Department of Mathematics

Details

Global Minimization and the Energy Landscape for a Variational Problem with Long-Range Interactions

by Rustum Choksi | McGill University
Time: 13:10  (Friday, Apr. 13, 2012)
Location: BA6183, Bahen Center, 40 St. George St.
Abstract:
Energy-driven pattern formation induced by competing short and long- range interactions is common in many physical systems. A nonlocal perturbation (of Coulombic-type) to the well-known Ginzburg- Landau/Cahn-Hilliard free energy gives rise to a mathematical paradigm with a rich and complex energy landscape. In the first half of this talk, we discuss rigorous asymptotic results concerning global minimizers. In the second part, we discuss a few hybrid numerical methods for accessing ground states.

Dates in this series

· Friday, Jan. 08, 2010: Temporal evolution of attractive Bose-Einstein condensate in a quasi 1D cigar-shape trap modeled through the semiclassical limit of the focusing Nonlinear Schrödinger Equation (Alex Tovbis)

· Friday, Jan. 15, 2010: Diffusion of waves in a random environment: problems and results (Jeffrey Schenker)

· Friday, Jan. 22, 2010: The Calderón Problem - From the Past to the Present (Leo Tzou)

· Friday, Jan. 22, 2010: 2D Schrödinger with a strong magnetic field: dynamics and spectral asymptotics near boundary  (Victor Ivrii)

· Friday, Feb. 19, 2010: Global existence for a free boundary problem with non-standard sources (Maria Gualdani)

· Monday, Feb. 22, 2010: Finding the Three Dimensional Structure of Molecules That We Don’t Know How To Crystallize (Shamgar Gurevich)

· Friday, Feb. 26, 2010: On Long Time Behavior of Coating Flow (Marina Chugunova)

· Friday, Mar. 05, 2010: Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation (Leonid Berlyand)

· Friday, Mar. 26, 2010: Eventual regularization of the slightly supercritical fractional Burgers equation (Chi Hin Chan)

· Friday, Apr. 09, 2010: On the solvability conditions for some non Fredholm operators (Vitali Vougalter)

· Tuesday, Apr. 13, 2010: Some problems in unique continuation (Alex Ionescu)

· Friday, Apr. 16, 2010: Smoother critical subsolutions to Hamilton-Jacobi Equation (Albert Fathi)

· Friday, Apr. 16, 2010: Entropy method for line energies (Radu Ignat)

· Friday, May. 14, 2010: Semiclassical evolution for the nonlinear Schrödinger equation (Alessandro Selvitella)

· Thursday, Jul. 08, 2010: Improving sharp Hardy-Sobolev inequalities by optimal remainder norms (Adele Ferone)

· Friday, Sep. 03, 2010: Ergodic methods in combinatorics (Nikos Frantzikinakis)

· Friday, Sep. 24, 2010: Stability of Federer curvature measures (Quentin Merigot)

· Friday, Oct. 01, 2010: Regularity of the extremal solution in fourth order nonlinear eigenvalue problems,  (Amir Moradifam)

· Friday, Oct. 08, 2010: Non-uniqueness of the Navier-Stokes equation in the hyperbolic setting (Magda Czubak)

· Friday, Oct. 15, 2010: 2D- and 3D-Magnetic Schrödinger Operator: Short Loops and Pointwise Spectral Asymptotics (Victor Ivrii)

· Friday, Oct. 22, 2010: Rigidity of Sobolev isometric embeddings (Robert Jerrard)

· Friday, Oct. 29, 2010: Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem (Gustav Holzegel)

· Friday, Nov. 05, 2010: Bursting shear flow as cycle-to-cycle homoclinic chaos (Lennaert van Veen)

· Friday, Nov. 05, 2010: Localization for the random displacement model (Michael Loss)

· Friday, Nov. 12, 2010: Double Bubbles from Euclidean to Gauss Space (Frank Morgan)

