# Partial Differential Equations

## PDE Aspects of Fluid Flows

## PIMS Workshop on Nonlocal Variational Problems and PDEs

## Nonlocal equations from various perspectives - lecture 3

## Nonlocal equations from various perspectives - lecture 2

## Nonlocal equations from various perspectives - lecture 1

## 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

Micro-ElectroMechanical Systems (MEMS) and Nano-ElectroMechanical Systems (NEMS) are now a well established sector of contemporary technology. A key component of such systems is the simple idealized electrostatic device consisting of a thin and deformable plate that is held fixed along its boundary , where is a bounded domain in The plate, which lies below another parallel rigid grounded plate (say at level ) has its upper surface coated with a negligibly thin metallic conducting film, in such a way that if a voltage l is applied to the conducting film, it deflects towards the top plate, and if the applied voltage is increased beyond a certain critical value , it then proceeds to touch the grounded plate. The steady-state is then lost, and we have a snap-through at a finite time creating the so-called pull-in instability. A proposed model for the deflection is given by the evolution equation

Now unlike the model involving only the second order Laplacian (i.e., ), 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.