BU/Brown PDE Seminar

BU/Brown PDE Seminar Archive

Each month, a joint PDE seminar between the Departments of Mathematics at Boston University and Brown University and the Division of Applied Mathematics at Brown University will be held. Below is a list of past seminars: to view abstracts, move your mouse over a title. To view the current and future schedule, please visit the main BU/Brown seminar page.

Spring 2011

• Wednesday, 9 February 2011, 3:30-5:30pm in Room 110 (182 George Street) at Brown

  Speakers	Titles
  Govind Menon (Brown University)	Complete integrability of shock clusteringRavi Srinivasan and I found a Lax pair that described a kinetic theory for shock statistics in scalar conservation laws with random initial data. More recent work shows that the kinetic equations are completely integrable. Precisely: they have a Hamiltonian structure, can be linearized via a matrix factorization, and solved by an inverse scattering theory.
  Margaret Beck (Boston University)	A mechanism for creating strange attractors in infinite-dimensional dynamical systemsIn systems that posses stable limit cycles and a property known as "shear," it is possible to apply time-periodic forcing and create a strange attractor, on which the dynamics can be characterized by an SRB measure. The resulting chaos is sustained (ie is not transient) and is phyically observable. This has been previously analyzed in finite dimensions, and in infinite dimensions for limit cycles that are created by Hopf bifurcations. We show how to extend this to more general limit cycles in infinite dimensions. This is join work with Will Ott.

• Wednesday, 2 March 2011, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University

  Speakers	Titles
  David Ambrose (Drexel University)	Two problems in interfacial fluid dynamicsFor many problems in interfacial fluid dynamics, such as the water wave or the vortex sheet with surface tension, existence of solutions for initial value problems and existence of traveling solutions has been established in recent years. In this talk, we will look at two other existence problems. First, we will explore the question of existence of weak solutions for interfacial Navier-Stokes flows, in the presence of surface tension. One signicant challenge in this problem is finding a weak formulation of the surface tension force, which for strong solutions is supported only at the interface and is given in terms of the curvature of the interface. Second, we will discuss existence of time-periodic interfacial flows. This includes joint work with Milton Lopes Filho, Helena Nussenzveig Lopes, Walter Strauss, and Jon Wilkening.
  Peter Lax (Courant Institute / NYU)	Hyperbolic partial differential equations and degenerate matricesThe occurrence of multiple characteristics for hyperbolic systems gives rise to singularities in solutions. This leads to topological and algebraic questions about degenerate symmetric matrices and their discriminants.

• Wednesday, 6 April 2011, 3:30-5:30pm in Room 110 (182 George Street) at Brown

  Speakers	Titles
  Benoit Pausader (Brown University)	The energy-critical nonlinear Schrödinger equation in different geometries.We prove global existence of the solutions of the energy-critical nonlinear Schrödinger equation in the hyperbolic space and in the three-dimensional Torus. The key, besides developping the necessary harmonic analysis is to consider the (Euclidean) scaling limits. This is a joint work with G. Staffilani and A. Ionescu.
  Sanjeeva Balasuriya (Connecticut College)	Abnormal Melnikov theoryMelnikov theory usually relates to the normal displacement of a homoclinic manifold under a perturbation. Here, I introduce a method of determining the tangential displacement of a manifold under a time-aperiodic perturbation in a non-area-preserving instance, while improving on existing normal displacement results. I will also present a class of non-standard applications of Melnikov methods, viz., determining the wavespeed and waveprofile perturbation to travelling waves in reaction-diffusion systems arising in ecology and combustion.

