Seminar Analysis and Mathematical Physics: Silja Haffter (EPFL)

Are weak solutions to the supercritical surface quasigeostrophic equation smooth a.e.?

The surface quasigeostrophic equation (SGQ) is a 2d physical model equation which emerges in meteorology. It has attracted the attention of the mathematical community since it shares many of the essential difficulties of 3d fluid dynamics: in the supercritical regime for instance, where dissipation is modelled by a fractional Laplacian of order less than 1/2, it is not known whether or not smooth solutions blow-up in finite time. On the other hand, the scheme of Leray still produces global-in-time weak solutions from any L^2-initial datum, but their regularity is poorly understood. In this talk, I will propose a nonempty notion of "suitable weak solution" for the supercritical SQG equation and prove that those solutions are smooth outside a compact set of quantifiable Hausdorff dimension; in particular they are smooth almost everywhere. I will also give a conjecture on what we believe to be an optimal dimension estimate.  This is a joint work with Maria Colombo (EPFL).

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