Seminars and Colloquia by Series

Oblique derivative problems for elliptic equations

Series
PDE Seminar
Time
Tuesday, December 2, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Gary M. Lieberman – Iowa State University
The usual boundary condition adjoined to a second order elliptic equation is the Dirichlet problem, which prescribes the values of the solution on the boundary. In many applications, this is not the natural boundary condition. Instead, the value of some directional derivative is given at each point of the boundary. Such problems are usually considered a minor variation of the Dirichlet condition, but this talk will show that this problem has a life of its own. For example, if the direction changes continuously, then it is possible for the solution to be continuously differentiable up to a merely Lipschitz boundary. In addition, it's possible to get smooth solutions when the direction changes discontinuously as well.

High-order numerical methods for nonlinear PDEs

Series
PDE Seminar
Time
Tuesday, November 25, 2008 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Bojan Popov – Texas A&M University

In this talk we will consider three different numerical methods for solving nonlinear PDEs:

  1. A class of Godunov-type second order schemes for nonlinear conservation laws, starting from the Nessyahu-Tadmor scheme;
  2. A class of L1 -based minimization methods for solving linear transport equations and stationary Hamilton- Jacobi equations;
  3. Entropy-viscosity methods for nonlinear conservation laws.

All of the above methods are based on high-order approximations of the corresponding nonlinear PDE and respect a weak form of an entropy condition. Theoretical results and numerical examples for the performance of each of the three methods will be presented.

On the formation of adiabatic shear bands

Series
PDE Seminar
Time
Friday, November 21, 2008 - 16:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Athanasios Tzavaras – Univeristy of Maryland
We consider a system of hyperbolic-parabolic equations describing the material instability mechanism associated to the formation of shear bands at high strain-rate plastic deformations of metals. Systematic numerical runs are performed that shed light on the behavior of this system on various parameter regimes. We consider then the case of adiabatic shearing and derive a quantitative criterion for the onset of instability: Using ideas from the theory of relaxation systems we derive equations that describe the effective behavior of the system. The effective equation turns out to be a forward-backward parabolic equation regularized by fourth order term (joint work with Th. Katsaounis and Th. Baxevanis, Univ. of Crete).

On Shock-Free Periodic Solutions for the Euler Equations

Series
PDE Seminar
Time
Tuesday, November 18, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Robin Young – University of Massachusetts, Amherst
We consider the existence of periodic solutions to the Euler equations of gas dynamics. Such solutions have long been thought not to exist due to shock formation, and this is confirmed by the celebrated Glimm-Lax decay theory for 2x2 systems. However, in the full 3x3 system, multiple interaction effects can combine to slow down and prevent shock formation. In this talk I shall describe the physical mechanism supporting periodicity, describe combinatorics of simple wave interactions, and develop periodic solutions to a "linearized" problem. These linearized solutions have a beautiful structure and exhibit several surprising and fascinating phenomena. I shall also discuss partial progress on the perturbation problem: this leads us to problems of small divisors and KAM theory. This is joint work with Blake Temple.

Uniqueness in the boundary inverse problem for elasticity

Series
PDE Seminar
Time
Tuesday, November 11, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Anna Mazzucato – Penn State University, State College
We discuss the inverse problem of determining elastic parameters in the interior of an anisotropic elastic media from dynamic measurements made at the surface. This problem has applications in medical imaging and seismology. The boundary data is modeled by the Dirichlet-to-Neumann map, which gives the correspondence between surface displacements and surface tractions. We first show that, without a priori information on the anisotropy type, uniqueness can hold only up to change of coordinates fixing the boundary. In particular, we study orbits of elasticity tensors under diffeomorphisms. Then, we obtain partial uniqueness for special classes of transversely isotropic media. This is joint work with L. Rachele (RPI).

A PDE and a stochastic model in cell polarity

Series
PDE Seminar
Time
Tuesday, October 21, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Sigurd Angenent – University of Wisconsin, Madison
I will discuss a few ways in which reaction diffusion models have been used to pattern formation. In particular in the setting of Cdc42 transport to and from the membrane in a yeast cell I will show a simple model which achieves polarization. The model and its analysis exhibits some striking differences between deterministic and probabilistic versions of the model.

