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arXiv · 1705.10842

Global solutions for the generalized SQG patch equation

Diego Córdoba·Javier Gómez-Serrano·Alexandru D. Ionescu
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Abstract

We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter $\alpha \in (1,2)$. The cases $\alpha = 0$ and $\alpha = 1$ correspond to 2d Euler and SQG respectively, and our choice of the parameter $\alpha$ results in a velocity more singular than in the SQG case. Our main result concerns the global stability of the half-plane patch stationary solution, under small and localized perturbations. Our theorem appears to be the first construction of stable global solutions for the gSQG-patch equations. The only other nontrivial global solutions known so far in the patch setting are the so-called V-states, which are uniformly rotating and periodic in time solutions.

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Diego Córdoba, Javier Gómez-Serrano, Alexandru D. Ionescu. 2017-05-30. Global solutions for the generalized SQG patch equation. https://arxiv.org/abs/1705.10842

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