Convex integration for a class of active scalar equations

TL;DR

This paper demonstrates that a broad class of active scalar equations, such as porous media and magnetostrophic turbulence models, have non-unique weak solutions, using convex integration techniques adapted from fluid dynamics.

Contribution

It extends convex integration methods to active scalar equations, establishing non-uniqueness results for a wide class of models.

Findings

Existence of non-unique weak solutions for active scalar equations.

Application of convex integration to models beyond fluid dynamics.

Broad class includes porous media and magnetostrophic turbulence models.

Abstract

We show that a general class of active scalar equations, including porous media and certain magnetostrophic turbulence models, admit non-unique weak solutions in the class of bounded functions. The proof is based upon the method of convex integration recently implemented for equations of fluid dynamics in [2,3].