Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications

TL;DR

This paper establishes new bounds and estimates for the Dirichlet fractional Laplacian in bounded domains, with applications to drift-diffusion equations and surface quasi-geostrophic equations, advancing understanding of nonlocal dissipation effects.

Contribution

It provides nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian, a novel contribution to the analysis of nonlocal operators in bounded domains.

Findings

Derived nonlinear lower bounds for the fractional Laplacian

Established commutator estimates for the operator

Proved global existence of weak solutions for critical SQG equations

Abstract

We prove nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. The applications include bounds for linear drift-diffusion equations with nonlocal dissipation and global existence of weak solutions of critical surface quasi-geostrophic equations.