CAREER: Statics and Dynamics of Singularities In Some Models From Material Science

New York University, New York NY

Investigators

Abstract

PI: Sylvia Serfaty, New York University DMS-0239121 This project, in the field of analysis and PDEs, is concerned with the analysis of statics and dynamics of singularities in some models from physics/material science. We have been and will continue to be particularly interested in two specific models: the Ginzburg-Landau model of superconductivity, and micromagnetics. In the area of micromagnetics, we studied previously some simplified two-dimensional model in the asymptotic regime corresponding to sharp "domain-wall" transitions. We established some optimal energy estimates, and exhibited optimal patterns for the transition profile, which fit very well with the "cross-tie wall" observations, and gave a first example of a non one-dimensional optimal profile in a vector-valued phase-transition problem. We intend to pursue this analysis towards more physically relevant models. In the area of the Ginzburg-Landau model, we previously focused on understanding the apparition, structure, and location, of vortices. We established a Gamma-convergence result deriving a limiting free-boundary problem, and proved convergence to some limiting vortex-densities, which we characterized, for minimizers as well as critical points. The project is to pursue further the analysis in the regime of high-applied fields, and turn towards the study of the associated dynamical models. We hope to establish the limiting dynamical laws obeyed by the vortices, first for a finite number of them, then for an infinite number of them, via some new estimates and an energetic approach to gradient-flow. Beyond that, we will be interested in understanding better the convergence of gradient flows for general Gamma-converging energies. This project, in the field of analysis and PDEs, is concerned with the analysis of statics and dynamics in some models from physics, in particular the Ginzburg-Landau model of superconductivity and microagnetics. In both cases the focus is on understanding, via rigorous mathematical proofs, the qualitative behavior of solutions to such models, and in particular explain the structure and the dynamics of singularities arising in some asymptotic regimes. The purpose of such research is thus two-fold. First to shed light on the understanding of the specific physical problems themselves: by obtaining asymptotic expansions and explicit formulas, one can explain physical experiments, confirm or disprove the validity of models, and by rigorous analysis one can one also derive reduced simplified models (for example reducing the dimension) that are easier to work and compute/simulate with. The second purpose is more mathematical. As mentioned, the underlying philosophy is to manage to reduce the original asymptotic problems to simpler limiting problems (of lower dimension) on which the core phenomena (here singularities) can be tracked down. This is the main philosophy of ``Gamma-convergence''. In order to perform such analysis, one needs to develop appropriate mathematical tools and provide the right settings to understand the phenomena. One also wishes to understand how much of the behavior is particular, and how much can be extended and understood as a more general mathematical phenomenon, with the hope that what is understood for one specific model may in turn help to understand others.