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

2024/09/08 by Elie Abdo, Abdo, Elie, Quyuan Lin +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2409.05209

openalex publication_date 2024/09/08 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28

Abstract

We investigate new properties of the fractional Dirichlet Laplacian on smooth bounded domains and establish fractional product estimates and nonlinear Poincaré 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.

Related

• Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications
• The Derivative Structure for a Quadratic Nonlinearity and Uniqueness for SQG
• Large solutions for fractional Laplacian operators
• Long time behavior and critical limit of subcritical SQG equations in scale-invariant Sobolev spaces
• A determining form for the subcritical surface quasi-geostrophic equation