• Wednesday, 27 April 2011, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University

  Speakers	Titles
  Irving Epstein (Brandeis University)	Reaction-Diffusion Patterns in Structured MediaI will look at two- and three-dimensional pattern formation in a reverse microemulsion consisting of nanometer diameter droplets of water containing the reactants of the Belousov-Zhabotinsky oscillating chemical reaction dispersed in oil. This system allows the physical structure (size, spacing) of the droplets and their chemical composition to be controlled independently, enabling one to generate a remarkable variety of stationary and moving patterns, including Turing structures, ordinary and antispirals, packet waves and spatiotemporal chaos. I will show examples of patterns and discuss two alternative mechanisms by which they may arise, one involving diffusion of different species at very different rates, the other via cross diffusion, whereby gradients in the concentration of one species influence the rate of diffusion of another species. I will also discuss pattern formation in one and two-dimensional arrays of micrometer diameter aqueous BZ droplets in oil prepared using microfludic techniques.
  Simon Tavener (Colorado State University)	A posteriori analysis and adaptive error control for operator decomposition approaches to coupled physics problemsA posteriori error estimation and adaptive error control based on the formulation and solution of an adjoint problem are well established for problems involving a single type of physics. For systems that exhibit sufficiently complicated physics or a range of scales so that they severely challenge standard solution techniques, operator decomposition provides an attractive way to decompose the problem into components with relatively simple physics or into behaviors that occur over a modest range of scales. Operator decomposition creates additional challenges for adaptive error control since errors in one component may limit the accuracy in another component, yet a global adjoint solution is typically not available. In this talk, I will describe an a posteriori analysis of operator decomposition methods for coupled elliptic systems. This analysis takes into account the accuracy with which individual components are solved as well as the global effects of operator decomposition. The estimates obtained provide the means for adaptive error control. Extensions of these ideas that are currently under investigation include coupled elliptic-parabolic and coupled parabolic systems arising in studies of cardiac physiology. Time-dependent problems present additional challenges and I will describe a one approach we have developed which we call "block adaptivity".

Fall 2010

• Wednesday, 15 September 2010, 3:30-5:30pm in Room 110 (182 George Street) at Brown

  Speakers	Titles
  Toan Nguyen (Brown University)	Stability of multi-dimensional viscous shocksWe present results on the long-time stability of multi-dimensional viscous shocks of a general class of symmetric hyperbolic--parabolic systems (possibly with variable multiplicities), including the compressible Navier-Stokes and MHD equations in dimensions d≥2. The results extend the existing ones of Zumbrun, by dropping a technical assumption previously assumed on structure of the so--called glancing set and relaxing an assumption on variable multiplicities of characteristic roots. The key analysis for us is to introduce a new argument for obtaining the L1→ Lp resolvent bounds in low-frequency regimes (of the eigenvalue equations), where we apply the construction of degenerate Kreiss' symmetrizers established by O. Gues, G. Metivier, M. Williams, and K. Zumbrun.
  Alethea Barbaro (UCLA)	Modeling Gang Rivalry Networks using an Agent-Based ModelWe propose an interacting particle model to simulate the formation of gang rivalries in the Hollenbeck policing district of eastern Los Angeles. The agents move according to a biased Levy process which dynamically interacts with an evolving rivalry network among the gangs in the system. Agents' choice of directional headings incorporates the location of their own gang's center of activity, as well as the location of the other gangs' centers. Geographic features of Hollenbeck such as freeways and the LA river are included into the model as semi-permeable boundaries and the number of agents in each gang reflects historical information from the LAPD.

• Wednesday, 6 October 2010, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University

