# CAMP/Nonlinear PDE's Seminar

Wednesdays at 4 PM in Eckhart 206.

## Spring 2001 Schedule

March 28,
Walter Strauss, Brown University
Magnetic Instabilities in a Collisionless Plasma

A collisionless plasma is modeled by the Vlasov-Maxwell equations, also called the collisionless Boltzmann equation with an external field. There is no dissipation. There exist many kinds of steady states, including homogeneous states, BGK states and magnetic states. Some are stable and some unstable. How do we distinguish the stable ones? There has been a lot of progress on this question recently. In this talk I will focus on a homogeneous state that is close to a maxwellian (=gaussian) but is anisotropic. Such a state is unstable under certain electromagnetic perturbations but is stable under purely electrostatic perturbations. The instability requires the spatial period to be larger than a certain precise critical value.

April 4,
Darryl Holm, Los Alamos National Laboratory
Lagrangian-mean and Eulerian-mean turbulence models

We are developing the geometric approach to dimension reduction especially for models of turbulence in fluids. We shall discuss Euler-Poincare formulations of Lagrangian-mean, and Eulerian-mean fluid equations for modeling turbulence. We use: Reynolds decomposition(s), Taylor hypotheses, Hamilton's principle, Averaged Lagrangians and Euler-Poincare equations to model and analyze the mean dynamical effects of fluctuations on 3D Lagrangian-mean and Eulerian-mean fluid motion. We shall show theoretical and numerical results for the Navier-Stokes alpha model of turbulence, and discuss its comparisons with experimental data.

April 11
Shi Jin, University of Wisconsin
Relaxation schemes for PDEs and mechanics

The classical solutions to many partial differential equations fail to exist in finite time even if the initial data are smooth. Classical examples include hyperbolic conservation laws and Hamilton-Jacobi equations. In such cases viscosity solutions have been introduced which allow the selection of physically relevant weak solutions beyond the singularity time. We introduce the relaxation approximation to such partial differential equations by replacing such equations with a semi-linear hyperbolic systems with stiff relaxations. While the viscosity regularization arises from the Navier-Stokes approximation to the Euler equations, the relaxation approximation is analogous to the regularization of the Euler equations by the more fundamental Boltzmann equation. The relaxation approximation introduces a physically natural way to select the entropy or viscosity solution to hyperbolic conservation laws and Hamilton-Jacobi equation. The semilinear nature of the relaxation system paves a new way to derive numerical schemes ( known as the relaxation schemes) that are simple, efficient, and Riemann solver free, with a high resolution, for problems involving shocks, fronts, or other type of discontinuous solutions.

April 18,
Robert Jerrard, University of Illinois, Urbana-Champaign
Gamma-limits for Gizburg-Landau functionals

We prove some theorems describing the Gamma-limits of Ginzburg-Landau functionals, both with and without magnetic field, in a variety of scalings. As corollaries we recover recent results of Sandier and Serfaty that describe the asymptotic behavior of energy-minimizing solutions of the Ginzburg-Landau equations for superconductors in an applied magnetic field, in the limit of extreme type II'' behavior. (joint work with H.M. Soner)

April 25,
Laszlo Erdos, Georgia Tech
How does Boltzmann equation emerge from quantum mechanics?

Classical fluid equations (Boltzmann, Fokker-Planck etc.) are widely used in electron kinetic theory and semiconductor modelling. However, their derivations are usually phenomenological and they rely on the assumptions that (i) the electron cloud can be viewed as a classical fluid; (ii) the fluctuations of the infinite degree of freedom in the "fluid" are independent ("propagation of chaos"), hence their cumulative effect is diffusive. Apart from a few particular models, none of these statements have ever been proven from first principles of quantum mechanics and they cannot be taken for granted. In fact, there are several quantum phenomena which contradict to classical "fluid" intuitions; e.g. tunnelling, resonance, Anderson-localization, propagation in periodic media. It is a deep fact of kinetic physics that in most kinetic problems the quantum correlations are not strong enough to destroy the classical fluid picture. In particular, we could derive irreversible Boltzmann type transport equations modelling conductance and diffusion as a scaling limit of reversible quantum dynamics. In this talk I will review some of the recent results in this field. This work is a collaboration with H.-T. Yau.

May 2,
Cancelled

May 9,
Peter Sternberg, University of Indiana,
A Second Variation Argument for the Non-existence of Permanent Currents

I will survey results over recent years on the existence of permanent currents within the context of the Ginzburg-Landau theory of superconductivity. Permanent currents are currents that circulate through a superconducting sample, sometimes for years, with only negligible resistance. Existence theories to date have relied on an assumption of either non-trivial topology or geometry of the sample. I will then present a new result on the non-existence of these currents in convex samples, obtained jointly with Shuichi Jimbo. The technique involves an analysis of the second variation of the energy. Before discussing elements of the proof, I will also survey related second variation non-existence arguments in the calculus of variations.

May 16,
Hatem Zaag, New York University,
One D behavior of N D solutions to a semilinear heat equation

We consider a blow-up solution $u$ to a semilinear heat equation with a non isolated blow-up point $a$. Under a non degeneracy condition, we prove that at the first order, $u$ looks like a 1D solution of the distance to the blow-up set. This gives $C^{1, \alpha}$ regularity for the blow-up set, asuming it is just continuous.

May 23, Cancelled on account of jury duty
Ray Pierrehumbert, University of Chicago,
My New Flame: Lattice models of ignition and combustion (and earthworm diffusion)

May 30,
Knut Solna, University of California, Irvine
Time-reversal and focusing in a random medium

We analyze the refocusing phenomenon in time-reversal acoustics. That is, a recorded signal is reversed in time and sent back into a random medium and approximately refocuses at the original source point. The refocusing resolution is improved by fine scale medium inhomogeneities. We discuss the role of scale separation and the statistical stability of the refocusing and the super-resolution phenomenon.

June 6,
Paul Wiegmann, University of Chicago
Integrable structure of the Dirichlet boundary problem

Conformal map of a domain of the complex plane evolves according to Toda integrable hierarchy under a deformation of the domain.

June 27,
Howard Cohl, Lawrence Livermore National Laboratory
The Potential of a Point Mass/Charge

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