Singular limits in compressible fluid dynamics
type: Item
instance of: scholarly article
title: Singular limits in compressible fluid dynamics
zbMATH Open document ID: 0829.76073
DOI: 10.1007/BF00387711
published in: Archive for Rational Mechanics and Analysis
publication date: 1996-01-17T00:00:00Z
review text: The nonstationary motion of a compressible fluid is considered in \(n\)- dimensional torus \((n \geq 2)\), \(\rho_t + U \nabla \rho + \rho \nabla U = 0\), \(\rho [U_t + (U \nabla) U]- \nu \Delta U - \mu \nabla (\nabla U) + \nabla p = 0\), \(\rho (x,0) = \rho_0 (x) + 1\), \(U(x,0) = U_0 (x)\). Here \(U\) is the velocity, \(\rho\) is the density of the fluid, \(p\) is the pressure which is considered as a function of the density \(\rho\) and the Mach number \(\lambda^{-1}\), \(p (\lambda, \rho) = \lambda^2 p(\rho)\). The behaviour of \((U,p)\) is studied as \(\lambda \to \infty\), \(\nu \to \overline \nu\) \(U_0 \to W_0\), \(\rho_0 \to 0\), \(\mu\) remains bounded. The limit equations are the equations of motion of an incompressible fluid with density 1 and viscosity \(\overline \nu \geq 0\): \(W_t - \overline \nu \Delta W + (W \nabla) W + \nabla \pi = 0\), \(\nabla W = 0\), \(W(x,0) = W_0\). The convergence of \((U, \rho)\) to \((U,1)\) is established in Sobolev spaces \(H^m\).
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 754029
zbMATH Keywords: dependence on the Mach number
zbMATH Keywords: \(n\)-dimensional torus
zbMATH Keywords: limit equations
zbMATH Keywords: incompressible fluid
zbMATH Keywords: convergence
zbMATH Keywords: Sobolev spaces
author: H. Beirão da Veiga
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
cites work: The incompressible limit of compressible fluid motions in a bounded domain
cites work: On the incompressible limit of the compressible Euler equation
cites work: Q3815024
cites work: An \(L^ p\)-theory for the \(n\)-dimensional, stationary, compressible Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions
cites work: Perturbation theory and well-posedness in Hadamard's sense of hyperbolic initial-boundary value problems
cites work: Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations
cites work: The initial-boundary value problem for the non-barotropic compressible Euler equations: Structural-stability and data dependence
cites work: On the singular limit for slightly compressible fluids
cites work: Singular limits in fluid dynamics
cites work: The motion of slightly compressible fluids viewed as a motion with strong constraining force
cites work: Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations
cites work: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids
cites work: Compressible fluid flow and systems of conservation laws in several space variables
cites work: Singular limits in the data space for the equations of magneto-fluid dynamics
cites work: The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit
cites work: Singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation
cites work: Fast singular limits of hyperbolic PDEs
cites work: The incompressible limit and the initial layer of the compressible Euler equation
description: scientific article; zbMATH DE number 754029