  Speakers	Titles
  Maria Westdickenberg (Georgia Institute of Technology)	Slow Motion of Gradient FlowsIn metastable systems, states tend to be trapped in "valleys" of the energy landscape for a very long time even though they are not near any stable state. A familiar example is the one-dimensional Allen Cahn equation: Initial data is drawn quickly to a "multi-kink" state and the subsequent evolution is exponentially slow. The slow coarsening has been analyzed by Carr and Pego; Fusco and Hale; Bronsard and Kohn; and X. Chen. The one-dimensional Cahn Hilliard equation is a conservative system that displays similar behavior. In general, what causes metastability in a gradient flow system? Our main idea is to convert information about the energy landscape (statics) into information about the coarsening rate (dynamics). We give sufficient conditions for a gradient flow system to exhibit metastability. We then apply this abstract framework to give a new analysis of the 1-d Allen Cahn equation. The central ingredient is to establish a certain nonlinear energy-energy-dissipation relationship. One benefit of the method is that it gives a natural proof of the fact that exponential closeness to the multi-kink state is not only propagated, but also generated. In other words, the initial data does not need to be close to the slow manifold. We will also discuss work in progress on the Cahn Hilliard equation. This work is joint with Felix Otto, Max Planck Institute for Mathematics.
  Martina Chirilus-Bruckner (Boston University)	Interaction of Pulses in Nonlinear Wave EquationsThe interaction of pulses, i.e. spatially and temporarily oscillating waves modulated by a spatially localized envelope, in nonlinear wave equations is described via an extended perturbation approach that provides explicit formulas for interaction effects such as a position shift or a shape deformation of the interacting pulses. The analysis involves a reduction to amplitude equations (based on Fourier or Bloch analysis) and a rigorous justification of these (using energy estimates or semigroup theory). The presented method is applicable to a wide class of nonlinear, dispersive equations in 1+1 dimensions. In particular, it can be carried out for nonlinear wave equations with periodic coefficients which arise, e.g., as model equations for light propagation in photonic crystals.

• Wednesday, 3 November 2010, 3:30-5:30pm in Room 110 (182 George Street) at Brown

  Speakers	Titles
  Nick Ercolani (University of Arizona)	A Burgers Model for Striped Pattern Formation in the Strong Bending RegimeWe will present a model for the formation of defects in patterns that arise in 2D, spatially extended physical systems whose principal bifurcation is from spatial homogeneity to semi-discrete ("striped") patterns. Our model stems from a variational extension of the Cross-Newell phase diffusion equation and incorporates perspectives from experiment (Rayleigh-Benard convection), simulation and analysis. The analysis takes advantage of a Cole-Hopf linearization for the variational equations and invites comparison to the known validity of the Burgers phase equation in 1D.
  Shreyas Mandre (Brown University)	Hydrodynamic stability of time dependent flowsHydrodynamic stability theory using normal modes, while a mature branch of fluid mechanics, is limited only to steady flows. Most flows observed in nature, however, are not steady. The linear stability of such time-dependent flows is usually carried out through a frozen coefficient analysis. In a frozen coefficient analysis one assumes the background flow to be frozen at a given instant of time, and then uses the traditional machinery developed for steady flows to determine stability. While acceptable for flows changing slowly with time, the frozen coefficient analysis is not necessarily applicable to flows varying rapidly with time. An example of such a flow is an impulsively started shear layer with thickness growing diffusively as t1/2. Because of the impulsive nature, the flow varies rapidly near t=0. I present a consistent theory for the linear stability of this flow. The theory reduces to the normal mode analysis for steady flows and to Floquet analysis for time-periodic flows. The formulation does not make any assumptions about the specific base flow considered here and can be applied to other time-dependent fluid flows.

• Wednesday, 1 December 2010, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University

  Speakers	Titles
  Luc Rey-Bellet (University of Massachusetts)	Game theory and statistical mechanicsI will discuss some of the connections between game theory and statistical mechanics. In particular we show that games can be interpreted in terms of equilibrium/non-equilibrium statistical mechanics; familiar games such as Rock-Paper-Scissor or Matching Pennies can be interpreted in terms of fluxes and entropy production. We also derive a number of mesoscopic equations starting from spatial games with long range interactions and we exhibit some interesting metastable effects for these equations.
  John Burke (Boston University)	Localized states in simple PDE models, without homoclinic snakingStable spatially localized states occur in many systems of physical interest. Many properties of these solutions can be understood in the context of relatively simple PDE models, such as the Swift-Hohenberg equation. Often, the localized states are organized in pairs of branches which undergo homoclinic snaking - on a bifurcation diagram, the branches intertwine, oscillating back and forth in a series of saddle-node bifurcations. The recent concentration of attention on solutions of this type can give the false impression that localized states are always associated with homoclinic snaking. In this talk I will give several examples of localized states which are nearly indistinguishable from those which undergo homoclinic snaking, but which have different behavior under continuation. This is joint work with Yiping Ma and Edgar Knobloch of UC Berkeley.

