Finite-time singularity formation for smooth inviscid SQG solutions

Determine whether smooth initial data for the inviscid surface quasi-geostrophic equation on the two-dimensional torus can develop a singularity in finite time.

Background

The paper studies the inviscid surface quasi-geostrophic equation, whose active scalar is transported by a velocity obtained through a singular integral operator. Although the paper focuses on nonuniqueness and flexibility for low-regularity momentum weak solutions, it explicitly identifies the regularity question for smooth solutions as unresolved.

Resolving this problem would establish whether smooth SQG dynamics remain globally regular or whether finite-time singularities can form, a central issue in the mathematical theory of active scalar equations.