Long time behavior for one-dimensional motion of a general barotropic viscous fluid
type: Item
instance of: scholarly article
title: Long time behavior for one-dimensional motion of a general barotropic viscous fluid
zbMATH Open document ID: 0704.76020
DOI: 10.1007/BF01053460
author: H. Beirão da Veiga
published in: Archive for Rational Mechanics and Analysis
publication date: 1989-01-01T00:00:00Z
review text: We study one-dimensional motion of a barotropic compressible viscous fluid in a bounded region with impermeable boundary, for a general pressure term. Without loss of generality we assume that the bounded region is the interval (0,1). In material Lagrangian coordinates, and after a normalization, the equations of motion become \[ (1)\quad v_ t- u_ q=0,\quad u_ t=\mu (v^{-1}u_ q)_ q-p(v)_ q+f(t,\int^{q}_{0}v(t,\xi)d\xi), \] where \(q\in (0,1)\), and \(t\geq 0\). Here the material Lagrangian coordinate q of a particle of fluid is the mass of the portion of fluid that occupies the region between the origin and the given particle. After the normalization, the total mass of the fluid is equal to 1. Hence the coordinate q runs over the interval (0,1). The external force f(t,x) is given as a known function of the Eulerian coordinates (t,x), \(x\in (0,1)\), where \(x=\int^{q}_{0}v(t,\xi)d\xi.\) In the above problem, the boundary conditions are \[ (2)\quad u(t,0)=u(t,1)=0\quad \forall t\geq 0, \] and the initial conditions are \[ (3)\quad u(0,q)=u_ 0(q),\quad v(0,q)=v_ 0(q),\quad q\in (0,1). \] Without loss of generality, we suppose that \[ (4)\quad \int^{1}_{0}v_ 0(q)dq=1,\quad m^{-1}\leq v_ 0(q)\leq m\quad \forall q\in (0,1). \] We assume that a real function \(p\in C^ 1(0,\infty)\) is given, such that \(p(1)=1\) and that \[ (5)\quad p'(s)<0\quad \forall s\in (0,+\infty). \] We investigate, in particular, the existence of global solutions satisfying the estimate (6) \(N_ 0^{-1}\leq v(t,q)\leq N_ 0\forall (t,q)\in Q_{\infty}\) \((N_ 0\geq 1)\) in the presence of external forces which do not become small for large values of t. I proved that there is a positive threshold \(r_ 0\), and real positive decreasing functions \(\rho\) (r), R(r), defined on \((0,r_ 0)\), satisfying \(\lim_{r\to 0}\rho (r)=\lim_{r\to 0}R(r)=\infty\), and such that the following condition holds: If \(| f(t,x)| \leq r\leq r_ 0\), and if \(\| u_ 0\| <\rho (r)\), \(\| (\log v_ 0)_ q\| <\rho (r)\), then th…
Mathematics Subject Classification ID: 76D99
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 4156085
zbMATH Keywords: one-dimensional motion
zbMATH Keywords: barotropic compressible viscous fluid
zbMATH Keywords: impermeable boundary
zbMATH Keywords: Lagrangian coordinates
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01053460
OpenAlex ID: W2134332051
description: scientific article; zbMATH DE number 4156085