




Partial Differential Equations / Équations aux dérivées
partielles
(Sponsored by the Pacific Institute for the Mathematical Sciences /
Parrainée par l'Institut Pacific pour les sciences mathématiques) (Richard Froese, Nassif Ghoussoub and Izabella Laba, Organizers)
 STEPHEN ANCO, Department of Mathematics, Brock University,
St. Catharines, Ontario
Wellposedness of the Cauchy problem for a
novel generalization of YangMills equations

We study a generalization of YangMills equations which is related to
wave maps (i.e. nonlinear sigma models) for Lie group targets.
Wave maps arise naturally in many areas of mathematical physics as a
geometrical nonlinear wave equation for a function of n ³ 1 space
variables and a time variable into a Riemannian target space. In the
case of Lie group target spaces, the wave map equation has a dual
formulation as a nonlinear abelian gauge field theory which we show can
be generalized to include a nonabelian YangMills interaction of the
gauge field. This yields a novel nonlinear geometrical wave equation
system combining features of both wave maps and YangMills equations.
We consider the Cauchy problem and prove existence, uniqueness, and
causality of (local in time) solutions with smooth compact support
initial data.
 CHANGFENG GUI, UBC and University of Connecticut
On some mathematical problems related to phase transition

Gradient theory of phase transitions has been studied by many
mathematicians. In this talk I will discuss some solved and unsolved
problems, particularly those regarding the basic configuration near
interfaces or junctions such as the De Giorgi conjecture.
 DIRK HUNDERTMARK, California Institute of Technology, California
An optimal L^{p} bound on the Krein spectral shift function

Let x_{A,B} be the Krein spectral shift function for a pair of
operators A, B, with C = AB trace class. We establish the bound

ó õ

F 
æ è

x_{A,B}(l) 
ö ø

dl £ 
ó õ

F 
æ è

x_{C,0}(l) 
ö ø

dl = 
¥ å
j = 1

[F(j)F(j1)]m_{j}(C), 

where F is any nonnegative convex function on [0,¥) with
F(0) = 0 and m_{j}(C) are the singular values of C. Specializing to
F(t) = t^{p}, p ³ 1 this improves a recent bound of Combes, Hislop,
and Nakamura.
 REINHARD ILLNER, Department of Mathematics and Statistics, University of Victoria,
Victoria, British Columbia V8W 3P4
Existence and use of kinetic equilibria in traffic dynamics,
diffusive granular flow and rarefied gases

The type and properties of equilibrium solutions in kinetic theory are
relevant in many applications, and notably in the closure of momentum
equations of the kinetic model. The prototype of this closure process
is the (formal) derivation of the compressible Euler equations from the
Boltzmann equation.
Kinetic models have recently been introduced with much success into the
fields of granular flow and traffic dynamics. In the case of granular
flow, it is known that the only equilibria are trivial, i.e., all
particles move with the same velocity. This changes if a diffusion term
D_{v} f is added to the collision term, for example by the
immersion of the system into a heat bath. One has to solve an equation
for a new type of kinetic equilibrium, called ``diffusive
equilibrium.'' We prove that diffusive equilibria do not exist for the
Boltzmann equation, but provide evidence for their existence for
granular flow.
A similar program is carried out for kinetic traffic flow models. We
show that the existence of nontrivial equilibria is closely related to
the nature of the braking and acceleration behavior of individual
drivers relative to their leading vehicles.
 ALEX IOSEVICH, ColumbiaMissouri
To be announced

 PETER PERRY, University of Kentucky, Lexington, Kentucky 405060027, USA
Zeta functions and determinants on hyperbolic manifolds of
infinite volume

Selberg's zeta function is the dynamical zeta function for geodesic
flow on a hyperbolic manifold. It is now known that for hyperbolic
manifolds of infinite volume without cusps, the divisor of the zeta
function is determined by eigenvalues and scattering resonances
together with the Euler characteristic of the manifold. We will discuss
a variety of applications of this result (due to Patterson and Perry in
even dimensions and Bunke and Olbrich in odd dimensions), including:
counting scattering resonances, counting closed geodesics, and defining
a `determinant of the Laplacian' whose zeros are the eigenvalues and
scattering resonances. The latter application (involving joint work
with David Borthwick) is especially interesting since standard
regularization tricks using spectral zeta functions appear to fail.
 DANIEL POLLACK, University of Washington, Department of Mathematics,
Seattle, Washington 981954350, USA
Gluing and wormholes for the Einstein constraint equations