· Friday, Nov. 19, 2010: Coarsening in energy-driven systems (Dejan Slepcev)

· Friday, Nov. 26, 2010: Hartree-Fock theory for atoms with closed shells (Marcel Griesemer)

· Monday, Nov. 29, 2010: Normally hyperbolic trapped sets and quasinormal modes for black holes (Maciej Zworski)

· Friday, Dec. 03, 2010: Interaction between internal and surface waves in a two layers fluid (Catherine Sulem)

· Friday, Dec. 17, 2010: The influence of large drift on front propagation (Mohammad El Smaily)

· Friday, Jan. 07, 2011: Mean field limits for interacting Bose gases and the Cauchy problem for Gross-Pitaevskii hierarchies (Thomas Chen)

· Friday, Feb. 04, 2011: Some problems concerning a Quasilinear Schrödinger Equation (Alessandro Selvitella)

· Friday, Feb. 11, 2011: Energy-based fracture evolution (Christopher Larsen)

· Friday, Feb. 18, 2011: On global well-posedness of defocusing energy-critical nonlinear Schrödinger equations on certain Riemannian manifolds (Alex Ionescu)

· Friday, Mar. 04, 2011: Infinitesimal inverse spectral geometry and applications in mathematical physics,  (Achim Kempf)

· Friday, Mar. 11, 2011: Numerical study of one-dimensional Bose-Einstein condensates in a random potential (Ziad Musslimani)

· Friday, Mar. 18, 2011: New conserved integrals for fluid flow in multi-dimensions (Stephen Anco)

· Friday, Apr. 08, 2011: Renormalization group approach to perturbation theory for PDEs (Walid K. Abou Salem)

· Friday, Jun. 10, 2011: Extensions of the disc algebra (Vassili Nestoridis)

· Friday, Jun. 17, 2011: Stability of planar fronts for a non--local phase kinetics equation with a conservation law in $D \le 3$ (Enza Orlandi)

· Friday, Jul. 15, 2011: Damped-driven Hamiltonian PDE (Sergei Kuksin)

· Friday, Jul. 15, 2011: Self-similar asymptotics of solutions to the Navier-Stokes system in two dimensional exterior domain (Christophe Lacave)

· Friday, Jul. 22, 2011: The theorem of characterization of the maximum principle (Julián López-Gómez)

· Friday, Sep. 09, 2011: A semiclassical proof of Quillen's theorem (Maciej Zworski)

· Friday, Sep. 16, 2011: Rayleigh-Benard convection: Bounds on the Nusselt number (Christian Seis)

· Friday, Sep. 16, 2011: Bounds on the cost of root finding  (Scott Sutherland)

· Friday, Sep. 23, 2011: Thin films for the Ginzburg-Landau model (Bernardo Galvao-Sousa)

· Friday, Sep. 30, 2011: The signature operator on stratified pseudomanifolds (Pierre Albin)

· Friday, Oct. 07, 2011: Breakdown Criteria for Nonvacuum Einstein Equations (Arick Shao)

· Friday, Oct. 14, 2011: Local well-posedness of the KdV equation with almost periodic initial data (Kotaro Tsugawa)

· Friday, Oct. 21, 2011: Anderson localization triggered by spin disorder (Daniel Egli)

· Friday, Oct. 28, 2011: From the Analysis of Einstein-Maxwell Spacetimes in General Relativity to Gravitational Radiation (Lydia Bierri)

· Friday, Nov. 11, 2011: Regularity of Eigenstates in Regular Mourre Theory (Matthias Westrich)

· Friday, Nov. 18, 2011: Sharp Stability Estimates in Geometric Variational Problems (Francesco Maggi)

· Friday, Nov. 25, 2011: Inverting the Attenuated X-Ray Transform (Nicholas Hoell)

· Friday, Dec. 02, 2011: Developments in High-Order Discontinuous Galerkin Methods for Hyperbolic Conservation Laws (Lilia Krivodonova)

