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

By Francisco Gancedo, Neel Patel. Annals of PDE, 2018

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We study patch solutions of a family of transport equations given by a parameter $$\alpha $$ α , $$0< \alpha <2$$ 0 < α < 2 , with the cases $$\alpha =0$$ α = 0 and $$\alpha =1$$ 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 $$H^{2}$$ H 2 patches in the half-space setting for $$0<\alpha < 1/3$$ 0 < α < 1 / 3 , allowing self-intersection with the fixed boundary. Furthermore, we are able to extend the range of $$\alpha $$ α for which finite time singularities have been shown in Kiselev et al. (Commun Pure Appl Math 70(7):1253–1315, 2017) and Kiselev et al. (Ann Math 3:909–948, 2016). Second, we establish that patches remain regular for $$0<\alpha <2$$ 0 < α < 2 as long as the arc-chord condition and the regularity of order $$C^{1+\delta }$$ C 1 + δ for $$\delta >\alpha /2$$ δ > α / 2 are time integrable. This finite-time singularity criterion holds for lower regularity than the regularity shown in numerical simulations in Córdoba et al. (Proc Natl Acad Sci USA 102:5949–5952, 2005) and Scott and Dritschel (Phys Rev Lett 112:144505, 2014) 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 Gancedo (Adv Math 217(6):2569–2598, 2008) and in Chae et al. (Commun Pure Appl Math 65(8):1037–1066, 2012), giving local existence for patches in $$H^{2}$$ H 2 for $$0<\alpha < 1$$ 0 < α < 1 and in $$H^3$$ H 3 for $$1<\alpha <2$$ 1 < α < 2 .

Subfields: Applied Mathematics, Mathematical Physics

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