LCDS Seminars 2008/2009

Monday, May 4, 2009

Lefschetz Center for Dynamical Systems Seminar

Abstract: The growth and form of a soft solid pose a range of problems that combine aspects of geometry and physics. I will discuss two examples of growth and form in the plant world motivated by qualitative and quantitative biological observations in the common Lily: (i) the shape of a freely growing pollen tube, and (ii) the undulating fringes on a leaf or petal. In each case, we will see how a combination of physical experiments, mathematical models and computations allow us to unravel the basis for the diversity and complexity of form and sharpens the search for structure.

Wednesday, April 29, 2009

Lefschetz Center for Dynamical Systems Seminar

Abstract: In order to describe slow modulations in time and space of spatially periodic solutions of pattern forming reaction-diffusion systems so-called phase diffusion equations and Cahn-Hilliard equations can be derived as formal approximation equations via multiple scaling analysis. If these phase diffusion equations are degenerate, there exist solutions showing a waiting time phenomenon. An example is the porous medium equation which can be derived as a degenerate phase diffusion equation for modulations of spatially periodic solutions of the real Ginzburg-Landau equation which have wave numbers close to the boundaries of the so-called Eckhaus-stable region. With the help of estimates between the formal approximations and the exact solutions of the original system we explain the extent to which these formal approximations are valid in different length and time scales. Furthermore, we show in which sense waiting time phenomena can occur in pattern forming systems.

Tuesday, April 28, 2009

Lefschetz Center for Dynamical Systems Seminar

Abstract: We consider one of the simplest possible models of non-equilibrium statistical mechanics: two coupled oscillators in contact with two Langevin heat baths. The twist is that one of the heat baths is at "infinite" temperature in the sense that no friction acts on the corresponding degree of freedom. We explore the question of the existence of a stationary state in this situation and its properties if it exists. In particular, we will see that the question "Is the corresponding degree of freedom at infinite temperature?" can have a surprising variety of answers.

Monday, April 27, 2009

Lefschetz Center for Dynamical Systems Seminar

Monday, April 20, 2009

Joint Lefschetz Center for Dynamical Systems/PDE Seminar

Wednesday, April 1, 2009

Boston University/Brown University PDE Seminar

Abstract: We prove the W_p^1,2-solvability of second order parabolic equations in nondivergence form in the whole space for p∈ (1,∞). The leading coefficients are assumed to be measurable in one spatial direction and have vanishing mean oscillation (VMO) in the orthogonal directions and the time variable in each small parabolic cylinder with the direction depending on the cylinder. This extends a recent result by Krylov for elliptic equations and removes the restriction that p>2.

Wednesday, April 1, 2009

Boston University/Brown University PDE Seminar

Abstract: I will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. I will begin by giving a precise account of the formulation of the surface water-wave problem and discussion of its distinct features. They include the dispersion relation, the nonlinearity, traveling waves and the Hamiltonian structure. I will describe the recent work of Hans Christianson, Gigliola Staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. If time permits, I will explore some open questions regarding long-time behavior and stability.

Monday, March 30, 2009

Lefschetz Center for Dynamical Systems Seminar

Abstract: We examine the topology and morphology of interfaces produced following phase separation in systems with conserved and nonconserved order parameters. These processes produce complex bicontinuous phases that have interfaces with spatially varying curvature. The conserved order parameter evolves by the Cahn-Hilliard equation and the nonconserved order parameter evolves by the Allen-Cahn equation. The morphology of these interfaces is determined using the interfacial shape distribution, the probability of finding a patch of interface with a given pair of principal curvatures, and the topology is quantified by the genus. The interface shape distribution is a function of the volume fraction of the phases and the dynamics used to produce the structures. However, these structures have the same scaled-genus, suggesting a universal value of the genus for a system undergoing self-similar coarsening. We also characterize the spatial correlations of the interfacial curvature. This analysis has indentified new characteristic length scales of these complex structures.

