Special Joint LCDS & PDE Seminar
**Please Note Special Day, Time and Place for This Week Only ***Originally scheduled for 11 a.m. |
Abstract:
We consider kinetic models describing two species
of particles interacting via a long range repulsive
potential and
a) with a reservoir at fixed
temperature,
b) by collisions.
The dynamics for the first model
conserves the total masses of the two species and
its sharp interface limit is described by a kind of
Mullins-Sekerka motion. The second dynamics models
the behaviour of a binary fluid and conserves masses,
momentum and energy. In the sharp interface limit
in this case the velocity field satisfies the
incompressible Navier-Stokes equations together with
a jump boundary condition for the pressure across the
interface which, in turn, moves with a velocity
given by the normal component of the velocity field.
Special Joint LCDS & PDE Seminar
**Please Note Special Day, Time and Place for This Week Only |
Brown Analysis Seminar
Special Lecture Series
Special Lecture Series
Division of Applied Mathematics Graduate Pizza Seminar
Abstract: Maps with unique fixed points are very useful. For example, we use them to prove existence and uniqueness for ordinary differential equations as well as existence and uniqueness of invariant distributions for irreducible markov chains. The contraction mapping theorem is a great way to show that your map has a unique fixed point. I am going to talk about a class of maps that are contractions in Hilbert's projective metric even though they don't appear to be contractions on the surface.
Brown University
Graduate School Dissertation Defense
PDE Seminar
Member, Fields of Mathematics and Applied Mathematics, Cornell University, Ithaca, NY | |
Abstract: Nonlinear (finite) elasticity is the central model of continuum solid mechanics. It has a vast range applications, including flexible engineering structures, biological structures - both macroscopic and molecular, and materials like elastomers and shape-memory alloys - everything from fighter jets to lingerie! Although the subject dates back to Cauchy, the current state of existence theory is quite poor. Local results, based upon the implicit function theorem, are well known. In 1977 Ball obtained deep results concerning the existence of absolute energy minima for a general class of conservative problems. To date, it is not known if such minima correspond to weak solutions of the momentum balance laws (Euler-Lagrange equations). Moreover, Ball's results do not address the existence of other types of equilibria (critical points).
We consider a general class of boundary value problems governing the equilibria of forced, three-dimensional nonlinearly elastic bodies. We impose strong ellipticity and pay careful attention to the constraint of local injectivity and the accompanying growth of the stored energy function. In particular, such physically realistic restrictions rule out the possibility of {\it {uniform}} ellipticity. This and the presence of traction boundary conditions typically preclude a conventional Leray-Schauder/Rabinowitz approach.
We present recent results for overcoming these difficulties in global continuation and bifurcation problems of nonlinear elastostatics. For pure displacement boundary-value problems, we show how to employ the Leray-Schauder degree. For a general class of "mixed" boundary value problems, we develop our own degree having all of the usual properties of the Leray-Schauder degree. In the absence of a-priori bounds, we fall short of a general existence theorem. Nonetheless, these are the first results in our subject addressing the existence of classical solutions "in the large". In particular, for displacement problems (Dirichlet boundary conditions), we present physically reasonable constitutive hypotheses ensuring the existence of unbounded branches of classical, injective solutions.
Department of Mathematics Colloquium
<--- 2004 Index