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A generalized 2-D Poincaré inequality. (English) Zbl 0978.46012

The authors prove a 2-D generalized Poincaré inequality in which the integrand functions is zero on a suitable arc contained in the domain. As an application it is shown that for the quasi-geostrophic equation of order four \[ \frac{\partial }{\partial t}\nabla ^{2}\psi +R\mathbf{J}[\psi ,\nabla ^{2}\psi ]+\frac{\partial \psi }{\partial x}=(\text{curl} \boldsymbol{\tau})_{z}+\epsilon \nabla ^{2}\nabla ^{2}\psi \] where \(\psi=\psi[x,y]\) is the stream function, \(\mathbf{J}\) is the Jacobian (or Poisson bracket) operator, \(\boldsymbol{\tau }\) is the wind stress, \(R\) and \(\epsilon\) are positive constants, \(t\) is time, and \(x,y,z\) are Cartesian coordinates. The set of boundary conditions \[ \begin{aligned} \frac{\partial }{\partial n}\nabla ^{2}\psi &=0\quad \text{on the coastline},\\ \nabla ^{2}\psi &=0\quad \text{on the sea boundary} \end{aligned} \] are compatible with general physical constraints dictated by the dissipation of kinetic energy.

MSC:

46E20 Hilbert spaces of continuous, differentiable or analytic functions
86A05 Hydrology, hydrography, oceanography
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47F05 General theory of partial differential operators
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