Wednesdays at 4:00 PM in Eckhart 308.
The asymptotic behavior, as the coefficient of the advection term approaches infinity, of the principal eigenvalue of an elliptic operator is determined. As an application a Lotka-Volterra reaction-diffusion-advection model for two competing species in a heterogeneous environment is investigated. The two species are assumed to be identical except their dispersal strategies: one disperses by random diffusion only, and the other by both random diffusion and advection along environmental gradient. When the advection is strong relative to random dispersal, both species can coexist. In some situations, it is further shown that the density of the species with large advection in the direction of resources is concentrated at the spatial location with maximum resources.
In this talk I will present an intriguing application of the braid group theory to the study of blow-up in a nonlinear heat equation $u_t = \Delta u + u^p$, where $p$ is supercritical in the Sobolev sense. The goal is to determine all the type II blow-up rates by analyzing the topological properties of certain braids.
The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of R^n. To each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. The results are new even for symmetric operators or in dimension 1. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new rearrangement technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.
Generalized equations of motion for the Weber-Clebsch potentials that reproduce Navier-Stokes dynamics are derived. These depend on a new parameter, with the dimension of time, and reduce to the Ohkitani and Constantin equations in the singular special case where the new parameter vanishes. Let us recall that Ohkitani and Constantin found that the diffusive Lagrangian map became noninvertible under time evolution and required resetting for its calculation. They proposed that high frequency of resetting was a diagnostic for vortex reconnection. Direct numerical simulations of the generalized equations of motion are performed. The Navier-Stokes dynamics is well reproduced at small enough Reynolds number without resetting. Computation at higher Reynolds numbers is achieved by performing resettings. The interval between successive resettings is found to abruptly increase when the new parameter is varied from zero to a value much smaller than the resetting interval.
Gaussian white noise turns out to be an invariant measure for KdV on the circle. We explain the precise meaning, and sketch two proofs based on results of Kappeler/Topolov and of Bourgain. If time permits we will also attempt to speculate on the physical meaning. This is joint work with Benedek Valko (Toronto).
Attempts to capture salient features of cellular flame instabilities have led to a variety of beautiful mathematical models of a very geometrical nature. The most famous of them is the Kuramoto-Sivashinsky (KS) equation. Its lesser relative is the Burgers-Sivashinsky (BS) equation (which is just a linearly forced Burgers equation). The linear dispersion relations for both equations admit exponential mode growth for a range of long waves. Nonetheless the equations are dissipative due to the nonlinear mixing. For the purposes of this talk, dissipativity is understood as the property that the eventual time evolution of solutions is confined to a bounded (actually compact) absorbing set. The principal subject of this talk is yet another, recently introduced model of quasi-steady evolution of cellular flames, the Quasi-Steady equation (joint work with M. Frankel, IUPUI). In a sense, QS is intermediate between BS and KS, as its dispersion relation coincides with that for BS for short waves, and is virtually identical to that of KS for long waves. Similarly to KS, QS demonstrate a very rich dynamical behavior (while BS has more or less trivial dynamics). The proof of dissipativity and generalizations to elliptic pseudo-differential operators will be discussed.
We discuss the large positive solutions of the nonlinear elliptic equation -Laplacian u = r exp u in D, u=0 on the boundary of D, and related equations. Here D is a smooth bounded domain. Note that this equation occurs in many places including combustion theory and catalysis theory. In particular, we sketch how some results known for D a ball generalize to arbitrary domains. For example, we show that the problem has infinitely many bifurcation points.
Some reaction-diffusion equations admit traveling wave solutions; these are simple models of a chemical reaction spreading with constant speed. Even in a random medium, solutions to the initial value problem may develop fronts propagating with a well-defined asymptotic speed. In this talk, I will describe recent results on reaction-diffusion fronts in random media, and I will focus on a particular randomly-excitable medium (in one spatial dimension) for which there exist generalized traveling waves. These are functions that solve the governing PDE for all time, and they generalize the notion of a traveling wave in a homogeneous medium. I will describe the construction of these traveling waves and discuss some of their properties.
