Vanishing viscosity in the barotropic \(\beta\)-plane
type: Item
instance of: scholarly article
title: Vanishing viscosity in the barotropic \(\beta\)-plane
zbMATH Open document ID: 0906.35072
author: Sanjeeva Balasuriya
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-09T00:00:00Z
review text: The initial boundary value problem associated with the inviscid barotropic potential vorticity equation \[ \Delta \psi^0_t+ \{\psi^0, \Delta\psi^0+ \beta y\}= 0\quad \text{in } \Omega, \tag{1} \] \[ \psi^0|_{\partial \Omega}=0, \quad\psi^0 |_{t=0}= \varphi(x,y). \] and its viscous analogue \[ \Delta\psi_t +\{\psi,\Delta \psi+\beta y\}= \varepsilon \Delta^2 \psi \quad \text{in } \Omega, \tag{2} \] \[ \psi |_{\partial \Omega} =\Delta \psi |_{\partial \Omega}=0, \quad \psi|_ {t=0} =\varphi (x,y). \] are studied, where the following notations are used: \(\{f,g \}= (\partial f/ \partial x)(\partial g/ \partial y)-(\partial f/ \partial y) (\partial g/ \partial x)\), \(\Delta\) is the Laplacian in the \((x,y)\) variables, \(\beta\) is a positive constant and \(\varepsilon\) is a positive parameter. It is shown that the velocity field \(u(x,y,t; \varepsilon)\) corresponding to \(\psi (x,y,t; \varepsilon)\) which solves (2) converges to the velocity \(u^0 (x,y,t)\) corresponding to the solution \(\psi^0 (x,y,t)\) of (1) as \(\varepsilon\) goes to zero in \(C^1 (\Omega)\) for finite times. Under additional assumptions on the smoothness of \(\psi^0\) the same result in \(C^3 (\Omega)\) is established.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76B47
Mathematics Subject Classification ID: 86A05
zbMATH DE Number: 1115505
zbMATH Keywords: viscous barotropic potential vorticity equation
zbMATH Keywords: inviscid barotropic potential vorticity equation
reviewed by: A. Cichocka
MaRDI profile type: Publication
OpenAlex ID: W2014661300
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DOI: 10.1006/JMAA.1997.5596
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description: scientific article; zbMATH DE number 1115505