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Fall 1999 - Spring 2000 CNA Seminars
  • 1:30 P.M., TUESDAY, August 31, 1999, in Room PPB 300

  • Philippe LeFloch
    Ecole Polytechnique
    Palaiseau, France
TITLE: "$L^1$-Continuous Dependence Property for Hyperbolic Systems of Conservation Laws"

ABSTRACT: Recently, the fundamental issue of the uniqueness and the $L^1$ Lipschitz continuous dependence of entropy solutions for nonlinear hyperbolic systems

$$ \partial_t u + \partial_x f(u) = 0, \qquad u(x,t) \in R^N, $$

has attracted a lot of attention. This activity was initiated in pioneering work by Bressan, Colombo, Crasta and Piccoli and continued with a fundamental contribution by Liu and Yang. This research culminated with two independent contributions by Bressan, Liu and Yang and by Hua and LeFloch, which now provide particularly simple proofs for the $L^1$-continuous dependence of entropy solutions.

In our proof, we follow the standard Haar's method, which was previously assumed to apply only to scalar conservation laws or to special systems of equations or, else, to piecewise smooth solutions. The main difficulty is to obtain the uniqueness of (discontinuous) solutions for a suitably linearized version of the nonlinear problem. Our approach is based on the notion of admissible averaging matrix and on fine properties of shock waves in approximate solutions of strictly hyperbolic systems. It uses the classification of the propagating discontinuities into: Lax shocks, slow or fast under compressive shocks and rarefaction shocks. The approach also applies to solutions with large amplitude.

  • 1:30 P.M., TUESDAY, September 14, 1999, in Room PPB 300

  • Wen-Ching Lien
    Carnegie Mellon University
    Department of Mathematical Sciences
    Pittsburgh, PA 15213
TITLE: "Stability of a Self-Similar 3-Dimensional Gas Flow"
  • 4:30 P.M., FRIDAY, September 24, 1999, in Wean Hall 7500. Refreshments will be served at 4:00 P.M., in Wean Hall 6220.

  • Michel Chipot
    Zurich, Switzerland
TITLE: ""Continuity of the Free Boundary for Problems in Dimension 2""
  • 3:30 P.M., FRIDAY, October 1, 1999, in Room PPB 300
    Please note change of time due to homecoming events.
  • Epifanio Virga
    University of Pavia, Italy
TITLE: "Dynamics of Vesicles Subject to Adhesion"

ABSTRACT: Vesicles are lipid membranes which play an essential role in the cell structure. In the vicinity of a rigid wall, which could well mimic a hard tissue, vesicles feel forces that would make them adhere to the wall. This lecture presents a mathematical model for the motion of vesicles subject to such adhesive forces; it is mainly concerned with dynamics in two space dimensions. This evolution of a vesicle is governed by a non-local equation: the normal velocity of a point dependson the whole shape. Several computer animations illustrate the main features of this equation.

  • 1:30 P.M., TUESDAY, October 12, 1999, in Room PPB 300

  • Michal Kowalczyk
    Department of Mathematical Sciences
    Carnegie Mellon University
TITLE: "Spikes and Bumps: Local Concentration Solutions in the Gierer-Meinhardt system "

ABSTRACT: The Gierer-Meinhardt system is a system of reaction-diffusion equations modeling the biochemical reaction of activator-inhibitor type. If the diffusion of the activator $d_A$ is small and the diffusion of the inhibitor $d_H$ large then one expects that the local increase in the concentration of the activator will be further amplified, forming regions of high concentrations of activator surrounded by the "sea" of essentially uniformly distributed inhibitor. In the most studied case $d_A<<1$ and $d_H=\infty$ the activator concentrates at isolated points (those solutions are called spikes). In the first part of the talk we consider the Gierer-Meinhardt system in the case when both $d_A$ and $d_H$ are finite with $d_A/d_H <<1$. We will derive the effective equations governing the dynamics of a single spike and show that, unlike in the case $d_H=\infty$ a single spike remains in the interior of the domain for all times. In the second part of the talk we concentrate on the case $d_A/d_H= O(1)$ in one space dimension. We show that there exists $\sigma>0$ such that if $d_A/d_H<\sigma$ then there are solutions to the Gierer-Meinhardt system for which the regions of high concentration of the activator form clusters of ``bumps". We will also discuss the connection between the Gierer-Meinhardt system, certain nonlinear Schrodinger equation and breaking of homoclinics.

