On the local existence and blow-up for generalized SQG patches

2018/11/01 by Gancedo, Francisco, Patel, Neel · 6 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.00530

Abstract

We study patch solutions of a family of transport equations given by a parameter α, 0< α<2, with the cases α=0 and α=1 corresponding to the Euler and the surface quasi-geostrophic equations respectively. In this paper, using several new cancellations, we provide the following new results. First, we prove local well-posedness for H² patches in the half-space setting for 0α/2 are time integrable. This finite-time singularity criterion holds for lower regularity than the regularity shown in numerical simulations in \citeCFMR and \citeScottDritschel for surface quasi-geostrophic patches, where the curvature of the contour blows up numerically. This is the first proof of a finite-time singularity criterion lower than or equal to the regularity in the numerics. Finally, we also improve results in \citeG and in \citeCCCGW, giving local-wellposedness for patches in H² for 0

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