Cavallini, Fabio; Crisciani, Fulvio A generalized 2-D Poincaré inequality. (English) Zbl 0978.46012 J. Inequal. Appl. 5, No. 4, 343-349 (2000). 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. Reviewer: Salvador Hernandez (Castellon) 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 Keywords:inequalities; quasi-geostrophic equations; boundary conditions; generalized Poincaré inequality; quasi-geostrophic equation; Poisson bracket; dissipation of kinetic energy PDFBibTeX XMLCite \textit{F. Cavallini} and \textit{F. Crisciani}, J. Inequal. Appl. 5, No. 4, 343--349 (2000; Zbl 0978.46012) Full Text: DOI EuDML