Navier-Stokes evolutions as self-dual variational problems

Series
PDE Seminar
Time
Tuesday, October 7, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Nassif Ghoussoub – University of British Columbia, Canada
We describe how several nonlinear PDEs and evolutions ­including stationary and dynamic Navier-Stokes equations­ can be formulated and resolved variationally by minimizing energy functionalsof the form I(u) = L(u, -\Lambda u) + \langle \Lambda u, u\rangle and I(u) = \Int^T_0 [L(t, u(t), -\dot u(t) - \Lambda u(t)) + \langle\Lambda u(t), u(t)\rangle]dt + \ell (u(0) - u(T) \frac{u(T) + u(0)}{2} where L is a time-dependent "selfdual Lagrangian" on state space, is another selfdual "boundary Lagrangian", and is a nonlinear operator (such as \Lambda u = div(u \otimes u) in the Navier-Stokes case). However, just like the selfdual Yang-Mills equations, the equations are not obtained via Euler-Lagrange theory, but from the fact that a natural infimum is attained. In dimension 2, we recover the well known solutions for the corresponding initial-value problem as well as periodic and anti-periodic ones, while in dimension 3 we get Leray solutions for the initial-value problems, but also solutions satisfying u(0) = \alpha u(T ) for any given in (-1, 1). It is worth noting that our variational principles translate into Leray's energy identity in dimension 2 (resp., inequality in dimension 3). Our approach is quite general and does apply to many other situations.

Variational Principles and Partial Differential Equations for Some Model Problems in Polycrystal Plasticity

Series
PDE Seminar
Time
Tuesday, September 30, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Marian Bocea – North Dakota State University, Fargo
The yield set of a polycrystal may be characterized using variational principles associated to suitable supremal functionals. I will describe some model problems for which these can be obtained via Gamma-convergence of a class of "power-law" functionals acting on fields satisfying appropriate differential constraints, and I will indicate some PDEs which play a role in the analysis of these problems.

Segregation of Granular Materials - Experiments, Modeling, Analysis and Simulations

Series
PDE Seminar
Time
Tuesday, September 23, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Michael Shearer – Department of Mathematics, North Carolina State University
Granular materials are important in a wide variety of contexts, such as avalanches and industrial processing of powders and grains. In this talk, I discuss some of the issues in understanding how granular materials flow, and especially how they tend to segregate by size. The segregation process, known scientifically as kinetic sieving, and more colorfully as The Brazil Nut Effect, involves the tendency of small particles to fall into spaces created by large particles. The small particles then force the large particles upwards, as in a shaken can of mixed nuts, in which the large Brazil nuts tend to end up near the lid. I'll describe ongoing physics experiments, mathematical modeling of kinetic sieving, and the results of analysis of the models (which are nonlinear partial differential equations). Movies of simulations and exact solutions illustrate the role of shock waves after layers of small and large particles have formed.

Derivation of shell theories from 3d nonlinear elasticity

Series
PDE Seminar
Time
Tuesday, September 9, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Marta Lewicka – School of Mathematics, University of Minnesota
A longstanding problem in the mathematical theory of elasticity is to predict theories of lower-dimensional objects (such as rods, plates or shells), subject to mechanical deformations, starting from the 3d nonlinear theory. For plates, a recent effort (in particular work by Friesecke, James and Muller) has lead to rigorous justification of a hierarchy of such theories (membrane, Kirchhoff, von Karman). For shells, despite extensive use of their ad-hoc generalizations present in the engineering applications, much less is known from the mathematical point of view. In this talk, I will discuss the limiting behaviour (using the notion of Gamma-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d mid-surface S. We prove that the minimizers of the 3d elastic energy converge, after suitable rescaling, to minimizers of a hierarchy of shell models. The limiting functionals (which for plates yield respectively the von Karman, linear, or linearized Kirchhoff theories) are intrinsically linked with the geometry of S. They are defined on the space of infinitesimal isometries of S (which replaces the 'out-of-plane-displacements' of plates), and the space of finite strains (which replaces strains of the `in-plane-displacements'), thus clarifying the effects of rigidity of S on the derived theories. The different limiting theories correspond to different magnitudes of the applied forces, in terms of the shell thickness. This is joint work with M. G. Mora and R. Pakzad.

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