Vincent Martinez, CUNY

Abstract: This talk discusses a family of active scalar transport equations characterized by increasingly singular constitutive laws. This family includes the 2D Euler and surface quasi-geostrophic (SQG) equations as members, and extrapolates beyond them. Although local well-posedness of the initial value problem in sufficiently regular settings are classical results for both the Euler and SQG equations, ill-posedness at critical regularity has only recently been established. For this talk, we consider various regularizations of this family in its most singular range that recover well-posedness results at the threshold regularity level, in spite of the apparent strongly quasilinear structure of the equations in this regime. This is joint work with M.S. Jolly and A. Kumar.