· Monday, Jan. 23, 2012: Wet drop impact (Robert Deegan)

· Friday, Jan. 27, 2012: Flux norm approach to finite-dimensional homogenization approximation with non-separated scales and high contrast (Leonid Berlyand)

· Friday, Feb. 03, 2012: Critical Sobolev Inequalities and Navier-Stokes Equations (Yun Wang)

· Friday, Feb. 10, 2012: On Singularity Formation Under Mean Curvature Flow (Gang Zhou)

· Friday, Feb. 17, 2012: Integro-differential equations and multiscale representations (Prashant Athavale)

· Friday, Mar. 02, 2012: Pursuit Laws for Mobile Robots (Bruce Francis)

· Friday, Mar. 16, 2012: Nonzero positive solutions of systems of elliptic boundary value problems (Kunquan Lan)

· Friday, Mar. 23, 2012: Wet drop impact (Robert Deegan)

· Friday, Apr. 13, 2012: Global Minimization and the Energy Landscape for a Variational Problem with Long-Range Interactions (Rustum Choksi)

· Friday, Jun. 01, 2012: Information Transmission over Optical Fibers Using the Nonlinear Fourier (Mansoor Yousefi)

· Tuesday, Jul. 31, 2012: The Kato square root problem on vector bundles with generalised bounded geometry (Lashi Bandara)

· Friday, Sep. 28, 2012: On a variational problem exhibiting concentration near a curve (Andres Contreras)

· Friday, Oct. 05, 2012: Global results for linear waves on expanding Schwarzschild de Sitter cosmologies (Volker Schlue)

· Friday, Oct. 12, 2012: Hamiltonian dynamics of a particle interacting with a wave field (Daniel Egli)

· Friday, Oct. 19, 2012: Null Cones to Infinity, Curvature Flux, and Bondi Mass (Arick Shao)

· Friday, Oct. 26, 2012: Coupled Nonlinear Oscillators: Linking Dynamical Systems Theory with Engineering Applications, (Antonio Palacios)

· Friday, Nov. 02, 2012: Short-time behaviour of hypoelliptic heat kernels on Lie groups (Joint Seminar with Inverse Problems and Image Analysis Seminar) (Abdol-Reza Mansouri)

· Friday, Nov. 09, 2012: Harmonic maps of conic surfaces (Jesse Gell-Redman)

· Friday, Nov. 16, 2012: Monge-Kantorovich for Problems in Signal Processing and Control (Joint with the Inverse Problems and Image Analysis Seminar) (Allen Tannenbaum)

· Friday, Nov. 23, 2012: TBA (Yaiza Canzani)

· Friday, Dec. 07, 2012: TBA (Walter Craig)

· Friday, Dec. 07, 2012: Vortex Filament Interactions (Walter Craig)

· Friday, Dec. 14, 2012: JOINT SEMINAR: ANALYSIS AND APPLIED MATH / INVERSE PROBLEMS AND IMAGE ANALYSIS SEMINAR -- The circular area signature for graphs of periodic functions (Jeffrey Calder)

· Friday, Jan. 11, 2013: Higher-order time asymptotics of fast diffusion in Euclidean space (using dynamical systems methods) (Robert McCann)

· Friday, Jan. 18, 2013: A Differential Equation For The Principal Directions And Radii Of A Surface (Daniel Mayost)

· Friday, Jan. 25, 2013: Axisymmetric critical points on the sphere of a nonlocal isoperimetric problem (Ihsan Topaloglu)

· Friday, Feb. 01, 2013: Blow-up for linear and non-linear wave equations on black hole spacetimes (Stefanos Aretakis)

· Friday, Feb. 08, 2013: Null singularities in general relativity (Jonathan Luk)

· Friday, Mar. 01, 2013: Continuous dissipative solutions of the Euler equations (Camillo De Lellis)

· Friday, Mar. 22, 2013: TBA (Keisuke Takasao)