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PDE/Appl Math Seminar, Spring 2006 - Abstracts

(Feb. 3) Felix Otto (Bonn) Multiscale analysis in micromagnetics

From the point of view of mathematics, micromagnetics is an ideal playground for a pattern forming system in materials science: There are abundant experiments on a wealth of visually attractive phenomena and there is a well--accepted continuum model. In this talk, I will focus on two specific experimental pattern for thin film ferromagnetic elements. One pattern is a ground state, the other pattern is a metastable state. Starting point for our analysis is the micromagnetic model which has three length scales and thus many parameter regimes. For both pattern, we identify the appropriate paramater regime and rigorously derive a reduced model via $\Gamma$--convergence. We numerically simulate the reduced model and compare to experimental data. This is joint work with A. DeSimone, R. V. Kohn, and S. Mueuller

(Feb. 9) Michael Westdickenberg (Bonn) The Entropy Condition for Hyperbolic Conservation Laws

Since weak solutions of hyperbolic conservation laws may be nonunique, typically an entropy condition is imposed to single out the relevant solution. We discuss how the entropy condition implies regularity and structure of entropy solutions of scalar conservation laws. For the one-dimensional system of isentropic Euler equations we show how the entropy condition gives global existence of entropy solutions with natural bounds.

(February 23) Zhiwu Lin (Courant Institute) Some recent results on stability of collisionless plasmas

A plasma is a completed inonized gas. In many applications such as in nuclear fusion or many space physical phenomena, the plasma is of high temperature and low density, and collisions can be ignored. The standard kinetic models for a collisionless plasma are Vlasov-Maxwell and Vlasov-Poisson systems. The Vlasov-Poisson system is also used to model galaxy dynamics, where a star plays the role of a particle. There exists infinitely many equilibria for Vlasov models and their stability is of central importance in physics. I will describe some of my recent works on nonlinear stability and instability of various Vlasov models.

(March 16) Sanjeeva Balasuriya (Sidney) Perturbing Optimally For Chaotic Mixing

What is the best perturbation to give to an integrable flow in order to optimize the resulting chaotic mixing? This question, interesting from a purely mathematical viewpoint, has topical relevance in the quest for optimizing mixing in micro-fluidic devices, whose flow is of low Reynolds number. This talk addresses this issue using recently developed methods for quantifying cross-separatrix flux using the Melnikov function and Fourier transforms, and tackles both two and three dimensions. The optimum perturbing flow protocol is characterized, and its strengths and weaknesses discussed. An application to cross-channel micro-mixers (in which the exact locations of the cross-stream channels is determined) is also presented.

(March 9) Giles Auchmuty (NSF and Houston) Finite Energy Solutions of 3D div-curl Boundary Value Problems

Div-curl boundary value problems model time-independent solutions of Maxwell's equations and also arise in fluid mechanics. They constitute a linear over-determined system of equations. In this talk I will describe

(i)solvability criteria for the existence of finite energy (L^2-)solutions,

(ii)the representation of solutions using scalar and vector potentials, and

(iii)the possible non-uniqueness of solutions; depending on the topology of the domain.

These results cover cases where either the normal, or tangential, component of the field is prescribed on the boundary. Also the case of mixed normal and tangential boundary data. The existence results are obtained using variational methods. For each of these cases, the uniqueness of the solutions depends on the differential topology of the domain; there is non-uniqueness when the domain is topologically non-trivial. To describe a well-posed problem extra integrals of the solution must also be specified - and these extra conditions have physical interpretations.

(March 30) Chuck Gartland (Kent State) Numerical Investigations of Electric-Field-Induced Transitions in Cholesteric Liquid Crystal Films

We consider thin films of a cholesteric liquid-crystal material subject to an applied electric field. In such materials, the liquid-crystal "director" (local average orientation of the long axis of the molecules) has an intrinsic tendency to rotate in space; while the substrates that confine the film tend to coerce a uniform orientation. The electric field encourages certain preferred orientations of the director as well, and these competing influences give rise to several different stable equilibrium states of the director field, including spatially uniform, translation invariant (functions only of position across the cell gap), and periodic (with 1-D or 2-D periodicity in the plane of the film). These structures depend on two principal control parameters: the ratio of the cell gap to the intrinsic "pitch" (spatial period of rotation) of the cholesteric and the magnitude of the applied voltage.

We report on numerical work (in progress) on the bifurcation and phase behavior of this system. The study was motivated by potential applications involving switchable gratings and eyewear with tunable transparency. We compare our results with recent experiments conducted in the Liquid Crystal Institute at Kent State University.