Wednesday, March 4, 2009

Boston University/Brown University PDE Seminar

Abstract: We report on an experimental and theoretical study of Langmuir layers, defined as a molecularly thin polymer layer on the surface of a subfluid. Langmuir layers can have multiple phases (e.g. fluid, gas, liquid crystal, isotropic or anisotropic solid); at phase boundaries a line tension force (the two-dimensional analogue of surface tension) is observed. We first consider two co-existing fluid phases; specifically a localized phase embedded in an infinite secondary phase. When the localized phase is stretched (by a transient stagnation flow), it takes the form of a bola consisting of two roughly circular reservoirs connected by a thin tether. This shape then relaxes to the minimum energy configuration of a circular domain. The tether is never observed to rupture, even when it is more than a hundred times as long as it is thin. We model these experiments by taking previous descriptions of the full hydrodynamics (primarily those of Stone & McConnell and Lubensky & Goldstein), identifying the dominant effects via dimensional analysis, and reducing the system to a more tractable form. The result is a free boundary problem where motion is driven by the line tension of the domain and damped by the viscosity of the subfluid. The problem has a boundary integral formulation that allows us to numerically simulate the tether relaxation; comparison with the experiments allows us to estimate the line tension in the system, often to within 1%. As time allows we will also report on some other phenomena observed in Langmuir systems, including collapse of gas phase bubbles, co-existence of three or more fluid phases, elastic buckling of surface layers, and formation of dogbone and labyrinth patterns due to dipolar repulsion in the layer.
This work is collaborative with Jacob Wintersmith (HMC 2006), George Tucker (HMC 2008), Elizabeth Mann (Kent State), Lu Zou (Kent State), J. Adin Mann, Jr. (CWRU) and James Alexander (CWRU).

Wednesday, March 4, 2009

Boston University/Brown University PDE Seminar

Abstract: Burgers equation arises in several surprising contexts. In this talk I will show how Burgers equation arises in the theory of random matrices. Dyson showed that the eigenvalues of random matrices from Gaussian ensembles satisfy a simple SDE. In a scaling limit, as the size of the matrices goes to infinity, we obtain a kinetic equation (first derived by Kerov) for the spectral measure whose Cauchy transform satisfies Burgers equation. No knowledge of random matrices will be presumed, and the talk will be largely self-contained.

Monday, February 23, 2009

Lefschetz Center for Dynamical Systems Seminar

Abstract: Many stochastic partial differential equation models arising in applications generate complex time-evolving patterns which are hard to quantify due to the lack of any underlying regular structure. The influence of stochasticity leads to variations in the detail structure of the patterns and forces one to concentrate on rougher common geometric features. In many of these instances, such as for example in phase-field type models in materials science, one is interested in the geometry of sublevel sets of a function in terms of their topology, in particular, their homology.
Recent computational advances make it possible to compute the homology of discrete structures efficiently and fast. Such methods can be applied to the above situation if the sublevel sets of interest are approximated using an underlying discretization of the considered evolution equation. Yet, this method immediately raises the question of the accuracy of the computed homology. In this talk, I will present a probabilistic approach which gives insight into the suitability of the above method in the context of random fields. We will obtain explicit probability estimates for the correctness of the homology computations, which in turn yield a-priori bounds for the suitability of certain grid sizes as well as information on the optimal location of sampling points.

Wednesday, February 18, 2009

Lefschetz Center for Dynamical Systems Seminar

Abstract: Correlated activity in neural tissue can impact the information carried by neural populations. However, there are few results that provide a mechanistic understanding of their generation and propagation. I will examine this question using versions of the integrate and fire model, extending previous results for single neurons to populations. I will also present new numerical methods for the simulation of networks of stochastic integrate and fire neurons which are orders of magnitude faster than Monte Carlo simulations.

Monday, February 2, 2009

Lefschetz Center for Dynamical Systems Seminar

Abstract: The one-dimensional nonlinear Schrodinger equation (1D NLS) emerges as a first order model in a variety of fields--from high intensity laser beam propagation to Bose-Einstein condensation to water waves theory. The 1D NLS is completely integrable, hence solvable, on the infinite line or with periodic boundary conditions. The realization that the integrable structure might not persist under small perturbations led to the study of the periodic 1D NLS perturbed by a slightly conservative periodic forcing. In this talk I will describe our studies in which we show co-existence of various types of perturbed solutions such as ordered, temporal chaotic and spatiotemporal chaotic solutions. The prediction of the initial profiles that evolve into the different types of perturbed solutions is performed by utilizing a novel geometrical phase space description of the integrable unperturbed equation. As a result, we identify three mechanisms of temporal chaos in the perturbed NLS: homoclinic chaos, hyperbolic resonance, and parabolic resonance. For the latter mechanism we show that it serves as a route from initial data near an unperturbed stable plane wave to a regime of spatiotemporal chaos. Statistical measures are employed to demonstrate that this spatiotemporal chaos is intermittent: there are windows in time for which the solution gains spatial coherence. This is a joint work with V. Rom-Kedar.