Initial data for Einstein's general relativistic field equations for
the gravitational field consist of a Riemannian metric g and a
symmetric tensor K specified on a threemanifold S^{3}, with
g and K satisfying, the vacuum constraint equations

 R_{g}  K^{2}_{g} + (trK)^{2} = 0 

 

In 1952, ChoquetBruhat proved that Einstein's vacuum field equations
G_{mn} º R_{mn}[1/2]R g_{mn} = 0, form a
locally well posed hyperbolic system. Thus for any choice of
(S^{3}, g, K) satisfying the vacuum constraint equations,
there exists an e > 0 and a Lorentz metric g defined on the
spacetime manifold S^{3}×(e, +e), with g
satisfying the Einstein equations, with g being the induced
intrinsic metric of the hypersurface S^{3}×{0}, and with
K being the induced second fundamental form. Using this result, one
can seek to understand solutions to the field equations by
understanding solutions to the vacuum constraint equations.
We will present a general gluing construction for solutions to the
vacuum constraint equations with trK = t constant. Using the
``conformal method'' of ChoquetBruhat, Lichnerowicz and York, this
involves solving a semicoupled system consisting of a linear elliptic
system for a vector field and a scalar semilinear elliptic equation for
the induced metric. Applications of the construction to connected sums
of spacetimes and the existence of wormholes will be given.
This is joint work with Jim Isenberg and Rafe Mazzeo.
 RANDALL PYKE, Ryerson Polytechnic University, Toronto, Ontario
Characterization of bound states for nonlinear wave and
Schrodinger equations

Bound states are solutions that are localized in space, uniformly in
time (some examples are solitons and traveling waves). Bound states
play a prominent role in in the scattering theory of these equations
since we expect that in the large time limit, a general solution will
converge (in a local norm) to a boundstate while radiating energy (the
convergence being driven by dispersion). However, except for special
equations (namely, integrable equations and equations with repulsive
nonlinearities), it is difficult to determine whether or not an
equation possesses bound states. We address this issue by first
proving that bound states are almost periodic in time. Then we can
apply previous established necessary conditions for the existence of
almost periodic solutions to state necessary conditions for the
existence of bound states.
 HART SMITH, University of Washington, Seattle, Washington, USA
Fundamental solutions for low regularity wave equations

We discuss recent results on absolute bounds for the Green kernel
associated to a general wave equation with metric of class C^{2}.
These bounds can be used to establish Strichartz type estimates for
such equations, as well as to establish L^{p} norm estimates for
spectral clusters on Riemannian manifolds with low regularity metrics.
 CATHERINE SULEM, Department of Mathematics, University of Toronto, Toronto,
Ontario
The waterwave problem and its longwave and modulational
limits

We review some of the basic mathematical results on solutions of the
waterwave problem, such as existence of traveling waves and the
wellposedness of the initial value problem. We also discuss rigorous
results which concern the various asymptotic scalings that lead to the
principal canonical equations of mathematical physics.
 GUNTHER UHLMANN, Department of Mathematics, University of Washington, Seattle,
Washington 98195, USA
Determining riemannian manifolds from the DirichlettoNeumann
map

We describe some recent results concerning the determination of an
isometric copy of a compact Riemannian manifold with boundary from the
DirichlettoNeumann map associated to the LaplaceBeltrami operator.
 MANWAH WONG, Department of Mathematics and Statistics, York University, Toronto,
Ontario M3J 1P3
The special Hermite semigroup

We give a formula for the oneparameter strongly continuous semigroup
generated by the special Hermite operator L in terms of
pseudodifferential operators of the Weyl type, i.e. Weyl
transforms, and use it to obtain an estimate for the solution of the
heat equation governed by L in terms of the initial data.