(April 6) Dmitry Dolgopyat (Maryland) Homogenization of advection-diffusion equation with small diffusivity

We consider a small diffusivity limit for advection diffusion equation on a 2D torus with area preserving vector field. In this case the result depends on the arithmetic properties of the rotation number of the corresponding vector field. Freidlin and Wenzell solved this problem for rational rotation numbers and conjectured the answer in general case. We present the proof of their conjecture for typical rotation numbers. This is a joint work with Leonid Koralov.

(April 20) Yann Brenier Derivation of particle, string and membrane motions from Born-Infeld electromagnetism

We derive classical particle, string and membrane motion equations from a rigorous asymptotic analysis of the Born-Infeld nonlinear electromagnetic theory. The Born-Infeld model is a modification of the classical Maxwell equations, designed in 1934 to cutoff, in a nonlinear fashion, the infinite electrostatic field generated by point charged particles, at a scale of order 10 to the -15 meters. Quickly shadowed by the success of Quantum Electrodynamics in the 40', this model has known a recent revival through the concept of Dirichlet-Branes in String Theory. Our asymptotic analysis starts by adding to the Born-Infeld equations the corresponding energy-momentum conservation laws and write the resulting system as a non-conservative symmetric system of ten first-order evolution PDEs. (Surprisingly enough, the extended system enjoys Galilean invariance, just like classical Fluid Mechanics.) Then, we show that four rescaled versions of the system have smooth solutions existing in the time interval where the corresponding limit problems have smooth solutions. These limits respectively describe the classical linear Maxwell equations (for weak fields), string motions (for large magnetic fields and moderate electric fields), particle motions (for large electromagnetic fields) and membrane motions. This is partly a joint work with Wen-An Yong, from Heidelberg.

(April 21) Yann Brenier String integration of some MHD equations

We first review the link between strings and some Magnetohydrodynamics equations. Typical examples are the Born-Infeld system, the Chaplygin gas equations and the shallow water MHD model. They arise in Physics at very different (from subatomic to cosmologic) scales. These models can be exactly integrated in one space dimension by solving the 1D wave equation and using the d'Alembert formula. We show how an elementary "string integrator" can be used to solve these MHD equations through dimensional splitting. A good control of the energy conservation is needed due to the repeted use of Lagrangian to Eulerian grid projections. Numerical simulations A good control of the energy conservation is needed due to the repeted use of Lagrangian to Eulerian grid projections. Numerical simulations in 1 and 2 dimensions will be shown.

(April 27) Daniel Coutand (Davis) Analysis of some interface and free surface problems

The first subject of my talk shall be on the well-posedness for the interface problem between a viscous fluid and an elastic solid. This is a two-phases problem, where each phase satisfies its own natural equation of evolution, and where the interaction between the two phases comes from the natural continuity of velocity field and normal stress across the unknown moving interface. The methods known in fluid moving boundary problems cannot handle the apparent incompatibility between the regularity of the two phases, which has led previous authors to consider the case where the solid satisfies a simplified law where the difficulties are not present. I shall present the new methods that where required in order to allow the treatment of classical elasticity laws (both linear and quasilinear hyperbolic) in this moving interface problem. I shall then explain how some of these ideas and some new tools preserving the transport structure of the Euler equations can provide the well-posedness for the free surface Euler equations with (or without) surface tension, without any restriction on the curl of the initial velocity.

(May 4) Ronghua Pan (Georgia Tech) On BV estimates for p-system with frictional damping

Abstract: In contrast to the success in nonlinear hyperbolic conservation laws, BV theory for nonlinear hyperbolic balance laws is widely open. Previous results require special structure of nonlinearity or strong dissipation from source terms. However, many physical systems have not yet been included in the existing results. Damped p-system is a simplest example where BV estimates are not available for generic small BV data. Based on some entropy techniques developed recently, we are able to solve this problem recently. A by-product verifies the validity of Darcy's law in large time asymptotic. This is a joint work with Constantine M. Dafermos.

(May 9) Bernardo Cockburn (Minneapolis) An adaptive method for Hamilton-Jacobi equations

In this talk we review our work on adaptive algorithms for Hamilton-Jacobi equations. We consider an adaptive version of the discontinuous Galerkin method for Hamilton-Jacobi equations proposed by Hu and Shu in 1999. It works as follows. Given the tolerance and the degree of the polynomial approximation of the approximate solution, the adaptive algorithm finds a mesh on which the approximate solution has a distance (in the unifrom norm) to the viscosity solution no bigger than the prescribed tolerance. The algorithm uses two main tools. The first is an a posteriori error estimate which is the main contribution of our work. The second is a new method that allows us to find a new mesh as a function of the old mesh and the ratio of the a posteriori error estimate to the tolerance. We display extensive numerical evidence that indicates that, for any given polynomial degree, the method achieves its goal with optimal complexity independently of the tolerance. This is joint work with Bayram Yenikaya (Sillicon Valley) and Yanlai Chen (U. of Minnesota).

Last modified: Mon Feb 20 11:02:31 EST 2006