# CAMP/Nonlinear PDE's Seminar

Wednesdays at 4 PM in Eckhart 206.

## Fall 2000 Schedule

September 27
Benoit Perthame, Ecole Normale Superieure, Paris
On the Helmholtz equation and its high frequency limit

October 4,
Mark Freidlin, University of Maryland
Averaging Principle for Deterministic and Stochastic Perturbations

Averaging principle for stochastic perturbations of dynamical systems with conservation laws is considered. Consideration of averaging of random perturbations, which is interesting by itself, sheds a new light on the classical averaging principle. Even pure deterministic perturbations of such systems lead, in general, to stochastic behavior of the slow component of the motion. This slow component can be described as a random process on a complex related to the collection of all first integrals of the system. In the case of one degree of freedom Hamiltonian system, the complex is the graph homeomorphic to the set of all connected components of the level sets of the Hamiltonian. We also will demonstrate how these results can be applied to asymptotic problems for reaction-diffusion in an 2D-incompressible fluid.

October 11
David Levermore, University of Maryland
Stokes and Acoustic Limits for the Boltzmann Equation

The endeavor to understand how fluid dynamical equations can be derived from kinetic theory goes back to the founding works of Maxwell and Boltzmann. Most of these derivations are well understood at several formal levels by now, and yet their full mathematical justifications are still missing. This talk will introduce this general problem and describe recent works in which the Stokes and acoustic limits are globally establish for the classical Boltzmann equation considered over any periodic spatial domain of dimension two or more.

October 18,
Divakar Viswanath, University of Chicago
Computation of Periodic Orbits and the Lindstedt-Poincare Technique

Periodic orbits are everywhere in the study of differential equations. Oscillatory phenomena in many areas of science are periodic solutions of differential equations. The mathematical significance of periodic orbits is considerable. For example, in Poincare's opinion, periodic orbits provided "the only breach" to attack problems in celestial mechanics that have mostly remained unresolved. This talk will present a fast and accurate algorithm for computing periodic orbits. The algorithm is based on the Lindstedt-Poincare technique in perturbation theory. Examples related to Hill's famous work on the motion of the Moon and to Hilbert's 16th problem demonstrate the effectiveness of this algorithm. The first of the two examples includes what is possibly the most accurate computation of the orbit of maximum lunation since its justly celebrated discovery by Hill in 1878.

October 25,
Evgeni Balkovsky, University of Chicago
Universal long-time properties of Lagrangian statistics in the Batchelor regime.

Abstract: We consider the transport of dynamically passive quantities in the Batchelor regime of a smooth in space velocity field. For the case of arbitrary temporal correlations of the velocity, we formulate the statistics of relevant characteristics of Lagrangian motion. This allows us to generalize many results obtained previously for strain $\delta$- correlated in time, thus answering a question about the universality of these results. We also apply these results to describe statistics of polymer elongations in turbulent flows.

November 1,
Yuri Lvov, Rensselaer Polytechnic Institute
Weak turbulence theory: development and novel applications

What semiconductor laser theory, fiber optics, surface water waves and acoustic waves have in common? Although these systems are seemingly disconnected and have quite different physical nature, they can be viewed as complex systems composed out of interacting particles or waves. There is a general theoretical framework for their statistical description, called weak turbulence theory. One can obtain a closed equation describing the time evolution of such systems, called kinetic equation. I will explain what classes of stationary solutions kinetic equation has, and how understanding of surface water waves can lead to better design of semiconductor lasers. I will also explain how one can hope to generalize weak turbulence theory for spatially inhomogeneous systems.

November 8,
Leslie Greengard, New York University,
Nonreflecting Boundary Conditions and High Order Methods for the Wave Equation

Time-domain modeling of acoustic or electromagnetic scattering in free space requires a reduction of the infinite physical domain to a finite computational domain. The exact nonreflecting boundary conditions - well-known to be nonlocal in both space and time - can be expressed as a convolution of the solution at the boundary from the time of quiescence to the present. The kernel of this integral operator, which depends on the computational domain, is presented for several boundary geometries. An efficient implementation of this formulation is discussed. We also present a new time-symmetric evolution formula for the scalar wave equation. It is simply related to the classical D'Alembert or spherical means representations, and can be used to develop stable, robust numerical schemes on irregular meshes.

November 15
Tom Hou, Caltech,
Singularity Formation in 3-D Vortex Sheets

We study singularity formation of 3-D vortex sheets using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contribution of the integral equation. Moreover, after applying a transformation to the physical variables, we found that this leading order 3-D vortex sheet equation de-generates into a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This rather surprising result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. We also show that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Furthermore by using the abstract Cauchy-Kowalewski theorem, we prove the long time existence of 3-D vortex sheets for analytic initial data. The existence time can be arbitrarily close to the singularity time predicted by the leading order equation when the initial condition is near equilibrium. A generalized Moore's approximation to 3-D vortex sheets is introduced. Detailed numerical study will be provided to support the analytic results, and to reveal the generic form of the three-dimensional nature of the vortex sheet singularity.

November 21, Tuesday, Colloquium
Sergei Kuksin, Heriot-Watt University
TBA

November 22,
Eitan Tadmor, University of California, Los Angeles
Detection of Edges in Spectral Data and Large Exponential Recovery Between Edges

We discuss the reconstruction of piecewise smooth data from (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth. The presence of jump discontinuities, however, is responsible for spurious O(1) Gibbs oscillations in the neighborhood of such jumps, and an overall deterioration to the unacceptable first-order convergence rate of spectral projections. The purpose is to regain the superior exponential accuracy in the piecewise smooth case, and this is achieved in two separate steps: (a) Localization. A detection procedure which based on appropriate choice of concentration factors which identify finitely many edges - both their location and their amplitudes. This is followed by (b) Mollification. We present a two-parameter family of spectral mollifiers which recover the data between the edges with exponential accuracy. We conclude with examples for applications in Computational Fluid Dynamics (-- formation of shocks), image and geophysical data processing.

November 29,
Scaling limits of Interacting Particle Systems

We will review the types of result known on the asymptotic behavior under scaling of interacting particle systems. We will use as examples the various types of simple exclusion processes. We will discuss both the typical behavior as well the estimation of probabilities for large deviations.

December 1, Friday, Colloquium
Yakov Sinai, Princeton University
Invariant measures for the Navier-Stoke system with random forcing and thermodynamical formalism.

December 6,
Antti Kupiainen, IAS, Princeton and Helsinki University
Ergodic properties of 2d turbulence

We consider the 2d Navier-Stokes equation with a random force which is white noise in time, and excites only a finite number of modes, for arbitary large Reynolds number. We prove that this Markov process has a unique invariant measure and is exponentially mixing in time. The proof uses ideas from Statistical Mechanics of 1d spin systems.

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