Session S37 - New Developments in Mathematical Fluid Dynamics

Thursday, July 15, 15:00 ~ 15:25 UTC-3

On well-posedness of the generalized SQG family in borderline spaces

Vincent Martinez

Hunter College, CUNY, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We review recent results regarding the issue of well-posedness of the family of generalized surface quasi-geostrophic equations, introduced by Chae, Constantin, et. al. 2011, in borderline Sobolev spaces. These equations are a family of active scalar equations, which include the 2D Euler equations as an endpoint and extrapolate beyond with an increasingly singular relation in the constitutive law between the scalar and its advecting velocity. In the most singular regime of velocities, the equations represent a genuinely quasilinear equation, whose coefficients are of higher-order than an arbitrarily small-order dissipative perturbation. We nevertheless show that this obstruction is only apparent due to the underlying commutator structure of the transport nonlinearity. To properly exploit this, however, a new approximation scheme by linear conservation laws is introduced that is able to accommodate this nuanced structure. We also address analogous results in the inviscid setting by considering suitably regularized velocities. In each case, thresholds for the regularizations are identified for which well-posedness can be guaranteed.

Joint work with Michael Jolly (Indiana University) and Anuj Kumar (Indiana University).

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