# 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.

The registration is now closed. For any inquiry please contact the organizers via email at gualdani@math.utexas.edu and spatrizi@math.utexas.edu.

Organizers: Maria Gualdani and Stefania Patrizi

Support: NSF CAREER 2019335, NSF RTG 1840314

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

Program:

Monday August 15 *** CHANGE *** Mary E. Gearing Hall (GEA) 100

9 am registration
9:30-10:30 A. Kiselev
10:30-11:00 Break and Discussion
11:00-12:00 A. Kiselev
12:00 Lunch
2:00-3:00 A. Kiselev
3:00 Discussion

Tuesday August 16 *** CHANGE *** Physics Mathematics Astronomy Building PMA 5.104
9:30-10:30 L. Silvestre
10:30-11:00 Break and Discussion
11:00-12:00 L. Silvestre
12:00 Lunch
1:30-2:30 A. Kiselev
2:30-3:00 Break and Discussion
3:00-4:00 A. Kiselev

Wednesday August 17 *** CHANGE *** Physics Mathematics Astronomy Building PMA 5.104
9:00-10:00 L. Silvestre
10:15-11:15 L. Silvestre
11:30-12:30 H. Tran
12:00 Lunch
2:00-3:00 I. Gamba
3:00-3:30 Break and Discussion
3:30-4:30 I. Gamba

Thursday August 18*** CHANGE *** Physics Mathematics Astronomy Building PMA 5.104
9:30-10:30 L. Silvestre
10:30-11:00 Break and Discussion
11:00-12:00 H. Tran
12:00 Lunch
1:30-2:30 H. Tran
2:30-3:00 Break and Discussion
3:00-4:00 I. Gamba

Friday August 19 *** CHANGE *** Physics Mathematics Astronomy Building PMA 5.104
9:30-10:30 H. Tran
10:30-11:00 Break and Discussion
11:00-12:00 H. Tran
12:00 Lunch
1:30-2:30 I. Gamba
2:30-3:00 Break and Discussion
3:00-4:00 I. Gamba

Titles and Abstracts:

Title: Problems in Kinetic Collisional Theory for Classical and Complex Interacting Particle Systems (Irene Gamba)

Abstract: We will discuss several problems on collisional kinetics characterized by non-local multilinear operators acting on probability densities.  The lecture will focus on several aspects of the theory of existence and uniqueness of solutions by means of studying ODE flows in the Banach spaces of observables or polynomial moments, and their associated $L^p_k$-theory in some of their cases and general Sobolev spaces for regularity and convergence to equilibrium. The methods relies in the calculation of statistical moment estimates that generate Ordinary Differential Inequalities enabling the propagation and generation of  $L^p_k$, $p\in [1,\infty]$, depending on the structure of the interactions. Such type of results extends to the generation and propagation of exponential moments as well as Sobolev norm, enabling regularity. The main focus will be in Boltzmann type equations for integrable angular transition functions (or cross sections) for the classical gas dynamic model for elastic or inelastic interactions, as well as the polyatomic model and a system of multispecies binary Boltzmann flows for disparate masses, including a discussion of the derivation of grazing collisions limits where the Landau equation arises. In addition, we will focus on how the ODE in Banach space techniques apply to other mean field models of quantum type modeling BEC-cold gas interactions for radial solutions, as well as some models of weak turbulence in deep ocean waves. If time permits, we will discuss some numerical techniques for conservative schemes and error estimates.

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:
Some mathematical problems in kinetic equations (Luis Silvestre)

Abstract: We will discuss some problems involving the Boltzmann and Landau equations. We apply some tools from nonlocal and hypoelliptic theory to study the regularity of solutions.

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).

1. A. Majda and A. Bertozzi, Vorticity and Incompressible Flow (Chapter 8)2. A. Bertozzi and Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-283. A. 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: 1. A. Kiselev and X. Luo, Illposedness of C^2 vortex patches, arXiv:2204.064162. A. Kiselev and X. Luo, On nonexistence of splash singularities for the $\alpha$-SQG patches, arXiv:2111.13794v23. A. Kiselev and C. Tan, The flow of polynomial roots under differentiation, arXiv:2012.09080