


Partial Differential Equations / Équations aux dérivées
partielles (Org: Victor Ivrii and/et John Toth)
 RICHARD BEALS, Yale University, New Haven, Connecticut, USA
Propagators for some degenerate hyperbolic equations
[PDF] 
Exact propagators for two classes of degenerate hyperbolic equations
can be obtained from explicit Green's functions for certain degenerate
elliptic equations. The latter, in turn, trace back to analogous
results for some subelliptic operators associated to weakly
pseudoconvex domains.
 NICOLAS BURQ, Université Paris SudOrsay, France
Nonlinear Schrödinger boundary value problems
[PDF] 
The purpose of this talk is to present some recent results obtained in
collaboration with P. Gérard and N. Tzvetkov (Université Paris
SudOrsay). On the well posedness of nonlinear Schródinger
equations on domains (with Dirichlet boundary conditions):
i¶_{t} u + Du + F(u) = 0, u_{¶W} = 0, u_{t=0} = u_{0} Î H^{s}_{0}(W). 

I will present two type of results:
1) In the ball, we show that if the non linearity has a gauge
invariance (typically F(u) = u^{p} u), the problem is not well posed,
even for initial data in some Sobolev spaces above the scaling critical
index.
2) If the domain is the exterior of a bounded obstacle satisfying a
nontrapping condition, we show that the problem is locally well posed
for any initial data in H^{1}_{0}(W) or L^{2}(W) (hence
globally well posed in the case of defocusing nonlinearities) for a
large class of non linearities.
 TANYA CHRISTIANSEN, University of Missouri, Columbia, Missouri 65211, USA
Pseudospectra in automorphic scattering
[PDF] 
We study surfaces with (hyperbolic) cusp ends. The generator, B, of
the LaxPhillips semigroup has spectrum given in terms of the
eigenvalues of the Laplacian and the poles of the scattering matrix.
We show that away from the continuous spectrum of the Laplacian, the
norm of the resolvent of B+1/2 is comparable (in the nonphysical
plane) to the norm of the scattering matrix. In particular, for the
modular surface this means that the norm of the resolvent of B+1/2 is
comparable to z(2s)^{1} when 0 < e £ Âs £ 1/2e. This is joint work with M. Zworski.
 JIM COLLIANDER, University of Toronto, Toronto, Ontario
Global existence and scattering for rough solutions of 3d
NLS
[PDF] 
This talk will describe a new result establishing global existence and
scattering of solutions for the cubic defocusing nonlinear Schrodinger
equation in three space dimensions. The main ingredient is a new
Morawetztype inequality which provides a global spacetime L^{4}
bound. This talk concerns joint work with Keel, Staffilani, Takaoka
and Tao.
 ANDREW COMECH, Department of Mathematics, University of North CarolinaChapel
Hill, Chapel Hill, North Carolina 27599, USA
Purely nonlinear instability of minimal energy standing waves
[PDF] 
For a variety of nonlinearities, the nonlinear Schroedinger equation is
known to possess localized quasistationary solutions (``standing
waves''). We prove that in the generic situation the standing wave of
minimal energy among all other standing waves is unstable. This case
was falling out of the scope of the classical paper by Grillakis,
Shatah, and Strauss on orbital stability of standing waves. An
interesting feature of the problem is the absence of (exponential)
instability in the linearized system; in this sense, the resulting
instability is ``purely nonlinear''. Essentially, the instability is
caused by higher algebraic degeneracy of zero eigenvalue in the
spectrum of the linearized system. The result can be generalized to
abstract Hamiltonian systems with U(1) symmetry.
 PETER GREINER, Department of Mathematics, University of Toronto, Toronto
Ontario M5S 3G3
Subelliptic PDE's and subRiemannian geometry
[PDF] 
Subelliptic PDE's induce the geometric ideas of subRiemannian
geometry. Using these subRiemannian geometric invariants I shall
construct some examples of fundamental solutions to subelliptic
PDE's.
 DMITRY JAKOBSON, Department of Mathematics, McGill University, Montreal,
Quebec H3A 2K6
