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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)

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STEPHEN ANCO, Department of Mathematics, Brock University, St. Catharines, Ontario
Well-posedness of the Cauchy problem for a novel generalization of Yang-Mills equations

We study a generalization of Yang-Mills 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 non-abelian Yang-Mills interaction of the gauge field. This yields a novel nonlinear geometrical wave equation system combining features of both wave maps and Yang-Mills 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 = A-B trace class. We establish the bound

ó õ F æ è |xA,B(l)| ö ø  dl £ ó õ F æ è |x|C|,0(l)| ö ø  dl = ¥ å j = 1  [F(j)-F(j-1)]mj(C),

where F is any non-negative 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 non-trivial equilibria is closely related to the nature of the braking and acceleration behavior of individual drivers relative to their leading vehicles.

ALEX IOSEVICH, Columbia-Missouri
To be announced

PETER PERRY, University of Kentucky, Lexington, Kentucky  40506-0027, 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  98195-4350, 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 three-manifold S³, with g and K satisfying, the vacuum constraint equations

¸g K - ÑtrK = 0 Rg - |K|2g + (trK)2 = 0

In 1952, Choquet-Bruhat 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 (S³, g, K) satisfying the vacuum constraint equations, there exists an e > 0 and a Lorentz metric g defined on the spacetime manifold S³×(-e, +e), with g satisfying the Einstein equations, with g being the induced intrinsic metric of the hypersurface S³×{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 Choquet-Bruhat, Lichnerowicz and York, this involves solving a semi-coupled 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 space-times 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 C². 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 water-wave problem and its long-wave and modulational limits

We review some of the basic mathematical results on solutions of the water-wave problem, such as existence of traveling waves and the well-posedness 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 Dirichlet-to-Neumann map

We describe some recent results concerning the determination of an isometric copy of a compact Riemannian manifold with boundary from the Dirichlet-to-Neumann map associated to the Laplace-Beltrami operator.

MAN-WAH WONG, Department of Mathematics and Statistics, York University, Toronto, Ontario  M3J 1P3
The special Hermite semigroup

We give a formula for the one-parameter strongly continuous semigroup generated by the special Hermite operator L in terms of pseudo-differential 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.