Vanishing viscosity in the barotropic \(\beta\)-plane (Q1378663)

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.