Spring 2010

• Wednesday, 3 February 2010, 3:30-5:30pm in Room 110 (182 George Street) at Brown

  Speakers	Titles
  Bob Rink (Vrije Universiteit Amsterdam)	Integrable continuum approximations for the Fermi-Pasta-Ulam lattice: a mechanism for metastability The Fermi-Pasta-Ulam experiment is famous for the unexpected recurrent behavior of long waves. Attempts to explain this observation usually involve an approximation by a KdV equation and a non-rigorous KAM argument. A rigorous justification of the KdV approximation for finite times was given only recently, independently by Bambusi-Ponno and Wayne-Schneider. In this talk I will show how to derive the KdV equation as a resonant normal form, and I will present an improvement to this first approximation. This is a step towards the justification of the so-called metastability scenario.
  Marta Lewicka (University of Minnesota)	Scaling laws and reduced theories of the prestrained elastic filmsThis talk will concern the analysis and the rigorous derivation of plate and shell models for thin films exhibiting residual stress at free equilibria. Examples of such structures include growing tissues (e.g. leaves). There, it is conjectured that the cell division results in the formation of non-Euclidean 'target metrics', leading to complicated morphogenesis of the tissue which tries to adapt itself to its internally imposed metric. A possible analysis of these phenomena uses the variational point of view. It departs from the model of 3d non-Euclidean elastic energy, which measures the pointwise deviation of the given deformation from orientation preserving realizations of the target metric. For metrics with non-zero Riemann curvature, the infimum of this energy is strictly positive at free equilibria, that is in the absence of boundary conditions or body forces. In this setting, we analyze the scaling of the energy minimizers in terms of the reference plate's thickness and rigorously derive the corresponding limiting theories, as the vanishing thickness Γ-limits. The theories are differentiated by the embeddability properties of the target metrics - in the same spirit as different scalings of external forces lead to a hierarchy of nonlinear elastic plate theories as recently displayed by Friesecke, James and Müller. Some new relationships with non-smooth isometric embeddings of 2d metrics (on the mid-plate) into R3 are also exhibited.

• Wednesday, 24 February 2010, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University

  Speakers	Titles
  Gigliola Staffilani (MIT)	Forward cascade for solutions to the cubic defocusing NLS equationWe consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any є ≪ 1, K ≫ 1, s >1, we construct smooth initial data u0 with ||u0||Ḣs < є, so that the corresponding time evolution u satisfies ||u(T)||Ḣs > K at some large time T. This growth occurs despite the Hamiltonian's bound on ||u(t)||Ḣ1 and despite the conservation of the quantity ||u(t)||L2.The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution's frequency support that simplifies the system of ODE's describing each Fourier mode's evolution. The second is a construction of solutions to these simpler systems of ODE's which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems.
  Tobias Schneider (Harvard University)	Extended and localized critical perturbations for the transition to turbulence in shear flowsPipe and plane Couette flow are shear flows that undergo transition to turbulence without mediation by a linear instability of the laminar profile. Triggering turbulence in these flows thus requires perturbations of finite amplitude that drive the system towards a chaotic saddle that supports turbulent dynamics and that coexists with the laminar fixed point in the state space of the system. Regions of laminar and turbulent dynamics in the system's state space are separated by the `edge of chaos' which extends the concept of basin boundaries to situations with transient turbulence. Using an iterated bracketing technique we can numerically trace the dynamics in this edge of chaos and determine the invariant relative attractors termed `edge states'. These invariant objects define the shape and size of critical perturbations required to trigger turbulence. In small, periodically continued domains critical perturbations that extend throughout the domain have been identified. However, in extended domains also localized invariant local attractors have recently been found. For plane Couette flow they are part of a snakes-and-ladders structure known from classical pattern-forming models like the Swift-Hohenberg equation. The localized solutions originate from bifurcations off the spatially periodic equilibria discovered by Nagata and others and retain their physical structure, demonstrating the relevance of exact solutions to turbulent flows in spatially extended domains, where localized perturbations are observed to induce spatially localized patches of turbulence which slowly invade the surrounding laminar flow.

