Bilinear estimates for functions with the Dirichlet boundary condition and an application to SQG (Mathematical Analysis in Fluid and Gas Dynamics)

Abstract

We discuss the validity of the bilinear estimate of functions satisfying the Dirichlet boundary condition on the two dimensional half space. For two functions f, g, we compare two nonlinearity of the standard product fg and the gradient off and the perpendicular component of the gradient of g, and we show that the first case needs a restriction for the regularity index, while the second case does not. We also introduce an application to the surface quasi-geostrophic equation with the critical dissipation. This paper is a survey of these results.