Far-field regularity for the supercritical quasi-geostrophic equation

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.

Keywords

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

2010 Mathematics Subject Classification

35Q35, 35R11, 76D03

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

Received 10 June 2017

Accepted 22 November 2017

Published 14 May 2018