In this talk, I will present some Liouville type results for linear operators with periodic coefficients. We also exhibit an explicit counterexample showing that the periodicity assumption cannot be relaxed by almost periodicity. We further present some Liouville type results for semilinear operators arising in population dynamics, obtained in collaboration with H. Berestycki and F. Hamel, as well as extensions to fully nonlinear operators.
When numerically solving a wave-type PDE, open boundaries must be used to prevent spurious reflections from the computational boundary. In this talk I introduce Phase Space Filtering, a new approach to this problem. The Time Dependent Phase Space Filter (TDPSF) algorithm consists of identifying and filtering outgoing waves in phase space based on knowledge of scattering theory. It can be extended to study wave equations in regions of phase space which are non-rectangular, and with long range potentials. It always stable, even for equations where the PML is not (e.g. the Euler equations with a jet flow).
I will describe how the exit measure from large balls for random walk in random environment converges to the uniform one, for d>=3 and an (isotropic in law) environment that is a small perturbation the fixed environment corresponding to simple random walk. Some plausible extensions will be pointed out.
Consider the Fisher-KPP equation, where the standard Laplacian is replaced by a fractional Laplacian. Front propagation will still occur, but the velocity will grow exponentially in time, rather than being asymptotic to a constant. This is in sharp constrast to what happens with the combustion type nonlinearity, where the dynamics is asymptotic to travelling waves - irrespective of the diffusion operator used in the equation. We will explain why exponential propagation holds; moreover we will discuss extensions and some sharp asymptotics. This is joint work with X. Cabre.
In the Boussinesq approximation a reaction-advection-diffusion equation for the temperature is coupled with the Stokes equation for the viscous fluid. We show that in two-dimensions on a bounded domain the resulting flow has improved mixing properties as a characteristic parameter, the Rayleigh number, becomes large. When there is no coupling, comparable flows do not improve mixing. In this talk we will explain the difference.
This talk presents stochastic variational integrators (SVIs) which are based on a Lagrangian description of stochastic Hamiltonian systems on manifolds. With dissipation these systems become the important class of mechanical systems governed by Langevin-type equations. SVIs provide an effective tool to simulate equilbrium, and for the first time, nonequilibrium dynamics of randomly forced and torqued mechanical systems on manifolds. The talk shows how one can use SVI theory to extend Verlet integrators to: mechanical systems with holonomic constraints at uniform temperature and rigid-body-type systems at uniform temperature. These are easy consequences of the fact that SVIs are derived from intrinsically defined objects. As an application of SVIs, the talk considers a ballistic pendulum --- a perturbation of the simple gravity pendulum that exhibits ballistic transport in its pendular degree of freedom.
I will describe two consequences of a stochastic Lagrangian formulation of the Navier-Stokes equations. The first is a (stochastic) particle system for the Navier-Stokes equations. On any fixed time interval, this system converges to the Navier-Stokes equations as the number of particles goes to infinity. However for a fixed number of particles, one finds that the system does not dissipate all it's energy after long time. Curiously, a similar system for the Burgers equations dissipates all it's energy as t goes to infinity. The second consequence I will describe is related to the Stochastic Navier-Stokes equations, a turbulence model developed by Rozovskii and Mikulevicius. A Lagrangian description of these equations can be developed in a manner similar to that for the Navier-Stokes equations. I will use this formulation to show how a Beale-Kato-Majda type result can be proved for the Stochastic Navier-Stokes equations. To the best of my knowledge, this result can not be obtained by regular (stochastic) PDE methods.
This talk is concerned with the existence of travelling fronts solutions for some reaction-diffusion in space-time periodic media. We prove that there exists a minimal speed c* such that there exist some pulsating travelling fronts of speed c if and only if c is higher than c*. It is possible to get a characterization of this speed that enables us to prove some new dependence results with respect to the coefficients.
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