Well-Posedness for Low Regularity Solutions to the g-SQG Equation with Regular Level Sets

TL;DR

This paper establishes local well-posedness for the generalized SQG equation with low regularity solutions that have regular level sets, and identifies conditions under which solutions cease to exist due to loss of level set regularity.

Contribution

It proves well-posedness for low regularity solutions of the g-SQG equation with regular level sets and characterizes the breakdown of solutions in terms of level set regularity loss.

Findings

Well-posedness in low regularity spaces with H"older continuity.

Solutions cease only when level sets lose H^2-regularity.

No finite-time blow-up due to level set collisions under certain conditions.

Abstract

We show that the generalized SQG equation on the plane is locally well-posed in spaces of low regularity solutions (essentially H\"older continuous with H\"older exponents depending on the equation parameter $\alpha\in(0,\frac 12)$) that have $H^2$ level sets (i.e., with $L^2$ curvatures). Moreover, for $\alpha\le\frac 16$ and initial data satisfying some additional hypotheses we show that the corresponding solutions can stop existing only when their level sets lose $H^2$-regularity, and hence not just due to level set collisions or "pile ups".