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

TL;DR

This paper establishes existence, uniqueness, and instant smoothing of solutions for the generalized SQG equations in critical Sobolev spaces, revealing solutions become analytic immediately, even with limited initial regularity.

Contribution

It provides the first results on well-posedness and smoothing effects for a quasilinear parabolic equation with higher-order coefficients in critical Sobolev spaces.

Findings

Solutions exist and are unique in critical Sobolev spaces.

Solutions immediately become Gevrey class (analytic).

The approach preserves the commutator structure despite low regularity.

Abstract

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.