords: stress tensor
zbMATH Keywords: Young function
zbMATH Keywords: Orlicz space
zbMATH Keywords: weak solution
author: Rostislav Vodak
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
full work available at URL: s00028-003-0139-2
OpenAlex ID: W1568948558
DOI: 10.1007/S00028-003-0139-2
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description: scientific article; zbMATH DE number 2103745