On the Dirichlet Fractional Laplacian and Applications to the SQG Equation on Bounded Domains

TL;DR

This paper explores the fractional Dirichlet Laplacian on bounded domains, develops new estimates, and applies these to analyze the long-term behavior of the SQG equation with boundaries, demonstrating the existence of a finite-dimensional global attractor.

Contribution

It introduces novel properties and estimates for the fractional Dirichlet Laplacian and applies them to establish the existence of a global attractor for the SQG equation with boundaries.

Findings

Established fractional product estimates and nonlinear Poincaré inequalities.

Proved the existence of a finite-dimensional global attractor for the forced SQG equation.

Analyzed the long-time dynamics of the SQG equation on bounded domains.

Abstract

We investigate new properties of the fractional Dirichlet Laplacian on smooth bounded domains and establish fractional product estimates and nonlinear Poincar\'e inequalities. We also use these tools to study the long-time dynamics of the surface quasi-geostrophic equation forced by some given time-independent body forces in the presence of physical boundaries and prove the existence of a finite-dimensional global attractor.