  • 1:30 P.M., TUESDAY, October 19, 1999, in PPB 300

  • Jan Kristensen
    Mathematical Institute
    Oxford, United Kingdom
TITLE: "Mazur's Lemma for Quasiconvexity"

ABSTRACT: Variational problems for multiple integrals on Sobolev spaces of vector-valued mappings are intrinsically connected to Morrey's notion of quasiconvexity. In this talk we discuss a version of Mazur's Lemma, which is valid in the context of quasiconvex functionals defined by means of the Lebesgue-Serrin extension procedure. As an application of the result we derive a new existence theorem for minimisers of quasiconvex integrals.

  • 1:30 P.M., TUESDAY, November 2, 1999, in PPB 300

  • Adelia Sequeira
    Department of Mathematics
    Instituto Superior Tecnico, Lisbon, Portugal
TITLE: "Mathematical Results and Numerical Methods for some non-Newtonian Fluid Models"

ABSTRACT: A wide range of models for viscoelastic flows are mixed nonlinear systems of PDE's, being elliptic-hyperbolic in the steady state and parabolic-hyperbolic in the unsteady state. This situation is difficult to treat numerically because most algorithms are designed for systems of equations of a single type. Efficient and accurate numerical schemes for solving these nonlinear systems must be based on their mixed mathematical structure in order to prevent numerical instabilities in problems which are mathematically well-posed. The aim of this lecture is twofold. First we present a short survey of recent mathematical results for the most common differential models of incompressible homogeneous viscoelastic non-Newtonian fluids, in the stationary case, {\sl e.g.} grade type Rivlin-Ericksen and Oldroyd type models. We address the questions of existence, uniqueness, stability and asymptotic behaviour (in space) of solutions, in several physically relevant flow geometries, by suitably decoupling the elliptic and hyperbolic parts in the system of equations. Finally, we conclude by discussing appropriate numerical schemes, using mixed finite element methods for the simulation of these models in 2D. The discussion will focus on a decoupled approach based on a Hood-Taylor method for the computation of the velocity-pressure field and a discontinuous Galerkin method for the computation of the components of the extra-stress tensor. The convergence of a fixed point algorithm is proved and error estimates are given.

  • 4:30 P.M., FRIDAY, November 12, 1999, in Wean Hall 7500. Refreshments will be served at 4:00 P.M., in Wean Hall 6220.

  • Anatole Katok
    Penn State
    University Park, PA
TITLE: "Hyperbolic, Elliptic and Parabolic Paradigms in Dynamical Systems: Overview and Open Problems"

ABSTRACT: It has been accepted by both mathematicians and scientists that a comprehensive description of the global orbit structure of all or "most" smooth dynamical systems is beyond the reach and very likely does not constitute a well--posed problem. Nevertheless, there are several leading paradigms which are applicable to broad classes of systems which appear in applications coming from both outside and inside mathematics. At a very crude level the four principal types of behavior for non--linear systems (hyperbolic, partially hyperbolic, elliptic, and parabolic) are modelled infinitesimally and locally on the corresponding effects in linear systems coupled with consequences of non--trivial recurrence. This general point of view allows to connect and contrast such disparate and deep areas as the theory of strange attractors and Pesin theory (hyperbolic), stable ergodicity (partially hyperbolic), KAM theory and fast periodic approximation (elliptic), Ratner rigidity theory and theory of rational billiards (parabolic). The talk will present the main features and typical examples from each area as well as a variety of challenging problems of current interest.

