Global solutions of the Navier-Stokes equations in thin three-dimensional domains
type: Item
instance of: scholarly article
title: Global solutions of the Navier-Stokes equations in thin three-dimensional domains
zbMATH Open document ID: 0933.35153
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-03-15T00:00:00Z
review text: The Navier-Stokes equations \[ \frac{\partial v}{\partial t}-\nu\Delta v+v\cdot\nabla v +\nabla p=f,\quad \nabla\cdot v=0 \tag{1} \] are considered in 3D-domain \(\Omega_\varepsilon=(0,l_1)\times (0,l_2) \times (0,\varepsilon)\). The equations (1) are associated with the periodic boundary conditions \[ v(t,0,x_2,x_3)=v(t,l_1,x_2,x_3),\quad v(t,x_1,0,x_3)=v(t,x_1,l_2,x_3),\quad v(t,x_1,x_2,0)=v(t,l_1,x_2,\varepsilon) \tag{2} \] and initial condition \[ v(0,x)=v_0(x).\tag{3} \] The additional average condition \[ \int_{\Omega_\varepsilon}v(t,x) dx=0 \tag{4} \] is required. The author describes the set of data \((v_0,f)\) for which the problem (1), (2), (3), (4) has a global regular solution. The global existence conditions intend in particular that \(\varepsilon\) is small enough.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 1295272
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: global solution
zbMATH Keywords: periodicity
author: Zhi-Min Chen
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6329
OpenAlex ID: W2049466382
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DOI: 10.1006/JMAA.1999.6329
description: scientific article; zbMATH DE number 1295272