On a \(\nu\)-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-type boundary condition
type: Item
instance of: scholarly article
title: On a \(\nu\)-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-type boundary condition
zbMATH Open document ID: 1200.35209
author: Hamid Bellout
author: Jiří Neustupa
author: Patrick Penel
published in: Discrete and Continuous Dynamical Systems
publication date: 2010-06-23T00:00:00Z
review text: The viscosity limit of the Navier-Stokes equations is studied. Let \((u,p)\) be a solution to the problem \[ \begin{aligned} &\frac{\partial u}{\partial t}+(u\cdot\nabla)u-\nu\Delta u+\nabla p=f, \quad \text{div}\,u=0,\quad x\in \Omega,\;0<t<T\\ &u(x,0)=u_0(x)\quad x\in \Omega,\\ &u\cdot n=0,\quad \text{curl}\,u \times n=\text{curl}\,(v+\nu w) \times n, \quad x\in \partial\Omega,\;0<t<T, \end{aligned}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb R^3\) with a smooth boundary. Let \((v,q)\) be a solution to the initial-boundary value problem for Euler equations \[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v+\nabla q=f, \quad \text{div}\,v=0,\quad x\in \Omega,\;0<t<T\\ &v(x,0)=u_0(x),\quad x\in \Omega,\\ &v\cdot n=0,\quad x\in \partial\Omega,\;0<t<T. \end{aligned}\tag{2} \] Let \((w,r)\) be a solution to the linear problem \[ \begin{aligned} &\frac{\partial w}{\partial t}+(v\cdot\nabla)w+(w\cdot\nabla)v+\nabla r=\Delta v, \quad \text{div}\,w=0,\quad x\in \Omega,\;0<t<T\\ &w(x,0)=0,\quad x\in \Omega,\\ &w\cdot n=0,\quad x\in \partial\Omega,\;0<t<T. \end{aligned}\tag{3} \] The authors construct a family of strong solutions \(\{u^\nu\}\) of the problem (1) on the time interval \((0,T)\), depending continuously on the viscosity coefficient \(\nu\) for \(0\leq \nu<\nu^*\). \(\{u^\nu\}\) tends to the solution \(v\) of the problem (2) for \(\nu\rightarrow 0+\).
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35Q31
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 5725438
zbMATH Keywords: Euler equations
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: zero viscosity limit
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: dcds.2010.27.1353
OpenAlex ID: W2041537056
DOI: 10.3934/DCDS.2010.27.1353
description: scientific article; zbMATH DE number 5725438