The Derivative Structure for a Quadratic Nonlinearity and Uniqueness for SQG

TL;DR

This paper proves the uniqueness of mild solutions in L^2 for the 2D surface quasi-geostrophic equation on bounded domains, introducing new methods for fractional Laplacians in nonlinear PDEs.

Contribution

It develops a novel approach to handle fractional Laplacians and second-order derivatives in nonlinear terms for the SQG equation on bounded domains.

Findings

Uniqueness of mild solutions in L^2 established

New method for fractional Laplacians in nonlinear PDEs

Approach to second-order derivatives for fractional Dirichlet Laplacian

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 $\mathbb R^2$, 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 derivativesfor the nonlinear term involving fractional derivatives of the Dirichlet Laplacian.