Communications in Mathematical Sciences

Volume 16 (2018)

Number 2

Far-field regularity for the supercritical quasi-geostrophic equation

Pages: 393 – 410



Igor Kukavica (Department of Mathematics, University of Southern California at Los Angeles)

Fei Wang (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)


We address the far field regularity for solutions of the surface quasi-geostrophic equation\[\begin{array}& \theta_t + u \cdot \nabla \theta + \Lambda^{2 \alpha} \theta = 0 \\u=\mathcal{R}^{\perp} \theta = (- \mathcal{R}_2 \theta , \mathcal{R}_1 \theta)\end{array}\]in the supercritical range $0 \lt \alpha \lt 1/2$ with $\alpha$ sufficiently close to $1/2$. We prove that if the datum is sufficiently regular, then the set of space-time singularities is compact in $\mathbb{R}^2 \times \mathbb{R}$. The proof depends on a new spatial decay result on solutions in the supercritical range.


quasi-geostrophic equation, eventual regularity, supercritical, weighted decay

2010 Mathematics Subject Classification

35Q35, 35R11, 76D03

The authors were supported in part by the NSF grant DMS-1615239.

Received 10 June 2017

Accepted 22 November 2017

Published 14 May 2018