• Wednesday, 17 March 2010, 3:30-5:30pm in Room 110 (182 George Street) at Brown:

  Speakers	Titles
  Arnd Scheel (University of Minnesota)	Coherent structures and Liesegang pattern formationLiesegang rings form in strikingly simple experimental setups. The resulting patterns, sometimes regular, sometimes irregular, are ubiquitous in nature --- yet mathematically poorly understood. We present models and results that lead to a conceptual understanding of existence and creation of such patterns, based on existence, stability, and bifurcation results for coherent structures in our models. A crucial role is played by pushed and pulled fronts in spinodal decomposition scenarios.
  Ian Tice (Brown University)	Rayleigh-Taylor instability for compressible fluidsWe consider the evolution of two compressible, immiscible fluids separated from one another by a free surface. If the fluids are acted on by a uniform gravitational field, then steady states with a denser fluid lying above a less dense fluid may be unstable. Such an instability, known as a Rayleigh-Taylor instability, is most easily seen at the linear level by constructing "normal mode solutions" that grow exponentially in time. In the case in which both fluids are inviscid, we will show how to use these normal mode solutions to prove ill-posedness for the linearized problem. In general there is no way to go from linear ill-posedness to non-linear ill-posedness, but for this problem it turns out to be possible because of some variational structure associated to the normal modes. We will show that the normal modes are sufficiently pathological to preclude a reasonable notion of well-posedness for the non-linear problem. We then consider the case of viscous fluids with or without surface tension. In this setting the construction of normal modes is complicated by the appearance of a non-standard eigenvalue problem. We will show how to solve such a problem using a family of modified variational problems. The variational structure is crucial because it then allows us to derive estimates for the growth of arbitrary solutions to the linearized problem in terms of the growth rate of the constructed normal modes. Along the way we will show that in a spatially periodic setting, surface tension is capable of preventing linear instability.

• Wednesday, 28 April 2010, 3:30-5:30pm in Room MCS 149 (111 Cummington Street) at Boston University

  Speakers	Titles
  Matthew Holzer (Boston University)	Stability of Traveling Pulses in a Reaction-Diffusion-Mechanics System.We study the existence and stability of traveling pulse solutions in a reaction-diffusion-mechanics model. The particular model we study is one of cardiac tissue where the electrical activity of the tissue is bi-directionally coupled to the mechanics of the underlying medium. We use geometric singular perturbation techniques to show that the model has a traveling pulse solution. The analysis is complicated by the passage of this solution near a non-hyperbolic fold point of a slow manifold. We then establish the spectral stability of this pulse solution. For the stability problem, we construct an analytic Evans Function and show that the only zero of this function is a regular zero at the origin.
  Scott McCalla (Brown University)	Radial solutions and the cessation of snaking for the multi-dimensional Swift-Hohenberg equationThe bifurcation structure and the cessation of snaking of localized radial patterns of the Swift--Hohenberg equation are explored through numerical computations as the dimension is varied. Our findings elucidate the connection between one-dimensional pulses and 2-pulses to planar spots and rings when the dimension is changed from one to two and then further to three. We also discovered a new class of planar localized spot solutions and discuss an analytical approach to establishing their existence.
  Blaz Mramor (Vrije Universiteit Amsterdam)	Aubry-Mather theory and Ghost circlesABSTRACT
  Christian Pfrang (Brown University)	Diagonalizing Random Matrices Using Integrable SystemsThe QR algorithm is known to be the time one map of an integrable Hamiltonian system on certain classes of matrices. I will explain this fact and introduce the related Toda algorithm. As a first step towards the probabilistic analysis of these algorithms we are trying to understand the time it takes to identify one of the eigenvalues up to a prespecified tolerance given an initial matrix drawn from the standard tridiagonal ensemble. Numerical simulations will be presented that point to the existence of scaling limits for this random time in case of both the QR and the Toda algorithm.