Critical points and L_{p} norms of eigenfunctions
[PDF] 
We discuss several results on critical points and L_{p} norms of
eigenfunctions of Laplacians on Riemannian manifolds.
 CHRIS SOGGE, John's Hopkins University
Eigenfunction estimates and applications
[PDF] 
I shall present a simple proof of sharp eigenfunction estimates for
manifolds with boundary. Using these and the finite propagation speed
for the wave equation one can prove some sharp estimates in harmonic
analysis, such as the L^{1} mapping properties of Riesz means. I shall
also discuss some open problems.
 CATHERINE SULEM, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
On asymptotic stability of solitary waves for
nonlinear Schrödinger equations
[PDF] 
We suppose that it possesses stable solitary wave solutions and we
investigate their asymptotic stability, that is the longtime behavior
of solutions whose initial conditions are close to a stable solitary
wave.
The method, initiated in [1], is based on the spectral decomposition
of solutions on the eigenspaces associated to the discrete and
continuous spectrum of the linearized operator near the solitary wave.
Under some hypothesis on the structure of the spectrum, we prove that,
asymptotically in time, the solution decomposes into a solitary wave
and a dispersive part described by the free Schrödinger equation. We
calculate explicitly the time behavior of the correction.
This is a joint work with V. Buslaev.
References
 [1]
 V. Buslaev and G. Perleman, Scattering for the
nonlinear Schrödinger equation: states close to a soliton.
St. Petersburg Math. J. 4(1993), 11111142.
 ANDRAS VASY, Department of Mathematics, MIT, Cambridge, Massachusetts 02139, USA
Manybody scattering and symmetric spaces
[PDF] 
I will talk about the use of methods from manybody scattering in the
study of the Laplacian on higher rank symmetric spaces. I focus on the
relationship of threebody scattering and SL(3,R)/SO(3,R). I will
describe the asymptotics of the Green's function at infinity (Taylor
series at infinity), extending results of Anker, Guivarch, Ji and
Taylor. I also describe the analytic continuation of the resolvent in
the spectral parameter through the continuous spectrum. The new
results presented are joint work with Rafe Mazzeo.
 JARED WUNSCH, Northwestern University, Evanston, Illinois, USA
The sojourn relation and the Schrödinger equation
[PDF] 
We discuss the construction of a parametrix for the timedependent
Schrödinger equation in nontrapping regions of a manifold X with
asymptotically conic ends. The construction, in the framework of the
``Legendrian distributions'' of MelroseZworski (generalized by Hassell and
Vasy) involves a phase function parametrizing a certain relation between
points in the cosphere bundle of X and the (rescaled) cotangent bundle of
the boundary of the compactification. We call this relation the `sojourn
time' owing to its similarity to the sojourn time in scattering theory
introduced by Guillemin. As a consequence, we are able to prove some new
results on propagation of singularities for the Schrödinger operator.
 STEVE ZELDITCH, John's Hopkins University
Quantum ergodicity of boundary values of eigenfunctions
[PDF] 
The purpose of my talk is to outline a proof of a new result obtained
jointly with Andrew Hassell (ANU) that L^{2}normalized boundary values
(i.e. Cauchy data) u_{j}^{\flat} of eigenfunctions of the
Laplacian on piecewise smooth convex domains W with corners and
with ergodic billiards are quantum ergodic. In other words, that
áA_{hj} u_{j}^{\flat},u_{j}^{\flat} ñ® 
ó õ

B^{*} ¶W

s_{A} dm_{B} in density one, 

for all semiclassical pseudodifferential operators on ¶W. The relevant notion of boundary values u_{j}^{b} depends on
the boundary condition B, as does the classical limit measure
dm_{B}. Our methods cover Dirichlet, Neumann, Robin and more
general boundary conditions. The proof is based on the analysis of
boundary layer potentials and their boundary restrictions as
quantizations of the billiard map.

