A generalized 2-D Poincaré inequality (Q1585223)

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.