Department of Mathematical Sciences

Colloquium Series

Purpose: To discuss mathematical research and interdisciplinary problems in all areas of mathematics and applied mathematics. Talks are welcome from faculty, graduate students, and outside speakers from academia and industry. Talks should be at a level accessible to graduate students. Students and faculty at all levels and from all departments are welcome to attend.

Date/Time Speaker Title/Abstract
Friday, March 18

3-4pm The MAC

Dr. Martin E. Glicksman

College of Engineering

Florida Institute of Technology

Title: Deterministic Pattern Formation

Abstract: The accepted physical basis for analyzing diffusion-limited pattern formation and related multiphase parabolic free-boundary problems is "selective amplification of noise". The appeal to noise modulation arises from the fact that crystallizing fluids, say water to ice, possess no obvious structural information at scales above the microscopic level of short-range molecular order, typically ~10-9 m. Familiar diffusion patterns, commonly observed in snowflakes, minerals, metallic dendrites and crystallized sub-structures, develop instead on much larger (mesoscopic) scales--hundreds, to millions, of times larger than molecules. Today's stochastic paradigm, i.e., pattern-formation-by-noise, is claimed as the only obvious 'source' of pattern-forming information available on suitable time and length scales. J.S. Langer's review, "Instabilities and pattern formation in crystal growth," Rev. Mod. Phys., 52, 1980, suggested that diffusion patterns are noise-mediated instabilities, and K. Kassner's monograph, "Pattern Formation in Diffusion-Limited Crystal Growth", World Scientific, Singapore, 1996, specifically supports micro-solvability, a pattern selection mechanism that choses from discrete dynamic free-boundary solutions on the basis of mathematical 'solvability.' In 2011, the speaker suggested that a ubiquitous natural energy field is responsible for 4th-order perturbations that deterministically stimulate free boundaries to develop patterns. Leibniz-Reynolds analysis shows explicitly that this perturbation field is (minus) the divergence of the capillary-mediated tangential flux, which arises on both moving and stationary solid-liquid interfaces in 2-D and 3-D. Such fields cause geometric responses (curvature changes) in response to variations in the interface potential. This thermodynamic action-reaction represents an unusual, but classical, variant of the Le Chatelier-Braun effect, where thermal departures from local equilibrium induce spontaneous geometric changes that non-linearly redirect an evolving system toward local equilibrium. Simulations of patterns using thermodynamically-consistent numerical models confirm quantitatively that inflection, branching, invagination, and splitting enhance pattern complexity, which in 2-D initiate at field roots, or 'Laplace points,' which may be derived for specified shapes using Reynolds theorem and field theory.

Friday, April 1

3-4pm The MAC

Dr. Veronica Quitalo

Department of Mathematics

Purdue University

SIAM Speaker

Title: Systems of partial differential equations arising from population dynamics and neuroscience: Free boundary problems as a result of segregation.

Abstract: In this talk we will present our latest results for two phase free boundary problems arising from population dynamics. We will focus on fully nonlinear systems with local interaction and linear systems with a (non local) long range interaction. In the long range model, the growth of a population ui at x is inhibited by the populations uj in a full area surrounding x. This will force the populations to stay at distance 1 from each other in the limit configuration, so the free boundary will be a strip along the support of the population with size exactly one. This is a joint work with Luis Caffarelli and Stefania Patrizi. We will also motivate the need of a system of partial differential equations that models the propagation of activity in the brain, ncorporating the role of the neurons as well as the volume propagation. This is a joint work with Aaron Yip and Zoltan Nadasdy.

Friday, April 29

3-4pm The MAC

Jonathan Goldfarb

Department of Mathematical Sciences

Florida Institute of Technology

Title: On the Optimal Control of the Free Boundary Problems for the Second Order Parabolic Equations

Abstract: We consider an inverse Stefan type free boundary problem (ISP), where information on the boundary heat flux and the density of heat sources are missing and must be found along with the temperature and free boundary. In this work, new variational formulation introduced in U.G. Abdulla, Inverse Problems and Imaging, 7, 2(2013), 307--340, is extended to a new setting and an optimal control problem is pursued where boundary heat flux, density of sources and the free boundary are components of the control vector, and the optimality criteria consists of the minimization of the L_2-norm declinations of the temperature measurement at the final moment, phase transition temperature, and final position of the free boundary. Existence of the optimal control and the Frechet differentiability of the cost functional in Besov-Hilbert-Sobolev spaces is proved under the minimal regularity assumptions on the data. The adjoint PDE problem is introduced and the explicit formula for the Frechet differential is derived. This result opens a way for the application of the iterative gradient type numerical methods of least computational cost in Hilbert spaces framework. Discretization through finite differences is pursued and the convergence of the sequence of discrete optimal control problems to the continuous optimal control problem both with respect to functional and control is proved. The major tool is the proof of two energy estimates in discrete Sobolev spaces for the semi-discrete PDE problem. The new approach allows one to tackle situations when the known density of sources is a measure given through the distributional derivative of an integrable function.

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