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

1:30-2:30

3:00-4:00

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

3:00-4:00

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

3:00-4:00

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:** Some mathematical problems in kinetic equations

**
Title: **Periodic homogenization of Hamilton-Jacobi equations: basic theory and some recent progresses

**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).

**Bibliography and additional info:**

**Kiselev:**

Background material:

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), 19-28 3. A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Inventiones Math. 167 (2007) 445-453 More recent results include: 1. A. Kiselev and X. Luo, Illposedness of C^2 vortex patches, arXiv:2204.06416 2. A. Kiselev and X. Luo, On nonexistence of splash singularities for the $\alpha$-SQG patches, arXiv:2111.13794v2 3. A. Kiselev and C. Tan, The flow of polynomial roots under differentiation, arXiv:2012.09080

**Silvestre:**

See the survey papers: