Nonlinear Partial Differential Equations

Date: November 9, 2021

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Hao Jia, University of Minnesota

Title: Linear vortex symmetrization: the spectral density function approach and Gevrey regularity

Abstract: The two-dimensional incompressible Euler equation is globally well-posed but the long time behavior is very difficult to understand due to the lack of global relaxation mechanism. Numerical simulations and physical experiments show that coherent vortices often become a dominant feature in two-dimensional fluid dynamics for a long time. The mathematical analysis of vortices, especially in connection to the so-called `vortex symmetrization` problem, has attracted a lot of attention in recent years. In this talk, after a quick review of recent developments in the study of nonlinear asymptotic stability of shear flows and the symmetrization problem for (the special case of) point vortices, we turn to the general vortex symmetrization problem and report a recent result with A. Ionescu for the linearized flow. The linearized problem has been analyzed before by Bedrossian-Coti Zelati-Vicol who obtained control on the profile of the vorticity field in Sobolev spaces with limited regularity. Our main new discovery is that in the vortex problem, unlike the shear flow case, it is no longer possible to obtain smooth control uniformly in time on a single modulated profile for the vorticity field. Rather, there are two such profiles. To address this issue (for future nonlinear applications), we propose instead to control a new object, the so-called `spectral density function`, which is naturally associated with the linearized flow and can be bounded, for the linearized flow at least, in the same Gevrey space as the initial data.