Non-unique weak solutions of forced SQG

TL;DR

This paper constructs non-unique weak solutions for the forced surface quasi-geostrophic (SQG) equation using a convex integration method, extending previous results to include external forcing and solutions with less regularity.

Contribution

It introduces a convex integration scheme adapted to the sum-difference system to produce non-unique solutions for the forced SQG equation, broadening the understanding of solution behavior.

Findings

Existence of non-unique weak solutions for forced SQG.

Construction of solutions with regularity $C_t^0C_x^{0-}$.

Extension of non-uniqueness results to forced cases.

Abstract

We construct non-unique weak solutions $\theta\in C_t^0C_x^{0-}$ for forced surface quasi-geostrophic (SQG) equation. This is achieved through a convex integration scheme adapted to the sum-difference system of two distinct solutions. Without external forcing, non-unique weak solutions $\theta$ in space $C_t^0C_x^{\alpha}$ with $\alpha<-\frac15$ were constructed by Buckmaster, Shkoller and Vicol, and Isett and Ma.