




Partial Differential Equations / Équations aux dérivées
partielles (Pengfei Guan, Organizer)
 STEPHEN ANCO, Brock

 DAN BURNS, Department of Mathematics, University of Michigan, Ann Arbor
Michigan 481091109, USA
Algebraicization of entire Grauert tubes

(This is joint work with Raúl Aguilar of Tufts University.) A Grauert
tube is a complex manifold (structure) defined on a neighborhood of
the zerosection in the (co)tangent bundle of an underlying real
analytic Riemannian manifold. The radius of existence, or convergence,
of this structure is the maximal radius in terms of the Riemannian
length function on tangent vectors on which the structure can be
defined. This radius is controled by subtle geometric and analytic
features of the Riemannian metric. Examples of entire tubes (tubes
with infinite radius) are rare, and we are trying to understand,
perhaps classify, this phenomenon. We prove two theorems in this
direction. First, we show that an entire Grauert tube is, in a natural
way, an affine algebraic variety. Second, for a tube of complex
dimension two, we show that this affine variety is either the product
of two punctured planes (for a flat metric on a real torus), or is the
affine quadric in complex three space, if the underlying real surface
is the two sphere. The first result relies heavily on Demailly's
theory of MongeAmpère measures and the fact that the length function
on tangent vectors is a solution of the homogeneous complex
MongeAmpère equation, while the second follows from the classical
theory of minimal models for projective algebraic surfaces.
 S. CHANILLO, Rutgers

 WALTER CRAIG, Brown

 NASSIF GHOUSSOUB, University of British Columbia, British Columbia
To be announced

 ZHENGCHAO HAN, Department of Mathematics, Rutgers University,
Piscataway, New Jersey 088548019, USA
Existence and compactness results of the Yamabe problem with
boundary

Let (M^{n},g_{0}) be a compact Riemannian manifold with boundary
¶M. We consider the problem of looking for a metric g on
M conformal to g_{0} and having constant scalar curvature c_{1} on
M and constant mean curvature c_{2} on ¶M. This is a
generalization of the Yamabe problem to manifolds with boundary.
Previous existence results on this problem were obtained by
J. Escobar in the case c_{1} = 0 and c_{2} appropriately normalized, and
in the case c_{2} = 0 or near 0 and c_{1} appropriately normalized. We
will present new existence and apriori estimates of
such metrics for any prescribed c_{2}(after c_{1} has been
appropriately normalized).
 VICTOR IVRII, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
Spectral asymptotics with sharp remainder estimates for operators
with low regularity condition

Weyl conjecture holds for secondorder operators with C^{1+s}
coefficients in domains with C^{1+s} boundary. This and similar
theorems, proven recently, are subject of my talk.
 HOWARD JACOBOWITZ, Rutgers University, Camden, New Jersey 08102, USA
On the existence of elliptic pseudodifferential operators

Given two complex vector bundles E, F over a compact manifold M,
we study the existence and classification of elliptic
pseudodifferential operators from sections of E to sections of F.
Using the Gysin sequence of the cosphere bundle of M, it is easy to
see the necessary condition that c_{i}(E) = c_{i}(F) for i < m and
c_{m}(E)  c_{m}(F) = k c where c_{i} denote the ith Chern class
and c is the Euler characteristic of M. In the case of line
bundles this condition is also sufficient. (This is joint work with
Gerardo Mendoza of Temple University).
 SONGYING LI, University of California, Irvine, California, USA, and University of
Maryland, Maryland, USA
Boundedness of the second derivatives of Green's operators on
H^{1} and BMO

Let W be a bounded domain in R^{n}. Let A(x) = (a_{jk}(x))
be an n×n symmetric matrix such that a_{jk} Î L^{¥}(W) and lI_{n} £ A(x) £ L I_{n} for a.e.
x Î W with 0 < l £ L < ¥. Let G [f] the
Green operator be the solution operator of the Dirichlet problem of
nondivergent form elliptic equation:

n å
j,k = 1

a_{jk}(x) 
¶^{2} u ¶x_{j} ¶x_{k}

= f in W and u = 0 on ¶W. 

We give sufficient, and almost necessary conditions on the smoothness
of a_{jk} and ¶W so that the second derivatives of the
Green's operator (¶^{2}_{i j} G[f]) are bounded on H^{1}(W)
and bounded BMO(W).
 XINAN MA, East China Normal University, China
To be announced

 ROBERT MCCANN, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
Kinetic equilibration rates for granular media

This joint work with José Carillo and Cedric Villani provides an
algebraic decay rate bounding the time required for velocities to
equilibrate in a spatially homogeneous flowthrough model representing
the continuum limit of a gas of particles interacting through slightly
inelastic collisions. The rate is obtained by reformulating the
dynamical problem as the gradient flow of a convex energy on an
infinitedimensional Riemannian manifold. An abstract theory is
developed for gradient flows which shows how degenerate convexity (or
even nonconvexity)if uniformly controlledwill quantify
contractivity of the flow.
 HAYATO NAWA, Nagoya University, Chikusaku, Nagoya 4648602, Japan
Blowup of solutions of a class of nonlinear Schrödinger equations

We consider the existence of singular solutions of the nonlinear
Schrödinger equations with local interactions, and their
asymptotic behavior.
 DMITRY PELINOVSKY, University of Toronto and McMaster University
Degenerate rational solutions of the KP hierarchy

Rational solutions of the KP hierarchy are reviewed by means of the
vertex operator methods. I will show how to construct and classify
general rational solutions of the lreduced and kconstrained KP
hierarchies as well as of their generalized counterparts.
 E. POLETSKY, Syracuse
To be announced

 JOEL SMOLLER, Department of Mathematics, University of Michigan, Ann Arbor,
Michigan 48109, USA
Nonformation of vacuum states for the compressible NavierStokes
equations

Under general conditions on the initial data, we prove that for the
compressible NavierStokes equations, if no vacuum is present at t = 0,
then no vacuum is present at all t > 0.
(This is joint work with David Hoff.)
 CATHERINE SULEM, Department of Mathematics, University of Toronto, Toronto, Ontario
Asymptotic stability of solitary waves for nonlinear
Schrödinger equations

We investigate the long time behavior of solutions whose initial
conditions are close to solitary waves. In particular, we compute
explicitly the leading terms, determine the rate of convergence as
well as the period of oscillations, and give uniform bounds on the
reminders.
 DANIEL TATARU, Northwestern University
On Strichartz type estimates and the nonlinear wave equation

We present some new estimates of Strichartz type for the wave equation
with nonsmooth coefficients. These estimates can in turn be used to
improve the local theory for nonlinear hyperbolic equations.
 JOHN TOTH, McGill
To be announced


