Vincent R. Martinez (Hunter College & Graduate Center, CUNY)
We study dissipative perturbations of the 2D generalized surface quasi-geostrophic (gSQG) equations. This family contains the 2D Euler equations in vorticity form at one endpoint, the SQG equation at its midpoint, and the Okhitani equation at the most singular endpoint of the family. Recent work of Bourgain/Li, Elgindi/Masmoudi, Cordoba/Zoroa-Martinez, and Jeong/Kim have established ill-posedness in this family at critical regularity. This talk will consider “mild" perturbations of the gSQG equation which recover well-posedness in a setting of critical regularity, but additionally confers a mild degree of regularity instantaneously as well. These perturbations encompass an intermediate case in contradistinction with strongly dissipative perturbations, which instantaneously confer Gevrey regularity and recover well-posedness at critical regularity (Jolly/Kumar/M 2021), and inviscid regularization, which do not regularize solutions, but nevertheless recover local well-posedness at critical regularity (Chae/Wu 2010). We show that in this intermediate regime that one may recover local well-posedness at borderline Sobolev regularity, as well as a global existence theory at the 2D Euler endpoint. Moreover, we provide a general existence theory for an entire class of such perturbations that is effectively sharp in light of the recent ill-posedness results. This is joint work with A. Kumar (Florida State University).
Well-posedness of mildly dissipative perturbations of a family of active scalar equations Sponsored by the Meyer Fund