Local Existence and Blow-Up For SQG Patches

Local Existence and Blow-Up For SQG Patches

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Neel Patel , University of Michigan
Fine Hall 322

The $\alpha$-patch problem for $0\leq\alpha\leq1$ studies the evolution of patch solutions to a family of transport equations, where $\alpha =0$ and $\alpha =1$ correspond to the Euler and the surface quasi-geostrophic equations respectively. Previously, finite time singularities have been demonstrated on the half-plane setting for the family of functions $0<\alpha<1/12$ by first establishing local well-posedness of the system for that range and then constructing a blow-up solution. In this talk, we will discuss new results extending local well-posedness of that half-plane system to $0<\alpha<1/3$ by a new cancellation in the patch contour equation. Furthermore, this new cancellation can be used to prove a singularity criterion of lower regularity than the numerical results for SQG patches and it can also be used to prove lower regularity local existence results for the extended family of equations $0< \alpha< 2$.