Wednesday, November 19, 2008

Boston University/Brown University PDE Seminar

Abstract: In this talk, I will consider long time asymptotics for hyperbolic-parabolic system of conservation laws. These equations govern the time evolutions of many physical systems in which both transport properties and diffusion are present, such as gas dynamics, magneto-hydro-dynamics and the like. It was previously known that first order asymptotics were mainly of parabolic type, and I will explain how the nonlinear interaction of those leading order diffusion waves necessary produce slowly decaying (in both space and time) higher order corrections. In sharp contrast with single conservation laws, this phenomenon occurs even for initial conditions of compact support. I will also explain how to reconcile these (seemingly) long tails with the conservation of finiteness for the spatial moments, and how these observations beg for the development of a theory mixing spatial dynamics and singular perturbation for PDE's in which spatial scales are time-dependent.

Wednesday, November 19, 2008

Boston University/Brown University PDE Seminar

Abstract: We give a convergent expansion of solutions of the two-dimensional, incompressible Navier-Stokes equations which generalizes the Helmholtz-Kirchhoff point vortex model to systematically include the effects of both viscosity and finite core size. The evolution of each vortex is represented by a system of coupled ordinary differential equations for the location of its center, and for the coefficients in the expansion of the vortex with respect to a basis of Hermite functions. The differential equations for the evolution of the moments contain only quadratic nonlinearities and we give explicit combinatorial formulas for the coefficients of these terms. We also show that in the limit of vanishing viscosity and core size we recover the classical Helmholtz-Kirchhoff point vortex model. This is joint work with R. Nagem, G. Sandri, and D. Uminsky.

Thursday, November 13, 2008

Lefschetz Center for Dynamical Systems Seminar

Abstract: In this lecture special emphasis will be placed on the dynamical systems aspect of the study of the Navier-Stokes equations on thin 3D domains. The focus will be on the role of the Navier (or slip) boundary conditions, and the connection with the 2D Reduced Problem, which arises as the thinness goes to 0.

Monday, November 10, 2008

Colloquium

Abstract: The current theory of global attractors for the Navier-Stokes equations on thin 3D domains is motivated by the desire to better understand the theory of heat transfer in the oceans of the Earth. (In this context, the thinness refers to the aspect ratio - depth divided by expanse - of the oceans.) The issue of heat transfer is, of course, closely connected with many of the major questions concerning the climate. In order to exploit the tools of modern dynamical systems in this study, one needs to know that the global attractors are "good" in the sense that the nonlinearities are Frechet differentiable on these attractors. About 20 years ago, it was discovered that on certain thin 3D domains, the Navier-Stokes equations did possess good global attractors. This discovery, which was itself a major milestone in the study of the 3D Navier-Stokes equations, left open the matter of extending the theory to cover oceanic-like regions with the appropriate physical boundary behavior. In this lecture, we will review this theory, and the connections with climate modeling, while placing special emphasis on the recent developments for fluid flows with the Navier (or slip) boundary conditions.

Monday, November 3, 2008

Lefschetz Center for Dynamical Systems Seminar

Abstract: I shall discuss some properties of solutions of the Navier-Stokes system written in the Fourier space.

Wednesday, October 29, 2008

Boston University/Brown University PDE Seminar

Abstract: Recently, differential equations involving both delayed and advanced arguments have appeared in an increasing number of models, originating from a wide variety of scientific disciplines. We present recent theoretical and numerical results concerning bifurcations from homoclinic and periodic solutions to such equations. In particular, we show how Lin's method can still be used in the infinite dimensional setting of MFDEs.

