Partial Differential Equations

PDE Aspects of Fluid Flows

Speaker: 
Vladimir Sverak
Date: 
Fri, Sep 23, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 
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. For more information, see the event webpage for this event.

Nonlocal equations from various perspectives - lecture 3

Speaker: 
Enrico Valdinoci
Date: 
Wed, Jun 15, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Workshop on Nonlocal Variational Problems and PDEs
Abstract: 
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

Speaker: 
Enrico Valdinoci
Date: 
Tue, Jun 14, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Workshop on Nonlocal Variational Problems and PDEs
Abstract: 
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

Speaker: 
Enrico Valdinoci
Date: 
Mon, Jun 13, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Workshop on Nonlocal Variational Problems and PDEs
Abstract: 
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

Speaker: 
Thomas Hou
Date: 
Fri, Feb 27, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 

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

Speaker: 
Phillip Griffiths
Date: 
Wed, Jun 19, 2013
Location: 
PIMS, University of British Columbia
Conference: 
Recent advances in Hodge theory
Abstract: 

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

Speaker: 
Nassif Ghoussoub
Date: 
Wed, Jul 8, 2009
Location: 
University of New South Wales, Sydney, Australia
Conference: 
1st PRIMA Congress
Abstract: 

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 $ \partial \Omega $, where $ \Omega $ is a bounded domain in $ \mathbf{R}^2. $ The plate, which lies below another parallel rigid grounded plate (say at level $ z=1 $) 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 $ l^* $, 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

$$\frac{\partial u}{\partial t} - \Delta u + d\Delta^2 u = \frac{\lambda f(x)}{(1-u^2)}\qquad\mbox{for}\qquad x\in\Omega, t\gt 0 $$
$$u(x,t) = d\frac{\partial u}{\partial t}(x,t) = 0 \qquad\mbox{for}\qquad x\in\partial\Omega, t\gt 0$$
$$u(x,0) = 0\qquad\mbox{for}\qquod x\in\Omega$$

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

Author: 
Michael P. Lamoureux
Date: 
Tue, Aug 1, 2006
Location: 
University of Calgary, Calgary, Canada
Conference: 
Seismic Imaging Summer School
Abstract: 
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.

Mathematics of Seismic Imaging

Author: 
William W. Symes
Date: 
Fri, Jul 1, 2005
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
These lectures present a mathematical view of reflection seismic imaging, as practiced in the petroleum industry.
Notes: 
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