Drift diffusion equations with fractional diffusion on compact Lie groups

TL;DR

This paper studies the well-posedness of diffusion equations involving fractional and subelliptic operators on compact Lie groups, with applications to fractional sub-Laplacians and models like quasi-geostrophic equations.

Contribution

It extends the analysis of diffusion equations to fractional and subelliptic operators on compact Lie groups, including new results on well-posedness and applications.

Findings

Established well-posedness for fractional subelliptic diffusion equations

Analyzed examples on SU(2) with fractional diffusion

Connected fractional diffusion to quasi-geostrophic models

Abstract

In this work we investigate the well-posedness for difussion equations associated to subelliptic pseudo-differential operators on compact Lie groups. The diffusion by strongly elliptic operators is considered as a special case and in particular the fractional diffusion with respect to the Laplacian. The general case is studied within the H\"ormander classes associated to a sub-Riemannian structure on the group (encoded by a H\"ormander system of vector fields). Applications to diffusion equations for fractional sub-Laplacians, fractional powers of more general subelliptic operators, and the corresponding quasi-geostrophic model with drift $D$ are investigated. Examples on SU(2) for diffusion problems with fractional diffusion are analysed.