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PDE/Applied Math Seminar

Schedule for Spring 2008

Talks are Thursdays at 3:30 pm in room 3206 MTH (the Math department colloquium room) unless noted otherwise.

If you want to receive e-mail announcements of talks please contact Dionisios Margetis (dio@math.umd.edu).

(Click here for previous semesters' schedules.)

January 31	NO SEMINAR
Abstract:
February 7	Alina Chertock Department of Mathematics -- North Carolina State University A Particle Method for the EPDiff Equation
We consider a model of active fluid transport described by an evolutionary equation, known as the the Euler-Poincar´e (EPDiff) equation. The EPDiff equation arises in many scientific applications. In particular, it appears in the nonlinear dynamic of shallow water waves, and coincides, for example, with the Camassa-Holm equation of shallow water in 1-D and 2-D, and with the averaged template matching equation for computer vision in higher dimensions. The EPDiff singular solutions are contact discontinuities, called peakons. The key feature of the peakons is that they carry momentum; so the wave front interactions they represent are collisions, in which momentum is exchanged. This is very reminiscent to the KdV solitons behavior in 1-D. We numerically investigate the EPDiff dynamics of contact interactions using particle methods. We show that he discretization by means of the particle method preserves the basic Hamiltonian, the weak and variational structure of the original problem and respects the conservation laws associated to the symmetry under the Euclidean group. We use this symmetry to reduce the dimensionality of the system, and thus to simplify the theoretical and, especially, numerical analysis of the dynamics of peakons and their interactions. This is a joint work with Jerrold E. Marsden, California Institute of Technology.
February 14	Steve Shkoller -- TALK POSTPONED -- RESCHEDULED FOR MAY 1 Free-boundary problems for the incompressible and compressible Euler equations

February 21	RESERVED FOR THE N. WIENER CTR FOURIER TALKS
Abstract:
SPECIAL APPLIED MATH SEMINAR (Note special day and time) MONDAY February 25 Room MTH 3206, 3:00PM	Mac Hyman Los Alamos National Laboratory Paradigm Shifts in Science-Based Simulations
Today's scientific world is experiencing a paradigm shift where the sophistication of mathematical models, the accuracy and efficiency of numerical algorithms, the robustness of computer software, and the power of computation have become so great that numerical simulations are now considered a third pillar, along with theory and experiment, in the triad of tools used for scientific discovery. The rate of advances in these fields, and our ability to simulate complex physical systems, will increasingly be the limiting factors in our ability to solve many of our most pressing scientific challenges. I will describe recent advances in mathematical models, numerical algorithms, software, and hardware that have allowed computer simulations of complex multidisciplinary problems to have unprecedented impact in guiding scientific discoveries.
SPECIAL PDE/APPLIED MATH SEMINAR (Note special day and time) WEDNESDAY February 27 Room MTH 3206, 11:00AM	Diego Cordoba Department of Mathematics -- Institute of Mathematics and Fundamental Physics Consejo Superior de Investigaciones Científicas, Spain Interfase evolution: the Hele-Shaw and Muskat problem
We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy~Rs law. The free boundary is given by the differences between the densities and viscosities of the fluids. This physical scenario is known as the two dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local well-posedness in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.
February 28	Xiaoyi Zhang Institute for Advanced Study, Princeton The characterization of minimal mass blow up solution of focusing mass critical NLS.
Let u be a global solution to the focusing mass critical nonlinear schrodinger equation for radially symmetrical H^1 initial data with ground state mass in dimension bigger or equal to 4. We prove that if u does not scatter, then up to symmetries, u is the solitrary wave. This together with the results from F. Merle shows that the pseudo-conformal blow up and the solitary wave are the only two minimal mass blow up solutions.
March 6	Derek Moulton Department of Mathematical Sciences -- University of Delaware Mathematical modeling of electrostatic interactions with mean curvature surfaces
We investigate the interaction of static electric fields with surfaces of zero and constant mean curvature. Such systems have as their driving components surface tension and electrostatic forces. The interaction of these forces is of key importance in many applications, including micro- and nanoelectromechanical systems, self-assembly, nanolithography, and microfluidic processes. The partcicular system we explore involves a minimal surface catenoid membrane deflected by an axially symmetric electric field. A model is formulated via variational techniques to describe equilibrium shapes of the membrane. A detailed analysis of the general solution set is performed, with emphasis on stability and the effect of dimensionless parameters. Experimental analysis is performed using soap-film bridges and a high voltage power source. The general effect of the electric field is determined and the utility of an electric field in such systems examined.
March 13	OPEN
Abstract:
March 20	SPRING BREAK -- NO SEMINAR