  • 1:30 P.M., TUESDAY, November 16, 1999, in Room PPB 300

  • Harumi Hattori
    Department of Mathematics
    West Virginia University
TITLE: "Entropy Rate Admissibility Criterion for a Phase Transition Problem"

ABSTRACT: The entropy rate admissibility criterion roughly says that among all solutions the entropy (actually energy in our context) decays as quickly as possible for the admissible solution. In the hyperbolic systems of conservation laws, this criterion is stronger than the entropy condition. On the other hand, in the phase transition problem we discuss, the system is a hyperbolic-elliptic mixed type since we use a nonmonotone stress-strain relation. It turns out that in such a system the entropy rate admissibility criterion is not compatible with the entropy condition. We discuss various consequence of this observation including the construction of global weak solutions.

  • 1:30 P.M., THURSDAY, December 2, 1999, in Room PPB 300

  • Michel Jabbour
    Department of Mathematical Sciences
    Carnegie Mellon University
  • 1:30 P.M., TUESDAY, January 25, 2000, in Room PPB 300

  • John Nagle
    Physics Department/Biological Sciences Department
    Carnegie Mellon University
TITLE: "Fluctuating Membranes "

ABSTRACT: The fundamental structural component in biomembranes, which are ubiquitous in life, is the lipid bilayer. The lipids themselves are in the class of surfactants, much studied by chemical engineers, and samples composed of lipid bilayers are smectic liquid crystals, one of the more interesting forms of condensed matter. Lipid bilayers of greatest relevance for biology are quite flexible. This flexibility poses interesting problems for structural determination and it affects the forces between membranes. These two topics are the focus of my current research that uses x-ray scattering and Monte Carlo simulations. The mathematics involves a continuum description at the nm to micron length scale. Unlike crystalline systems which have long range order, these systems have logarithmically diverging correlation functions; this is called quasi long range order. Instead of having delta-function Bragg peaks, the scattering has long power law tails. These systems are analogous to systems at a critical point and it is the dimensionality of the system that is critical.

  • 1:30 P.M., TUESDAY, February 1, 2000, in Room PPB 300

  • Christian Klingenberg
    Applied Mathematics,
    Wuerzburg University
    presently visiting JPL and Applied Math
    California Institute of Technology
TITLE: "Mathematical Modeling and Numerical Simulation of Star Formation"

ABSTRACT: We consider astrophysical jet flow associated with star formation. This is modeled by a system of conservation laws. The non-linear nature of these differential equations requires numerical discretization in so-called conservation form. For our astrophysical problem this puts us into a quandary: internal variables like pressure and temperature can no longer be computed accurately. We propose the following way out: Embed the astrophysical jet model into a more complete model. There it is readily possible to compute the internal variables. Then we project these variables back to the original model. We can prove that the translation of this into a numerical procedure leads to reliable solutions. I will make this talk accessible to graduate students. It will be illustrated with pictures showing actual astronomical observations and numerical simulations.

  • 1:30 P.M., TUESDAY, February 8, 2000 in Room PPB 300

  • Lukasz Kruk
    Department of Mathematical Sciences
    Carnegie Mellon University
  • 1:30 P.M., TUESDAY, February 29, 2000 in Room PPB 300

Roger Lui
Department of Mathematical Sciences
Carnegie Mellon University
  • 1:30 P.M., TUESDAY, April 4, 2000 in Room PPB 300

Dehua Wang
Department of Mathematics
University of Pittsburgh
Pittsburgh, PA
  • 1:30 P.M., TUESDAY, April 13, 2000 in Room PPB 300

  • Guy Bouchitte
    Department de Mathematiques, Universite de Toulon.
    Carnegie Mellon University
  • 1:30 P.M., TUESDAY, April 18, 2000 in Room PPB 300

  • Luigi Ambrosio
    Scuola Normale Superiore,
    Scuola Normale Sup. Pisa.
  • 1:30 P.M., TUESDAY, April 25, 2000 in Room PPB 300

  • Luisa Mascarenhas

  • 4:30 P.M., FRIDAY, April 28, 2000, in WeH 7500. Refreshments will be served at 4:00 P.M., in Wean Hall 6220.

  • Patrick Gerard
    Department of Mathematics
    Universite de Paris Sud.