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This paper studies the dissipative generalized surface quasi-geostrophicequations in a supercritical regime where the order of the dissipation is smallrelative to order of the velocity, and the velocities are less regular than theadvected scalar by up to one order of derivative. We also consider anon-degenerate modification of the endpoint case in which the velocity is lesssmooth than the advected scalar by slightly more than one order. The existenceand uniqueness theory of these equations in the borderline Sobolev spaces isaddressed, as well as the instantaneous smoothing effect of their correspondingsolutions. In particular, it is shown that solutions emanating from initialdata belonging to these Sobolev classes immediately enter a Gevrey class. Suchresults appear to be the first of its kind for a quasilinear parabolic equationwhose coefficients are of higher order than its linear term; they rely on anapproximation scheme which modifies the flux in such a way that preserves theunderlying commutator structure lost by having to work in the critical spacesetting, as well as delicate adaptations of well-known commutator estimates toGevrey classes.

On the existence, uniqueness, and smoothing of solutions to the generalized SQG equations in critical Sobolev spaces