2009-2010 Analysis and Applied Math Seminar

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Of related interest in Toronto (and sometimes cross-listed):

The seminar meets regulary on Fridays, 2:10-3pm, 6183 Bahen Center.

For more information, please contact one of the organizers: Almut Burchard (almut [at] math), James Colliander (colliand [at] math), Gideon Simpson (simpson [at] math), If you will be speaking, details on how to use the equipment in BA 6183 can be found here.

Previous Year's Seminars: 2008-09, 2007-08, 2006-07, 2005-06, 2004-05, 2003-04, 2002-03.


Contents

Calendar

January 27 (2010), Carlen, 14:10 - 15:00. Fields Institute Room 230 Fields Colloquium in Applied Mathematics

Title: Rate of relaxation to stable profiles for some fourth order evolution equations equations
Abstract: We will explain recent work on obtaining strong stability results, with rate of relaxation bounds, on stationary profiles for a class of forth order equations of thin film and Cahn-Hilliard type. The talk is based on joint work with Carvalho, Orlandi, and Suleyman.

January 8 (2010), Ivrii, 14:10 - 15:00. BA6183

Title: 2D Schrödinger with a strong magnetic field: dynamics and spectral asymptotics near boundary
Abstract: We consider Magnetic Schrödinger operator H=\bigl(h\nabla - \mu \mathbf{A}(x)\bigr)^2 +V(x), \qquad \mathbf{A}(x)=(-\frac{1}{2}x_2, \frac{1}{2}x_1) (or more general one) and derive rather sharp asymptotics of \int \psi (x)\, e(x,x,0)\,dx as \mu\to\infty and h\to 0 where (e(x,y,λ) is a Schwartz kernel of spectral projector of H and ψ(x) is a cut-off function.
Corresponding classical dynamics associaated with operator in question inside of domain is a cyclotron movement along circles of radius \asymp \mu^{-1} combined with slow drift movement (with a speed \asymp \mu^{-1}) along level lines of V(x).
However near boundary dynamics consists of hops along it; this hop dynamics could be torn away from the boundary and become an inner dynamics and v.v.
This classical dynamics has profound implications for spectral asymptotics (with remainder estimate better than O(h − 1) and up to O(μ − 1h − 1)).
We consider also the case of superstrong magnetic field (as \mu h\ge 1) when classical dynamics is at least applicable but the difference between Dirichlet and Neumann boundary conditions are the most drastic.

December 3, Chow, 16:10 - 17:00. McLennan 102, Physics/Fields Colloquium

Title: The physics of obesity
Abstract: The past few decades have seen a surge in the incidence of obesity in the developed world. Changes in body weight that can lead to obesity are known to result from imbalances between the energy derived from food and the energy expended to maintain life and perform physical work. However, measuring and quantifying this relationship has proved to be difficult. Here, I will show how simple ideas from thermodynamics and nonlinear dynamics can be used to provide a general theoretical description of how body weight will change over time. The theory can then be used to answer open questions (and dispel some myths) regarding weight loss and gain.

December 2, Chow, 15:10 - 16:00. Fields Institute, Physics/Fields Colloquium

Title: Kinetic theory of coupled oscillators
Abstract: Coupled oscillators arise in contexts as diverse as the brain, synchronized flashing of fireflies, coupled Josephson junctions, or unstable modes of the Millennium bridge in London. Generally, such systems are only analysed for a small number of oscillators or in the infinite oscillator, mean-field limit. The dynamics of a large but finite network of coupled oscillators are largely unknown. Here, I will show how concepts from the kinetic theory of gases and plasmas can be applied to a system of coupled oscillators to infer the large scale collective behavior from the small scale dynamics. Calculations are facilitated by perturbative methods developed for quantum field theory.

November 27, Metayer, 13:10 - 14:00. Bahen 6183

Title: Rheology of confined granular flows: scale

invariance, glass transition and friction weakening

Abstract: We study fully-developed, steady granular flows confined between parallel flat frictional sidewalls using experiments and numerical simulations. Above a critical rate, sidewall friction on the flow stabilizes the underlying heap at an inclination larger than the angle of repose. The shear rate is constant and independent of inclination over much a flowing layer. In the direction normal to the free surface, the solid volume fraction increases on a characteristic scale equal to half the flowing layer depth. Beneath a critical depth at which internal friction is invariant, grains exhibit creeping and intermittent cage motion similar to that in glasses, causing gradual weakening of friction at the walls.

November 27, Galvao-Sousa, 14:10 - 15:00. Bahen 6183

Title: Variational Methods for Phase Transitions
Abstract: In this talk, I will present some results in phase transitions. First, for the liquid-liquid phase transitions, we study the effect of taking higher-order terms and transitions on the boundary as well as in the interior. Then, I will show some results on a thin-film model for solid-solid phase transitions.

