Finite-time singularity of the inviscid surface quasi-geostrophic equation
In plain words
The surface quasi-geostrophic equation describes temperature carried by a flow it drives itself, and it behaves like a two-dimensional version of the 3D Euler equations. Whether a smooth temperature pattern can form an infinitely sharp front in finite time is unknown.
Precise statement
Inviscid SQG: d theta/dt + u . grad theta = 0, $u = (-R_2 \theta, R_1 \theta)$ with $R_j$ the Riesz transforms, on $R^2$ or $T^2$, smooth (e.g. $C_c^\infty$) initial $\theta$, no forcing. Prove global regularity, or give smooth data with integral_0^T ||grad theta(t)||_inf dt = infinity at finite T. Answer: yes or no with proof.
What would settle it
A proof of global regularity or a verified finite-time singularity from smooth finite-energy data without forcing.
Status in the literature
Unverified note
Frontal blowup was proposed by Constantin, Majda and Tabak (1994); blowup from smooth compactly supported data with a suitable time-dependent force, inside the Sobolev well-posedness regime, was proven for generalized SQG with $\gamma \text{ in } (0,1)$ (Cordoba, Dominguez, Lucas-Manchon and Martinez-Zoroa, arXiv 2608.17192, 2026), not for unforced SQG.