Fall 2009

• Wednesday, 23 September 2009, 3:30-5:30pm in Room 110 (182 George Street) at Brown University:

  Speakers	Titles
  Gregor Kovacic (Rensselaer Polytechnic Institute)	Fokker-Planck Description for Noisy Neuronal Network DynamicsKinetic theory provides a coarse-grained alternative to the integrate-and-fire neuronal network description. In the limit of infinitely short conductance responses, a Boltzmann-type differential-difference equation can be derived for the probability density function of the neuronal voltage. A Fokker-Planck and a mean-field equation can be derived in the limit of small and vanishing conductance fluctuations, respectively. The talk will present detailed solutions to these equations, describing both the steady asynchronous and synchronously-oscillating states of the network, and will also discuss the effects of the network architecture. The mean-field provides exact solutions for the steady asynchronous state. For scale-free neuronal networks, it can be used to argue that the distributions of the firing rates and neuronal activity correlations are also scale free. The steady asynchronous state is also described by asymptotic solutions of the Fokker-Planck equation, using the size of the neuronal conductance fluctuations as the small parameter. in addition, the Fokker-Planck equation can also be used to describe the likelihood and temporal period of synchronous network oscillations, in which all the neurons fire in unison. The likelihood of synchrony is computed combinatorially using the network oscillation period and the voltage probability distribution. The oscillation period is found from a first-passage-time problem described by a Fokker-Planck equation, which is solved analyticaly via an eigenfunction expansion. The voltage probability distribution is found using a Central-Limit-Theorem-type argument via a calculation of the voltage cumulants. Differences between oscillations in all-to-all coupled and scale-free networks will also be discussed.
  Peter van Heijster (Brown University)	Front interactions in a three-component system

• Wednesday, 21 October 2009, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University:

  Speakers	Titles
  James Nolen (Duke University)	Stability and fluctuations for traveling waves in an inhomogeneous mediumI will talk about generalized traveling waves for scalar reaction diffusion equations in an inhomogeneous environment. As the name suggests, these solutions generalize the traditional notion of a traveling wave, although the wave profile is not fixed and the wave speed may not be well-defined. These solutions are stable attractors. If the environment has a certain statistical structure, then the asymptotic wave speed is well-defined, and the interface moves like a Brownian motion with positive drift.
  Thierry Gallay (University of Grenoble)	Three-dimensional stability of Burgers vorticesBurgers vortices are explicit solutions of the three-dimensional Navier-Stokes equations which are often used to model the vortex filaments observed in turbulent flows. Despite obvious limitations, this model describes in a correct way the fundamental mechanisms which are responsible for the persistence of coherent structures in three-dimensional turbulence. In this perspective, an important problem is to determine the stability of Burgers vortices with respect to perturbations in the largest possible class. This question has been open for almost three decades, and rigorous answers have been obtained so far for small Reynolds numbers only, or in the particular case of two-dimensional perturbations. In this talk, I shall show how a detailed analysis of the linearized operator allows to prove the stability of Burgers vortices with respect to three-dimensional perturbations, for any given value of the circulation number. This is a joint work with Yasunori Maekawa (Kobe University).

• Wednesday, 18 November 2009, 3:30-5:30pm in Room 110 (182 George Street) at Brown University:

  Speakers	Titles
  Jonathan Rubin (University of Pittsburgh)	The geometry of neuronal oscillator recruitmentOur ability to perceive and respond to transient external stimuli arises from the brain's ability to generate sustained, spatially localized activity in response to transient inputs. Mathematical models designed to represent this process in networks of neurons must provide a mechanism for recruitment of neurons into an active group as well as a mechanism to limit the spread of activity and hence maintain its localization. Analysis of such models often focuses on how different localization features, such as long-range inhibition, contribute to the existence and stability of sustained, localized activity patterns. In this talk, I will discuss work on the recruitment of neuronal oscillators, in contexts where desynchronization of inputs or competition complicate outcomes. I will consider discrete models, most of which are not specific to neurons, as well as a continuum limit. This work involves a geometric perspective, and the talk assumes no background knowledge of neuroscience.
  Vadim Zharnitsky (University of Illinois)	Near-linear dynamics in KdV with periodic boundary conditionsKdV equation is a standard model of weakly nonlinear long waves on the surface of shallow water. It will be shown that in KdV with periodic boundary conditions, high frequency solutions evolve almost as the linear ones for large time. For KdV (or some other dispersive equations) on the real line such behavior could be expected due to the dispersive decay. While on the circle (i.e. periodic boundary conditions) such dispersive decay is not possible, the dispersion manifests itself in averaging out nonlinearity over high frequency solutions. This result is obtained by the application of normal form transformations in the appropriate spaces. The integrability properties of KdV are not used, so similar results could be obtained for other KdV like equations. The interaction of these high frequency solutions with a cnoidal wave will be discussed, too. This work has been motivated by an attempt to explain some phenomena in nonlinear optics and fluid dynamics. This is a joint work with M.B. Erdogan and N. Tzirakis (also University of Illinois).