November 20, Zworski, 14:10 - 15:00. Bahen 6183

Title: Effective dynamics of double solitons for perturbed mKdV
Abstract: We show that an interacting double soliton solution to the perturbed mKdV equation is close in $H^2$ to a double soliton following an effective dynamics obtained as the Hamilton's equations for the restriction of the mKdV Hamiltonian to the submanifold of solitons. The interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to soliton stability is central in the proof. (joint work with J Holmer and G Perelman)

November 6, Oh, 14:10 - 15:00. Bahen 6183

Title: KdV with measures as initial data, and stochastic KdV (SKdV) with additive and multiplicative noise
Abstract: Bourgain '97 proved global well-posedness of the periodic KdV with measures as initial data, assuming that the total variation is sufficiently small. His argument was based on the nonlinear analysis on the second iteration of the integral formulation, assuming an a priori bound on the Fourier coefficients. With the complete integrability of KdV, he then proved such an a priori control.
In this talk, we first discuss the nonlinear analysis on the second iteration without the complete integrability or smallness assumption on the total variation. This answers a question posed by Bourgain at least in the local-in-time setting. Then, using a stochastic version of such nonlinear analysis, we discuss local well-posedness of SKdV with additive space-time white noise. Finally, we consider SKdV with multiplicative noise in L^2(T). By a sequence of transformations, we reduce it to a system of mean-zero SKdV and a SDE (for the mean of the original solution), which is then shown to be globally well-posed.

November 4, Lieb, 14:10 - 15:00. Fields Institute Room 230 Fields Colloquium in Applied Mathematics

Title: A second look at the second law of thermodynamics
Abstract: The increase of entropy was regarded as perhaps the most perfect and unassailable law in physics and it was even supposed to have philosophical import. Einstein, like most physicists of his time, regarded the second law of thermodynamics as one of the major achievements of the field, and it entered his work in several ways. The essence of the second law is the statement that all processes can be quantified by an entropy function whose increase is a necessary and sufficient condition for a process to occur. As a fundamental physical law no deviation, however tiny, is permitted and its consequences are far-reaching. Current wisdom regards the second law as a consequence of statistical mechanics but the entropy principle, which was discovered before statistical mechanics was invented, ought to be derivable from a few logical principles without recourse to Carnot cycles, ideal gases and other assumptions about such things as 'heat', 'hot' and 'cold', 'temperature', 'reversible processes', etc. Like conservation of energy (the first law), the existence of a law so precise and so model-independent must have a logical foundation that is independent of the details of the constitution of matter. In this lecture the foundations of the subject and the construction (with J. Yngvason) of entropy from a few simple principles will be presented. (No previous familiarity with the subject is required.)
A summary can be found in: "A Guide to Entropy and the Second Law of Thermodynamics", Notices of the Amer. Math. Soc. vol 45 571-581 (1998). This paper received the American Mathematical Society 2002 Levi Conant prize for ``the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding five years.

October 30, Zwiers, 14:10 - 15:00, BA6183

Title: Blowup of the Cubic Focusing Nonlinear Schrodinger Equation on a Ring
Abstract: I prove there exist solutions to the three-dimensional cubic focusing nonlinear Schrodinger equation that blowup on a circle, in the sense of L2 concentration on a ring, bounded H1 norm outside any surrounding toroid, and growth of the global H1 norm with the log-log rate.
My proof follows a bootstrapping scheme. I will argue that, when there is sufficient decay outside a surrounding toroid, the dynamic near the circle of concentration is essentially two-dimensional. For appropriate data, I show the robust two-dimensional log-log blowup regime occurs. Merle & Raphael's precise description of this singular behaviour allows me to prove an unusual persistance of regularity away from the circle of concentration, which re-establishes the sufficient decay outside the surrounding toroid. The sticking points in this argument will be exposed and discussed.

October 23, Lee, 14:10 - 15:00, BA6183

Title: Continuity of Optimal Control Costs and its application to Weak KAM Theory
Abstract: In this talk, we will discuss a version of weak KAM theorem corresponding to certain optimal control problems. The proof of such theorem relies on the continuity of the corresponding optimal control cost. I will give conditions which guarantees this continuity and show that the condition is sharp. This is a joint work with A. Agrachev.

October 21, Spencer, 16:10 - 17:00, BA6183, UofT Math Colloquium

Title: Statistical Mechanics, Random Band Matrices and Hyperbolic Symmetry
Abstract: Random band matrices are a generalization of random matrices in which matrix elements are concentrated in a band about the diagonal. In physics, spectral properties of these matrices are reformulated in terms of certain statistical mechanics models with hyperbolic symmetry. This talk will discuss a simplified version of one of these models. The simpler model has the advantage that it is closely related to a random walk in a highly correlated random environment. In 3 dimensions, a phase transition is proved reflecting a spectral transition for certain matrix ensembles. This is joint work with M. Disertori and M. Zirnbauer.

October 16, Abou-Salem, 14:10 - 15:00. Bahen 6183

Title: Dimensional reduction of the mean-field dynamics of bosons in strongly anisotropic harmonic potentials
Abstract: I discuss recent results on the spatial dimensional reduction of the effective mean-field dynamics of many-body bosonic systems in strongly anisotropic harmonic potentials. In particular, the dynamics in the limit of strong anisotropy is effectively described by the nonlinear Hartree equation that is restricted to Euclidean submanifold of the original configuration space. I also discuss extending the analysis to complete Riemannian submanifolds whose Ricci curvature is bounded from below.

