A generalized 2-D Poincaré inequality

Abstract

Two 1-D Poincaré-like inequalities are proved under the mild assumption that the integrand function is zero at just one point. These results are used to derive a 2-D generalized Poincare inequality in which the integrand function is zero on a suitable arc contained in the domain (instead of the whole boundary). As an application, it is shown that a set of boundary conditions for the quasi geostrophic equation of order four are compatible with general physical constraints dictated by the dissipation of kinetic energy.

Keywords

• Inequalities
• Quasi-geostrophic equations
• Boundary conditions