# Summer Program in Partial Differential Equations 2022 August 15-19, 2022

This summer program provides 5 days of concentrated study of topics in Analysis of Partial Differential Equations at the graduate level.

The school will offer 4 courses on topics at the forefront of current research in Partial Differential Equations. Lectures will be followed by daily discussion sessions. During discussion sessions students will be divided in small groups and work on sets of problems related to the courses topics.

For organizational purposes the school will be able to accomodate only a limited number of participants.

Registration is open at this link.

Organizers:
Maria Gualdani and Stefania Patrizi

Irene Gamba (UT Austin), Alexander Kiselev (Duke Univ.), Luis Silvestre (Univ. of Chicago), Hung Tran (Univ. Madison, Wisconsin).

Titles and Abstracts:

Title:
Two stories in PDEs (Alexander Kiselev)
Abstract: I will discuss two recent developments in fluid mechanics type PDE. In the first part I will talk about patch solutions to 2D Euler and modified SQG equations. For these solutions, the problem reduces to interface evolution, and I will overview the available results on regularity and formation of singular structures. The focus will be on the recent ill-posedness construction of a patch with continuous initial curvature that loses continuity immediately but regains it at every integer time (without being periodic), based on a joint work with Xiaoyutao Luo (Duke). In the second part I will describe a fluid-type PDE that has been conjectured by Stefan Steinerberger to describe evolution of polynomial roots under differentiation. The PDE is very similar to but somewhat more complex than Euler alignment equation used to model collective motion in biology. I will prove that for a class of periodic polynomials of large degree, this PDE indeed closely tracks the dynamics of zeroes under differentiation. This part is based on joint work with Changhui Tan (University of South Carolina).

Title: Periodic homogenization of Hamilton-Jacobi equations: basic theory and some recent progresses (Hung Tran)

Abstract: I will first give an introduction to front propagations, Hamilton-Jacobi equations, and homogenization theory. I will then describe the basic theory and some recent progress in periodic homogenization of Hamilton-Jacobi equations. We show that the optimal rate of convergence is $O(\varepsilon)$ in the convex setting. We then give a minimalistic explanation that the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts in two dimensions (if time permits).

Background material: A. Majda and A. Bertozzi, Vorticity and Incompressible Flow (Chapter 8)A. Bertozzi and Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophicequation, Inventiones Math. 167 (2007) 445-453More recent results include: A. Kiselev and X. Luo, Illposedness of C^2 vortex patches, arXiv:2204.06416A. Kiselev and X. Luo, On nonexistence of splash singularities for the $\alpha$-SQG patches, arXiv:2111.13794v2A. Kiselev and C. Tan, The flow of polynomial roots under differentiation, arXiv:2012.09080