Global existence for some transport equations with nonlocal velocity

TL;DR

This paper establishes the global existence of weak solutions for various one- and multi-dimensional transport equations with nonlocal velocity fields, including models related to geophysical flows and porous media.

Contribution

It proves global weak solution existence for several nonlocal transport equations with rough initial data, extending previous results to more complex models.

Findings

Global weak solutions exist for 1D surface quasi-geostrophic equation

Global weak solutions exist for incompressible porous media equation

Results apply to both dissipative and non-dissipative models

Abstract

In this paper, we study transport equations with nonlocal velocity fields with rough initial data. We address the global existence of weak solutions of an one dimensional model of the surface quasi-geostrophic equation and the incompressible porous media equation, and one dimensional and $n$ dimensional models of the dissipative quasi-geostrophic equations and the dissipative incompressible porous media equation.