• Wednesday, 2 December 2009, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University:

  Speakers	Titles
  Robert Ghrist (University of Pennsylvania)	Floer theory for braidsWhat do you get when you mix scalar parabolic PDEs, braids, the Maximum Principle, Morse Theory, and nonlinear Cauchy-Riemann? Floer theory for braids; with applications to parabolic & Hamiltonian dynamics.
  John Gibson (Georgia Institute of Technology)	Invariant solutions and state-space dynamics in plane Couette flowIt has recently become possible to compute precise equilibrium, traveling wave, and periodic orbit solutions to pipe and plane Couette flow at Reynolds numbers above the onset of turbulence. These invariant solutions capture the complex dynamics of "coherent structures", in low-Reynolds turbulent flows and provide a framework for understanding such flows as dynamical systems. We present a number of weakly unstable equilibria, traveling waves, and periodic orbits of plane Couette flow and visualizations of their physical and state-space dynamics. What emerges is a picture of low-Reynolds turbulence as a walk among a set of weakly unstable invariant solutions.

Spring 2009

• Wednesday, 1 April 2009, 3:30-5:30pm in Room 110 (182 George Street) at Brown University:

  Speakers	Titles
  Hongjie Dong (Brown University)	Parabolic equations with variably partially VMO coefficients We prove the Wp1,2-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.
  Mikyoung Vera Hur (MIT)	Dispersive properties of surface water wavesI 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.

• Wednesday, 4 March 2009, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University:

  Speakers	Titles
  Andrew Bernoff (Harvey Mudd College)	Domain Evolution and Relaxation in Langmuir Films 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).
  Govind Menon (Brown University)	Dyson's brownian motion, Kerov's kinetic equation and Burgers equationBurgers 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.

Fall 2008

• Wednesday, 19 November 2008, 3:30-5:30pm in Room 110 (182 George Street) at Brown University:

  Speakers	Titles
  Guillaume van Baalen (Boston University)	Long tails in the long time asymptotics of quasilinear hyperbolic-parabolic systems of conservation lawsIn 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.
  Eugene Wayne (Boston University)	A generalization of the Helmholtz-Kirchoff model of vortex motionWe 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.

• Wednesday, 29 October 2008, 3:30-5:30pm in Room MCS 137 (111 Cummington Street) at Boston University:

  Speakers	Titles
  Hermen Jan Hupkes (Brown University)	Lin's Method and Homoclinic Bifurcations for Functional Differential Equations of Mixed TypeRecently, 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.
  Aaron Hoffman (Boston University)	Monotone fronts and minimum wave speeds in a class of ``upwind'' lattice differential equationsWe study monotone traveling wave solutions connecting 0 and 1 for a class of N dimensional lattice differential equations of the form uv′ = −uv + g(uv−e1,uv−e2,⋯, uv−eN); v ∈ ZN and which contain un′ = −un + (un−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.

• Wednesday, 17 September 2008, 3:30-5:30pm in Room 110 (182 George Street) at Brown University:

  Speakers	Titles
  John Burke (Center for BioDynamics, Boston University)	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.
  Justin Holmer (Brown University)	Breathing patterns in nonlinear relaxation 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.

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• Department of Mathematics at BU
• Department of Mathematics at Brown
• Division of Applied Mathematics at Brown