Wednesday, October 29, 2008

Boston University/Brown University PDE Seminar

Abstract: We study monotone traveling wave solutions connecting 0 and 1 for a class of N dimensional lattice differential equations of the form
u_v′ = −u_v + g(u_v−e1,u_v−e2,⋯, u_v−eN); v ∈ Z^N
and which contain
u_n′ = −u_n + (u_n−1)²; n ∈ Z
as an example. We establish the existence of a minimum wave speed c_* such that for c ≥ c_* there is a unique monotone traveling wave connecting 0 and 1. The critical wave speed c_* depends on the direction of propagation in a higher dimensional lattice as well as the gradient of the convex envelope of the nonlinear coupling function g evaluated at the spatially homogeneous equilibrium (1,1,⋯,1). Our techniques are grounded in the direct analysis of a differential-delay equation; our intuition is guided by spatial dynamics.

Monday, October 6, 2008

Lefschetz Center for Dynamical Systems Seminar

Abstract: Burgers turbulence (Burgers equation with random initial data or forcing) is a nonlinear, out of equilibrium system with applications ranging from cosmology to interface dynamics. In the forceless case, shocks act as particles that cluster through ballistic aggregation and the system exhibits coarsening. Motivated by previous results in the limit cases of Levy process and white noise data, we will demonstrate that 1-D Burgers turbulence with spectrally negative Markov initial data is a completely integrable system. Specifically, we demonstrate that (i) the entropy solution remains of this type and (ii) the time evolution of the solution's infinitesimal generator is given by a Lax pair. These results also hold true in the case of general 1-D scalar conservation laws. The evolution equation is shown to have a rich family of solutions as evidenced by a highly nontrivial, explicit solution derived in the 1980's through entirely probabilistic methods.

Monday, September 29, 2008

Joint Algebra Seminar and Lefschetz Center for Dynamical Systems Seminar

Abstract: Arithmetic dynamics is the study of dynamical systems from a viewpoint derived from the classical theory of Diophantine equations and arithmetic geometry. I will explain this correspondence and describe some of the main results and conjectures in the area, in particular those related to rationality of periodic points and integrality of wandering points. Additional topics, as time permits, will include dynamical canonical heights, dynamical analogues of theorems of Faltings and Raynaud, reduction modulo p, and dynamical analogues of cyclotomic fields and cyclotomic units.

Monday, September 22, 2008

Lefschetz Center for Dynamical Systems Seminar

Abstract: Time-periodic shocks in systems of viscous conservation laws are shown to be nonlinearly stable. The result is obtained by representing the evolution associated to the linearized, time-periodic operator using a contour integral, similar to that of strongly continuous semigroups. This yields detailed pointwise estimates on the Green's function for the time-periodic operator. The evolution associated to the embedded zero eigenvalues is then extracted. Stability follows from a Gronwall- type estimate, proving algebraic decay of perturbations.

Wednesday, September 17, 2008

Boston University/Brown University PDE Seminar

Abstract: Snakes and Ladders: localized states in the Swift-Hohenberg equationStable spatially localized structures occur in many systems of physical interest. Examples can be found in the fields of optics, chemistry, fluid mechanics, and neuroscience to name a few. In this talk I will focus on one particular model, the Swift-Hohenberg equation, which arises in many of these applications. This equation contains a remarkable wealth of localized states, organized in a 'snakes-and-ladders' structure; a large number of these localized states are simultaneously stable. The talk will include an overview of the results for this model in both one and two spatial dimensions. The goal is to present a physical understanding of the mathematical and numerical results. Despite the simple model used in this analysis, there is evidence that the localized states observed in some experiments are organized in similar structures. Results will be presented for the example of natural doubly diffusive convection.

Wednesday, September 17, 2008

Boston University/Brown University PDE Seminar

Abstract: In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous media one observes "breathing" in the sense of the amplitude of the wave going up and down on a much faster scale than the motion of the wave. We investigate this phenomenon in the simplest case of stationary waves in which the evolution corresponds to relaxation to a nonlinear ground state. The particular model is the popular δ₀ impurity in the cubic nonlinear Schrödinger equation on the line. We give asymptotics of the amplitude on a finite but relevant time interval and show their remarkable agreement with numerical experiments.

Monday, September 8, 2008

Lefschetz Center for Dynamical Systems Seminar