  This paper studies the dissipative generalized surface quasi-geostrophic
equations in a supercritical regime where the order of the dissipation is small
relative to order of the velocity, and the velocities are less regular than the
advected scalar by up to one order of derivative. We also consider a
non-degenerate modification of the endpoint case in which the velocity is less
smooth than the advected scalar by slightly more than one order. The existence
and uniqueness theory of these equations in the borderline Sobolev spaces is
addressed, as well as the instantaneous smoothing effect of their corresponding
solutions. In particular, it is shown that solutions emanating from initial
data belonging to these Sobolev classes immediately enter a Gevrey class. Such
results appear to be the first of its kind for a quasilinear parabolic equation
whose coefficients are of higher order than its linear term; they rely on an
approximation scheme which modifies the flux in such a way that preserves the
underlying commutator structure lost by having to work in the critical space
setting, as well as delicate adaptations of well-known commutator estimates to
Gevrey classes.
