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PDE/Applied Math Seminar - Abstracts Fall 2004

(Oct. 7) Dongho Chae Global regularity for the 2D Boussinesq equations with partial viscosity terms

In this talk we present the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also show that as diffusivity(viscosity) goes to zero the solutions of the fully viscous equations converges strongly to those of zero diffusion(viscosity) equations. (Oct. 28) Roger Moser The Modica-Mortola functionals and the Willmore energy

An asymptotic theory for Modica-Mortola functionals (which are used in the van der Waals-Cahn-Hilliard theory of phase transitions in order to model the energy of phase interfaces) gives rise to a generalized area functional. We study a higher order asymptotic problem related to these functionals. Here the expected limit functional is a generalization of the Willmore -functional. An analysis of the problem under a monotonicity assumption supports this conjecture.

(Nov. 11) Jonathan Mattingly Ergodicity of the 2D Navier-Stokes equations under very degenerate forcing

Consider the two dimensional Navier-Stokes equation subject to random excitation. I will give essentially optimal conditions on the structure of the sochastic forcing under which the dynamics posses a unique statistical steady state. The conditions use information on the geometry of the forcing and are independent of the viscosity (or Reynolds number). Hence the results hold for a fixed forcing as the viscosity is made smaller.

These very recent results are the culmination of a long project under taken by the speaker, his collaborators and others in the community to understand the ergodic theory dissipative SPDEs when only a few degrees of freedom are stochastically excited. The result amounts to a version of H�rmander's "sum of squares" theorem for hypo-elliptic operators in an infinite dimestional setting. The talk will also touch on the tools from Malliavin calculus and anticipative stochastic processes used to prove the result. This is joint work with Etienne Pardoux and Martin Hairer building on earlier joint works with Ya Sinai and Weinen E.

(Nov. 18) Michael Vogelius Non-linear Elliptic Boundary Value Problems related to Corrosion Modeling

I shall discuss the solution structure and the blow-up phenomena associated with two dimensional boundary value problems of the form $$ \Delta u = 0~~\hbox{ in } \Omega~~~~,\frac{\partial u }{\partial {\bf n} } =Du+\lambda f(u)~~\hbox{ on } \partial \Omega~~, $$ for certain $f$ that are odd, non-decreasing, and with $f'(0)=1$. Special emphasis will be placed on functions $f$ of an exponential character:~in particular $f(u)=\sinh(u)=(e^u-e^{-u})/2$.

As will be shown, the solution structure is extremely different depending on whether $\lambda<0$ (generically: finitely many solutions) or $\lambda>0$ (generically: infinitely many solutions). Non-trivial solutions in general blow up as $\lambda \rightarrow 0$, but again: the nature of the blow-up for exponential $f$ is completely different depending on whether $\lambda \rightarrow 0_-$ or $\lambda \rightarrow 0_+$. Some of the character of the blow-up for positive $\lambda$ is reminiscent of phenomena associated with the Ginzburg-Landau equations.

Our general analysis has been significantly aided by the discovery of surprisingly simple explicit solutions and by significant computational experimentation. I shall discuss both of these aspects in some detail.

(Nov. 19) Michael Vogelius Electromagnetic imaging for small inhomogeneities

Electromagnetic Imaging in this context refers to the identification of internal characteristics of a medium based on boundary (or near-field) measurements of the electric and/or magnetic fields. After a brief review of some of the main mathematical results in Electromagnetic- and Impedance Imaging (from the last 20 years, or so) I shall proceed to discuss some very recent, extremely efficient representation formulas that lead to a surprisingly accurate identification of the size, and the location of relatively small inhomogeneities. These representation formulas take into account polarization effects, and they may be derived by variational techniques related to $H-$ (or $\Gamma-$) convergence. The magnitude of the polarization effects may be estimated in ways that are very reminiscent of effective media bounds (of the Hashin-Shtrikman type). A precise assessment of the polarization effects is very important for highly accurate size estimates. Finally, these representation formulas lend themselves very naturally to the application of reconstruction methods of a linear sampling- or MUSIC (MUltiple SIgnal Chararcterization) character. On this matter I shall discuss some general ideas, and implementation issues, as well as provide examples of computational reconstructions.

(Dec. 9) Joy Ko The Search for Singular Solutions to the Landau-Lifshitz (Gilbert) Equations

The Landau-Lifshitz and Landau-Lifshitz Gilbert equations are the basic evolution equations in micromagnetics, a continuum model for magnetic behavior in ferromagnetic materials. In the setting where the magnetic behavior is determined by the Dirichlet energy, these equations are a hybrid Sch\"rodinger map flow and harmonic map heat flow into the unit sphere $S^2$. The question of singularity formation for finite energy data for these equations is open, but the search is motivated by the fact that under a suitable transformation, these equations are reminiscent of the cubic nonlinear Schr\"odinger equation for which singular solutions abound. Analytical attempts have met impasses but have yielded insight which underlies current numerical investigations (in collaboration with S. Bartels).

Last modified: Mon Nov 1 11:01:15 EST 2004