October 13, Moody, 14:10 – 15:00. Bahen 6183 Fields Colloquium in Applied Mathematics

Title: Symmetry, diffraction, and the homometry problem
Abstract: Diffraction has been the mainstay of experimental crystallography for nearly a hundred years. Recent interest in quasicrystals and aperiodic tilings has brought fresh insights into the nature of diffraction and its relation to symmetry, especially in the case of pure point diffraction.
In this talk I will try to make a case for diffraction as an encoding of symmetry and then delve into the famous inverse problem of unravelling the information about a structure from information about its diffraction.
The diffraction is a measure. Which pure point measures can occur as diffraction patterns and given such a measure how does one find and classify all the structures that could have produced it? This is the homometry problem. In answering it we arrive naturally in the setting of certain stochastic processes. The complexity of the classification revolves around the set of extinctions in the diffraction.
The talk will be aimed at a general mathematical audience.

October 9, Ball, 14:10 – 15:00. Bahen 6183

Title: The Q-tensor theory of liquid crystals
Abstract: The lecture will survey what is known about the mathematics of the de Gennes Q-tensor theory for describing nematic liquid crystals. This theory, despite its popularity with physicists, has been little studied by mathematicians and poses many interesting questions. In particular the lecture will describe the relation of the theory to other theories of liquid crystals, specifically those of Oseen-Frank and Onsager/Maier-Saupe. This is joint work with Apala Majumdar and Arghir Zarnescu.

October 2, Fried, Friday 14:10-15:00. Bahen 6183

  • Eliot Fried (McGill University)
Title: Features and challenges of the Navier--Stokes-αβ equations
Abstract: The Navier--Stokes-α equation regularizes the Navier--Stokes equation by including additional dispersive and dissipative terms. The former term is proportional to the divergence of the corotational time-rate of the symmetric part of the gradient of the filtered velocity. The latter term is proportional to the bi-Laplacian of the filtered velocity. Both terms involve factors of α2, where, roughly, α represents the characteristic size of the smallest resolvable eddy.
Combining dispersion and dissipation yields a model with at least some attractive features. In particular, the Navier--Stokes-α equation possesses circulation properties analogous to those of the Navier--Stokes equation and allows for simulations with less artificial damping than those arising from more conventional subgrid-scale and Reynolds stress models.
One drawback concerns boundary conditions. Except for flows in periodic domains, the additional dissipative term entering the Navier-- Stokes-α equation leads to the need for additional boundary conditions. Unfortunately, the averaging method used to derive the Navier--Stokes-α equation does not provide such conditions. The absence of physically meaningful boundary conditions limits the applicability of the model.
Using a framework for fluid-dynamical theories with gradient dependencies, we have derived a flow equation that includes the Navier--Stokes-α equation as a special case. Aside from α, this equation involves an additional length scale β. For β = α, our flow equation reduces to the Navier--Stokes-α equation. Our formulation also yields boundary conditions at walls and free surfaces.
We will consider the effects of α and β on the energy spectrum and the alignment between the filtered vorticity and the eigenvalues of the filtered stretching tensor in three-dimensional homogeneous and isotropic turbulent flows in a periodic cubic domain, including the limiting cases of the Navier--Stokes-α and Navier--Stokes equations. We will also discuss open mathematical challenges associated with the model.

September 18, Eden, Friday 14:10-15:00. Bahen 6183

  • Alp Eden (Bogazici University, Itanbul)
Title: From a generalized Davey-Stewartson system to the almost cubic nonlinear Schrödinger equation
Abstract: In this talk, I will try to highlight some of the central results in our work on a generalized Davey-Stewartson system. In the purely elliptic case, the generalized Davey-Stewartson system can also be considered as an almost cubic nonlinear Schrödinger equation that shares many similarities with the two dimensional cubic nonlinear Schrödinger equation. The class of almost cubic nonlinear Schrödinger equations also includes the usual Davey-Stewartson system in the elliptic-elliptic case, as well as some cases of the Zakharov-Schulmann equations.
The usual analysis of the cubic nonlinear Schrödinger equation rests on how the local cubic nonlinearity acts between different function spaces allowing the use of the Strichartz inequalities. Although the nonlinearity of the almost cubic nonlinear Schrödinger equation is non-local in nature, its similar action on various function spaces allow a similar analysis for the local well-posedness in various Sobolev spaces. The problem of demarcation of the focusing and defocusing cases of the generalized Davey-Stewartson system when viewed this way becomes a much more transparent problem. It is no surprise that this is also the demarcation for the existence of standing waves, whose existence will also be discussed.

July 02, Abou-Salem, Thursday 14:10 - 15:00, Bahen 6183

Title: Mean Field Bosons in Confining Traps
Notes
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