The derivative structure for a quadratic nonlinearity and uniqueness for SQG

Abstract

We study the two-dimensional surface quasi-geostrophic equation on a bounded domain with a smooth boundary. Motivated by the three-dimensional incompressible Navier-Stokes equations and previous results in the entire space $R^2$, we demonstrate that the uniqueness of the mild solution holds in $L^2$. For the proof, we provide a method for handling fractional Laplacians in nonlinear problems, and develop an approach to derive second-order derivatives for the nonlinear term involving fractional derivatives of the Dirichlet Laplacian.