(Sep. 24) Chunjing Xie
Title: Classical Solutions of Two Dimensional Inviscid Rotating
Shallow Water System
(Oct. 1) Barbara Keyfitz
Title: The sonic line as a free boundary: Stability under perturbations ---
(Oct. 8) Todd Kapitula
Title: Orbital stability of periodic waves for generalized KdV ---
(Oct. 15) Jeff Rauch
Title: Finite Speed and Uniqueness in the
Cauchy Problem
for Symmetrizable Hyperbolic Systems ---
(Oct. 22) Folkmar Bornemann
Title: Fredholm determinants versus Painlevé equations:
a numerical perspective with applications ---
(Oct. 29) Cedric Villani
Title: Landau damping: relaxation without dissipation ---
(Nov. 12) Qing Han
Title: The Linearized System for Isometric Embeddings and Its Characteristic
Variety ---
In this talk, we will discuss the global existence and
asymptotic behavior of classical solutions for two dimensional inviscid
Rotating Shallow Water system with small initial data subject to the
zero-relative-vorticity constraint. One of the key steps is a reformulation
of the problem into a symmetric quasilinear Klein-Gordon system, for which
the global existence of classical solutions is then proved with combination
of the vector field approach and the normal forms. We also probe the case of
general initial data and reveal a lower bound for the lifespan that is
almost inversely proportional to the size of the initial relative vorticity.
This is a joint work with Bin Cheng.
The study of self-similar solutions of multidimensional conservation laws
leads to systems of equations that change type. Change of type occurs either
across a transonic shock or at a sonic line. Often the sonic line appears as
a free boundary in the formulation of the problem. Some recent numerical
(and experimental) discoveries of a new kind of shock reflection ('Guderley
Mach reflection') lead to interesting and still unresolved questions
concerning the nature of the self-similar solutions in this generic case.
In this talk, I will present some analysis of a simple model for this
phenomenon, using the transonic small disturbance equation. The simplified
problem seems amenable to analysis, but we are just beginning to make
progress. This is a report on current joint work with Allen Tesdall and
Kevin Payne.
We first investigate the (in)stability of spatially periodic waves
to the generalized KdV equation for various power
nonlinearities when the perturbation has the same period as that of the
wave. Solutions of the integrable modified KdV equation are studied
analytically in detail, as well as small solutions for higher-order pure
power nonlinearities. The stability question for KdV has been answered when
the period of the perturbation is the same as that of the underlying cnoidal
wave. However, up until now the question of the orbital stability of these
waves with respect to periodic perturbations whose period is an integer
multiple of the wave period was still open, as in this case the wave is not
a local minimizer of a constrained energy. By using the integrable structure
associated with KdV we are able to show that these energetically unstable
waves are indeed orbitally stable. This is joint work with Bernard
Deconinck.
Precise finite speed, in the sense of that the domain of influence is
a subset of the union of
influence curves through the support of the initial data is proved
for hyperbolic systems
symmetrized by pseudodifferential operators in the spatial variables.
From this, uniqueness
in the Cauchy problem at spacelike hypersurfaces is derived by a H
olmgren style duality
argument. Sharp finite speed is derived from an estimate for
propagation in each direction.
Propagation in a fixed direction is proved by regularizing the
problem in the orthogonal
directions. Uniform estimates for the regularized equations is proved
using pseudodiffential
techniques of Beals-Fefferman type.
Explicit solutions in integrable systems (such as in
inverse scattering, in the Ising chain, in infinite
dimensional representation theory, and in random
matrix theory) can frequently be expressed as operator
determinants and their derivatives. However, their
numerical evaluation has generally been thought to rely
on alternative analytic expressions, most famously in
terms of the Painlevé transcendents. We will discuss a
simple and yet extremely effective numerical method for
operator determinants and its application to the
computation of probability distributions in random
matrix theory.
This is lecture 3 in a series on related problems arising in the
classical kinetic theory of plasmas. A recurring theme is that messy
physics sometimes leads to the discovery of beautiful mathematical
phenomena.
It is not necessary to have attended lecture N in order to understand
lecture N+1.
Lecture 1 will start with a presentation for non-experts of a few classical
equations and problems in classical kinetic theory, especially plasma
dynamics.
I will discuss some of the most famous physical phenomena, and give a brief
overview of results developed in lectures 2 and 3.
To conclude this lecture I'll present an unexpected application
of the kinetic theory of plasma to the resolution (around the turn of
the century) of a conjecture.
Lecture 2 is devoted to the large-time analysis of the Vlasov-Fokker-Planck
equation with smoothed interaction. This is the opportunity to present
a general nonlinear "hypocoercivity theory", and a new approach to
hypoelliptic regularity.
Lecture 3 investigates Landau's surprising collisionless damping,
classically treated in the linear regime. Understanding the
perturbative, nonlinear regime was an open problem for half a century.
I shall present the solution which I recently devised with Clement Mouhot.
We prove a conjecture of Bryant, Griffiths, and Yang concerning the
characteristic variety for the determined isometric embedding system.
In particular, we show that the characteristic variety is not smooth
for any dimension greater than 3. This is accomplished by introducing
a smaller yet equivalent linearized system, in an appropriate way,
which facilitates analysis of the characteristic variety.