# Partial Differential Equations

## Initial value problems viewed as generalized optimal transport problems with matrix-valued density fields

The initial value problem for many important PDEs (Burgers, Euler, Hamilton-Jacobi, Navier-Stokes equations, systems of conservation laws with convex entropy, etc…) can be often reduced to a convex minimization problem that can be seen as a generalized optimal transport problem involving matrix-valued density fields. The time boundary conditions enjoy a backward-forward structure of “ballistic” type, just as in mean-field game theory.

## PDE Aspects of Fluid Flows

We explain some of the recent results in concerning PDEs describing fluid flows, as well as some of the difficulties. Model equations will also be discussed.

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## PIMS Workshop on Nonlocal Variational Problems and PDEs

## Nonlocal equations from various perspectives - lecture 3

We would like to give a detailed presentation of some equations which exhibit some nonlocal phenomena. Often, the nonlocal effect is modelled by a diffusive operator which is (in some sense) elliptic and fractional. Natural example arise from probability, geometry, quantum physics, phase transition theory and crystal dislocation dynamics. We will try to discuss some of the mathematical tools that are useful to deal with these problems, explain in detail some of the main motivations, describe some recent results on these topics and list some open problems.

## Nonlocal equations from various perspectives - lecture 2

We would like to give a detailed presentation of some equations which exhibit some nonlocal phenomena. Often, the nonlocal effect is modelled by a diffusive operator which is (in some sense) elliptic and fractional. Natural example arise from probability, geometry, quantum physics, phase transition theory and crystal dislocation dynamics. We will try to discuss some of the mathematical tools that are useful to deal with these problems, explain in detail some of the main motivations, describe some recent results on these topics and list some open problems.

## Nonlocal equations from various perspectives - lecture 1

We would like to give a detailed presentation of some equations which exhibit some nonlocal phenomena. Often, the nonlocal effect is modelled by a diffusive operator which is (in some sense) elliptic and fractional. Natural example arise from probability, geometry, quantum physics, phase transition theory and crystal dislocation dynamics. We will try to discuss some of the mathematical tools that are useful to deal with these problems, explain in detail some of the main motivations, describe some recent results on these topics and list some open problems.

## Blowup or no blowup? The interplay between theory and computation in the study of 3D Euler equations

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. We first review some recent theoretical and computational studies of the 3D Euler equations. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. We then present strong numerical evidence that the 3D Euler equations develop finite time singularities. To resolve the nearly singular solution, we develop specially designed adaptive (moving) meshes with a maximum effective resolution of order $10^12$ in each direction. A careful local analysis also suggests that the solution develops a highly anisotropic self-similar profile which is not of Leray type. A 1D model is proposed to study the mechanism of the finite time singularity. Very recently we prove rigorously that the 1D model develops finite time singularity.This is a joint work of Prof. Guo Luo.

## Non Classical Flag Domains and Spencer Resolutions

This talk has two parts. The common themes are the very interesting properties of flag domains and their quotients by discrete subgroups present only in the non-classical case. The first part will give a general overview of these properties, especially as they relate to several of the other talks being presented at this conference. The second part will focus on one particular property in the non-classical case. When suitably localized, the Harish-Chandra modules associated to discrete series -- especially the non-holomorphic and totally degenerate limits (TDLDS) of such -- may be canonically realized as the solution space to a holomorphic, linear PDE system. The invariants of the PDE system then relate to properties of the Harish-Chandra module: e.g., its tableau gives the K-type. Conversely, the representation theory, especially in the case of TDLDS, suggest interesting new issues in linear PDE theory.

## On Fourth Order PDEs Modelling Electrostatic Micro-Electronical Systems

Now unlike the model involving only the second order Laplacian (i.e., $d = 0$), very little is known about this equation. We shall explain how, besides the above practical considerations, the model is an extremely rich source of interesting mathematical phenomena.

## The Mathematics of PDEs and the Wave Equation

We look at the mathematical theory of partial differential equations as applied to the wave equation. In particular, we examine questions about existence and uniqueness of solutions, and various solution techniques.