Short Courses given by:

- Eric Baer: Symmetrization techniques with applications to isoperimetric problems and PDEs
- Donghyun Lee: Basic mathematical models of fluid dynamics
- Jessica Lin: An introduction to weak solutions for Nonlinear PDEs
- Hung Tran: Some fixed point results and applications to ODEs, PDEs

**About this Program:** We will be organizing a summer school in Summer 2017 partially supported by the NSF RTG in Analysis and Applications for current undergraduate students at the University of Wisconsin--Madison. Students will be expected to be available from May 15th-June 9th.

**Topics and Structure of the Program:** Our goal is to present lectures on special topics in (nonlinear) PDE that are typically not a part of the standard undergraduate PDE curriculum.

These lectures will be supplemented by daily discussion sections/problem sessions, and participants will be expected to contribute towards developing lecture notes and giving presentations.

All lectures and problem sessions will take place in Van Vleck B235. An up-to-date schedule is here:

**Participant Support:** We will be able to provide financial support for participants who are (1) US Citizens or Permanent Residents and (2) Rising Seniors or younger.

ANYONE who wishes to participate in the summer school (with or without funding) needs to fill out the following application available here.

Presentation Schedule (Friday, 6/9 in 901 VV):

10:00 AM- **Adam Christopherson**

Title: Nonlinear Schrodinger Equation (NLS)

Abstract: In this presentation, we discuss the Nonlinear Schrodinger Equation (NLS), starting with a derivation of the linear solution using the Fourier transform. We also prove energy conservation, and we use this result to prove that the solution undergoes finite time blowup under certain assumptions.

10:25 AM- **Aoran Wu**

Title: The Sobolev inequality and Some Applications

Abstract: Sobolev inequality is a very useful tool that connects a function and it’s derivative. This presentation would include a common proof of Sobolev inequality. Another important inequality between norm infinite of a function and norm 2 of it will be proved by Sobolev inequality and Moser’s iteration.

10:50 AM- **Matthew Frazier**

Title: Energy Decay in Navier-Stokes Fluids

Abstract: An energy estimate of the Navier-Stokes equation tells us that, due to viscosity, the kinetic energy of a Navier-Stokes fluid will be monotonically decreasing. One immediate question is whether the energy will converge to some positive value or decay to zero. I will review the energy estimate and develop tools that allow a time dependent energy estimate, which tells us that the kinetic energy exponentially decreases to zero as expected.

11:15 AM- **Hangyu Pi**

Title: Fixed point methods and Application to ODEs and PDEs part1 : Well-posedness of a Specific ODE System

Abstract: In class, Prof.Tran talked about local existence of this ODE using Picard–Lindelöf theorem. Using Gronwall’s inequality, I got the global uniqueness.Then I apply the finite difference scheme to define a series of piece-wise linear functions and found that the limit of these functions is a solution to the ODE , which gives global existence. Together, we have the global well-posedness of this system.

11:15 AM- **Daotong Ge**

Title: Fixed point Methods and Application to ODEs and PDEs – Part 2: A Simple Reaction-Diffusion PDE.

Abstract: From Hangyu’s proof, we see a way to proof the well-posedness of a given ODE. Using the same idea, we extend the proof to a simple reaction-diffusion PDE.

**Lunch**

1:30 PM- **Moses Wong**

Title: The Vanishing Viscosity Method for Hamilton-Jacobi Equations

Abstract: The vanishing viscosity method is a general technique to show the existence of weak solutions. In this talk, I will outline how to build viscosity solutions for Hamilton-Jacobi Equation using the vanishing viscosity technique.

1:55 PM- ** Barton Gattis**

Title: An Introduction to Probabilistic Interpretations of PDEs

Abstract: This presentation is meant to provide an introduction to the probabilistic interpretation of solutions to partial differential equations. In particular, we discuss a probabilistic interpretation of the solution to the discrete Dirichlet problem and derive some properties of the solution in the discrete case. Along the way, we discuss multivariate random walks and discrete harmonic functions.

2:20 PM- ** Zongbo Cai **

Title: Prandtl’s Boundary Layer Theory

Abstract: An introduction to the boundary layer theory and derivation of boundary layer equations.

For questions about the program, please contact Jessica Lin (jessica@math.wisc.edu).