March 27 Aziz Lecture	Richard James Dept. of Aerospace Engineering & Mechanics -- University of Minnesota Objective molecular dynamics
Perhaps the most important deformations in solid mechanics are those that represent the bending, twisting and extension of beams. The most important flows in fluid mechanics are viscometric flows. In both cases these are the motions that, when compared with the corresponding experiments, are used to measure the material constants. We give a universal molecular level interpretation of these motions. It is argued that all these motions are associated at molecular level with a time-dependent invariant manifold of the equations of molecular dynamics. The presence of this manifold can be used to simplify molecular-level computations. Its presence also suggests a modification of the principle of material frame-indifference, a cornerstone of nonlinear continuum mechanics. Interesting links to theories of turbulence, to the kinetic theory of gases (i.e., the Boltzmann equation), to the dynamics of nanostructures, and to the Langevin equation will be briefly discussed. Joint work with Traian Dumitrica and Kaushik Dayal.
April 3	Nikolaos Tzirakis Department of Mathematics -- University of Illinois at Urbana-Champaign Correlation Estimates and applications to Nonlinear Schroedinger equations
In this talk I will show how one can obtain new interaction Morawetz type (correlation) estimates in one and two dimensions. These estimates correspond to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. The framework is quite general and we can simultaneously obtain the global a priori estimates that have been obtained before in dimensions higher than three. In higher dimensions the proof uses commutator vector operators acting on the conservation laws of the equation. In one dimension we use the Gauss-Weierstrass summability method acting on the conservation laws. I will then show several applications of the new estimates to the semilinear Schroedinger equation.
JOINT STATISTICS/PDE/NWC EVENT April 10, Rm MATH 3206 (Note different, prolonged duration) 3:30PM-5:00PM	DOCUMENTARY FILM SCREENING: "Wolfgang Doeblin - a mathematician rediscovered'' a film by Agnes Handwerk and Harrie Willems See also this Statistics Announcement Duration: 82 min

April 17 APPLIED MATH COLLOQUIUM	Robert V. Kohn Courant Institute -- New York University Cloaking by Change of Variables
We say a region of space is "cloaked" with respect to electromagnetic measurements if its contents -- and even the existence of the cloak -- are inaccessible to such measurements. One recent proposal for achieving cloaking takes advantage of the coordinate-invariance of Maxwell's equations. I shall explain this scheme, including its mathematical basis and its apparent limitations.
April 24	OPEN
Abstract:
JOINT NUMERICAL ANALYSIS/PDE APPL. MATH SEMINAR (Note day and time) TUESDAY April 29 Room MTH 3206, 3:30PM--4:30PM	Houman Owhadi Applied & Computational Mathematics California Institute of Technology From Stochastic Variational Integrators to Ballistic Diffusion at Uniform Temperature
We present a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. Analogously to discrete mechanics and variational integrators, this theory leads to structure preserving numerical integrators for noisy mechanical systems by extremizing a discrete stochastic action. In this talk, we will focus on a particular one, designed to approximate the solutions of Langevin equations with uniform friction and noise. This new integrator is characterized by the following two properties: * It exactly preserves the exponential rate of decay of the symplectic form. * It inherits a near Boltzmann-Gibbs measure preservation property from the near energy preservation property of its associated noise/friction- free symplectic Euler integrator. By applying our integrator to a fluctuation-driven magnetic motor we exhibit a paradigm for isothermal, mechanical rectification of stochastic fluctuations. The central idea is to transform energy injected by (random/thermal) fluctuations into rigid-body rotational kinetic energy. We show that although directed motion is not possible at uniform temperature, thermal noise can be used to obtain ballistic diffusion. This is a joint work with Nawaf Bou-Rabee.
May 1	Steve Shkoller Department of Mathematics -- University of California, Davis Free-boundary problems for the incompressible and compressible Euler equations
We describe a new method for treating free boundary problems in perfect incompressible and compressible fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface 3D Euler equations for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of a perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order. We will describe the geometry behind the a priori nonlinear estimates and the approximation schemes that must be developed in order to prove existence of solutions. This is joint work with D. Coutand and H. Lindblad.
May 8	OPEN
Abstract:
May 15	Henrique Versieux Instituto de Matematica Pura e Aplicada (IMPA), Brazil Numerical analysis of a steepest-descent PDE model for surface relaxation below the roughening temperature
We study the numerical solution of a PDE describing the relaxation of a crystal surface to a flat facet. The PDE is a singular, nonlinear, fourth order evolution equation. It can be seen as the gradient flow of a convex but non-smooth energy with respect to the $H^{-1}$ inner product. Our numerical scheme uses implicit discretization in time and a mixed finite-element approximation in space. The singular character of the energy is handled using regularization, combined with a primal-dual method that remains robust as the regularization parameter tends to zero. We study the convergence of this scheme, both theoretically and numerically.

How to reach the Math Department by car and public transportation

Special accomodations for individuals with disabilities can be made by calling in advance (301) 405-5048. It would be appreciated if we are notified at least one week in advance.

For further information contact D. Margetis at dio@math.umd.edu.