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Organization
Index Activities |
LCDS Events and Seminars, 2010
Wednesday, February 3, 2010
Abstract: 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.
Wednesday, February 3, 2010
Abstract: This 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, February 24, 2010
Abstract: We 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.
Wednesday, February 24, 2010
Abstract: Pipe 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, March 17, 2010
Abstract: Liesegang 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.
Wednesday, March 17, 2010
Abstract: We 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.
Monday, March 22, 2010
Abstract: The regular contraction of the heart keeping us all alive is controlled by the propagation of electrical waves through the heart. When things go wrong and this regular propagation is disrupted, arrhythmias arise often leading to death. The study of these waves has led to a wealth of novel mathematics in the areas of dynamical systems, asymptotics, and pattern formation. In this talk I will explain the basics of excitable media and then cover, in a mostly non-technical way, the mathematical insights that have been gained by studying these waves. I will conclude with the potential these results have for low-voltage defibrillation strategies.
Monday, April 5, 2010
Abstract: There has been a great deal of work in recent years on the asymptotics of solutions to scalar parabolic PDEs whose solutions remain bounded or blow up in finite time. In this talk I will present recent results addressing the intermediary case of slowly non-dissipative PDEs, whose solutions may undergo infinite-time grow-up. Many of the central methods used to study equations in the original two categories must be redefined and reconstructed in order to be applied to slowly non-dissipative equations. I will introduce the distinguishing characteristics of a slowly non-dissipative PDE, and discuss two new constructs, the Completed Inertial Manifold and the Non-Compact Global Attractor, which are crucial to understanding the asymptotic behavior of solutions to such equations.
Monday, April 26, 2010
Abstract: It is well known that for the subcritical semilinear heat equation, negative initial energy is a sufficient condition for finite time blowup of the solution. We show that this is no longer true when the energy functional is replaced with the Nehari functional, thus answering negatively a question left open by Gazzola and Weth (2005). Our proof proceeds by showing that the local stable manifold of any non-zero steady state solution intersects the Nehari manifold transversally. As a consequence, there exist solutions converging to any given steady state, with initial Nehari energy either negative or positive. Joint work with: F. Dickstein, N. Mizoguchi and Ph. Souplet
Wednesday, 28 April 2010
Abstract: 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.
Wednesday, 28 April 2010
Abstract: The 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.
Wednesday, 28 April 2010
Wednesday, 28 April 2010
Abstract: The 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.
Tuesday, September 7, 2010
Abstract: We consider the evolution of an interface, modeled by a parabolic equation, in a random environment. The randomness is given by a distribution of smooth obstacles of random strength. To provide a barrier for the moving interface, we construct a positive, steady state supersolution. This construction depends on the existence, after rescaling, of a Lipschitz hypersurface separating the domain into a top and a bottom part, consisting of boxes that contain at least one obstacle of sufficient strength. We prove this percolation result. Furthermore, we examine the existence of a solution propagating with positive velocity in a random field with non-bounded random obstacle strength. This work shows the emergence of a rate independent hysteresis in systems subject to a viscous microscopic evolution law through the interaction with a random environment. Joint work with N. Dirr (Bath University) and M. Scheutzow (TU Berlin).
Wednesday, September 15, 2010
Abstract: We 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, September 15, 2010
Abstract: We 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.
Monday, September 20, 2010
Abstract: We consider the U(1)-invariant Klein-Gordon equation in discrete space-time, with the nonlinearity concentrated at one point. We show that solitary waves form the weak global attractor for this equation. That is, for large positive or negative times any finite energy solution converges to the set of all solitary waves. The convergence takes place in localized (weighted) norms. Important points in the proof are the Titchmarsh convolution theorem for functions supported on a circle and the existence theory for the nonlinear Klein-Gordon equation in discrete space-time. This is a joint work with Alexander Komech, Vienna University and IITP, Moscow.
Wednesday, October 6, 2010
Abstract: The 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, October 6, 2010
Abstract: In 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.
Wednesday, October 13, 2010
Abstract: Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this talk, I discuss the stability of periodic traveling wave solutions to the full set of nonlinear equations via a non-local formulation of the water wave problem for a one-dimensional surface due to Ablowitz, Fokas and Muslimmani. Transforming the non-local formulation into a traveling coordinate frame, we obtain a new equation for the stationary solutions in the traveling reference frame as a single equation for the surface in physical coordinates. We develop a numerical scheme to determine non-trivial traveling wave solutions by exploiting the bifurcation structure of this new equation. Specifically, we use the continuous dependence of the amplitude of the solutions on their propagation speed. Finally, we numerically determine the spectral stability of the periodic traveling wave solutions by extending Fourier-Floquet analysis to apply to the associated linear non-local problem. In addition to presenting the full spectrum of this linear stability problem, we recover past well-known results such as the Benjamin-Feir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wave-length of the perturbation.
Wednesday, November 3, 2010
Abstract: Hydrodynamic 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, November 3, 2010
Abstract: We 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.
Wednesday, November 10, 2010
Wednesday, December 1, 2010
Abstract: Stable 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.
Wednesday, December 1, 2010
Abstract: I 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.
Past SeminarsListings of past seminars presented by LCDS are archived here:
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| Last change: Feb. 3, 2012 |
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