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.
Organizers: Maria Gualdani and Stefania Patrizi
Lecturers (advanced courses): Irene Gamba (UT Austin), Alexander Kiselev (Duke Univ.), Luis Silvestre (Univ. of Chicago), Hung Tran (Univ. Madison, Wisconsin).
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).
Bibliography and additional info:
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), 19-28 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: A. Kiselev and X. Luo, Illposedness of C^2 vortex patches, arXiv:2204.06416 A. Kiselev and X. Luo, On nonexistence of splash singularities for the $\alpha$-SQG patches, arXiv:2111.13794v2 A. Kiselev and C. Tan, The flow of polynomial roots under differentiation, arXiv:2012.09080