Wednesdays at 4 PM in Eckhart 202.
This seminar is organized in collaboration with Computation in Science, MRSEC and EFI Theory seminars. See the Computation in Science seminar home page for an abstract.
This seminar is organized in collaboration with Computation in Science, MRSEC and EFI Theory seminars. See the Computation in Science seminar home page for an abstract.
Smoluchowski's coagulation equation is a fundamental mean-field model for the agglomeration of clusters. I'll describe comprehensive results obtained recently with Govind Menon regarding dynamic scaling of solutions for certain `solvable' rate kernels. We develop a framework for dynamic scaling analysis that ties together probability and dynamical systems theory in inspiring and profitable ways. Our results apply in particular to Burgers' turbulence model, a basic model for understanding statistics of solutions of nonlinear PDEs. Via work of J. Bertoin, we obtain a complete classification of universality classes for dynamic scaling of shock size distributions for the inviscid Burgers equation, with initial velocity that is random with stationary, independent increments with no positive jumps.
Hamiltonian PDE such as the nonlinear Schroedinger equation, nonlinear wave equations and others, have solutions which behave like classical Hamiltonian mechanics would dictate, with however infinitely many degrees of freedom. I'll describe some basic results in this direction, including questions of resonances, recurrence times, Nekhoroshev stability, and KAM theory for PDE.
This is a joint work with F. Rezakhanlou (UC, Berkeley) and S.R.S. Varadhan (Courant Institute, NYU). We prove a homogenization result for a class of stochastic Hamilton-Jacobi-Bellman equations. We assume that the Hamiltonian is superlinear and convex with respect to the gradient and stationary and ergodic with respect to the spatial variables. In some special cases this homogenization result is closely related to large deviations for quenched path measures for the Brownian motion in random media.
In a joint work with L. A. Caffarelli (UT Austin) and K.-A. Lee (Seoul Nat. University) we study flame propagation in periodic excitable media. The model under consideration is a nonlinear singular equation leading to a free boundary problem as the width of the flame goes to zero. We investigate the effect of homogenization on those models under the assumption that the width of the flame is much smaller than the length of the oscillations (in other words, we aim at the homogenization of the free boundary problem).
with Kurt L. Polzin (WHOI), and Esteban G. Tabak (NYU).
Wave turbulence formalism for long internal waves in a stratified
fluid is developed, based on a new, natural Hamiltonian description.
A kinetic equation, appropriate for descrbing spectral energy
transfer, is derived, and a one-parameter family of its power-law
stationary solutions corresponding to a direct energy cascade toward
the short scales is found with the aid of numerical quadrature. These
solutions include the high wavenumber limit of the Garrett-Munk
spectrum of long internal waves in the ocean. Several other ocean
spectral measurements appear to be well approximated by these
power-law solutions, including by one, different from the Garrett-Munk
spectrum, which was obtained analytically by using a similarity
argument.
Dissipation time is a time scale designed to measure the speed at which a noisy system approaches its equilibrium. In this talk I will introduce and discuss classical and quantum dissipation times of systems with compact (2d-dimensional torus) classical phase-spaces. The primary emphasis will be on chaotic dynamics including Anosov diffeomorphisms and generalized cat maps. In recent years a considerable progress has been made in semiclassical analysis of quantum counterparts of these systems. The main point of the study is to find the traces of chaoticity of original systems in their quantum versions. In particular one is interested in the longest time scale (Ehrenfest time) on which the quantum-classical correspondence holds. I will relate these results to the notion of quantum dissipation time and if time permits to the problems of quantum decoherence and the computations of quantum dynamical entropy.
In the first half of the talk we present a general theory of scaling limits for waves in randomly inhomogeneous media and derive 6 different radiative transfer equations. We show that these scaling limits have the self-averaging property and are statistically stable. We then focus on a special case related to optical propagation in atmospheric turbulence and study its implications in time reversal. We show, among other things, a duality relation between the forward spread of a wave beam and its time-reversed focal spot size.
In the past few years, Bose-Einstein condensates have become an exciting new playground for nonlinear waves. Bright and dark solitons, vortices, soliton trains and vortex lattices have become experimentally accessible in this novel setting. In this talk, we will focus on mean field models of such waves in the context of the Gross-Pitaevskii equation and its variants. We will highlight some of the solitary wave forming instability mechanisms and will then focus on the patterns consisting of trains or lattices of such coherent structures. When possible, we will give comparisons of analytical predictions and numerical findings with experimental results. We will conclude with a number of future challenges and open questions in this intriguing journey.
We consider the inversion of the two-dimensional attenuated Radon transform (AtRT) from full or partial measurements. The AtRT is routinely inverted in SPECT (Single Photon Emission Computed Tomography), a popular medical imaging technique. We show that two spatially independent source terms can be reconstructed from the AtRT. This is based on an extension of the recent Novikov formula and on recasting the inversion as a Riemann Hilbert problem. Next we consider the reconstruction of one spatially dependent source term from half of the angular measurements ($180^\circ$ measurements). We show that under a smallness condition on the gradient of the known absorption map, compactly supported source terms can uniquely be reconstructed. An iterative procedure is presented. Finally we consider a fast, robust, and accurate technique to compute and invert the AtRT. The numerical technique is based on a generalization of the fast slant stack algorithm, which performs very well to compute and invert the classical Radon transform. The technique is very accurate for moderate values of the absorption map. Modifications are proposed in the case of larger absorption maps. Numerical simulations complement the theory and show the robustness of the method.
At each site $x$ of $Z_d$, put a (random) probability measure $\omega_x$ on $\{-d,\ldots,-1,1,\ldots,d\}$, and then start a random walk that, when at site $x$, jumps to site $x+e$ with probability $\omega_x(e)$ for $|e|=1$. This model of a random walk in a random environment (RWRE) is arguably the simplest model of motion in random media one can think of. Many surprising properties of the RWRE have been discovered. For example, in dimension 1, the RWRE can exhibit sub-diffusive behavior, transience with zero average speed, and aging. Many challenges remain (for example, no criterion for transience or recurrence is known for $d\geq 2$). I will report on recent progress in the study of RWRE and on some surprising counter-examples to natural conjectures.
I will go through methods for making predictions when the data are insufficient or when the complexity of the problem is too high for complete resolution. The methods are based on the Mori-Zwanzig decomposition of statistical mechanics. The methods encounter the "curse of circularity": one can get very good reduced models when the full solution is known. Some of the ways this can be beaten rely on Kadanoff's construction of the renormalization group, which also suggests new and more general ways of looking at the prediction problem.
I will describe a way in which one can combine ideas from dynamical systems theory and kinetic theory to describe the long-time behavior of solutions of the Navier-Stokes equations. In two dimensions this leads to a very complete description of the behavior of solutions whose initial vorticity is at least slightly localized.
For questions, contact Eduard Kirr at ekirr@math.uchicago.edu
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