LCDS Seminars 2008

2008

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

Past Seminars

Listings of past seminars presented by LCDS are archived here:

• 2010
• 2009
• 2008
• 2007
• 2006
• 2005
• 2004
• 2003
• 2002
• 2001
• 2000