


Qualitative Behaviour and Controllability of Partial Differential Equations / Comportement qualitatif et controlabilité des EDP (Org: Holger Teismann, Acadia University)
 DAVID AMUNDSEN, Carleton University
Resonant Solutions of the Forced KdV Equation

The forced Kortewegde Vries (fKdV) Equation provides a canonical
model for evolution of weakly nonlinear dispersive waves in the
presence of additional effects such as external forcing or variable
topography. While the symmetries and integrability of the underlying
KdV structure facilitate extensive analysis, in this generalized
setting such favourable properties no longer hold. Through physical
and numerical experimentation it is known that a rich family of
resonant steady solutions exist, yet qualitative analytic insight into
them is limited. Based on hierarchical perturbative and matched
asymptotic approaches we present a formal mathematical framework for
construction of solutions in the small dispersion limit. In this way
not only obtaining accurate analytic representations but also
important a priori insight into the response of the system as it
is detuned away from resonance. Specific examples and comparisons in
the case of a fundamental periodic resonant mode will be presented.
Joint work with M. P. Mortell (UC Cork) and E. A. Cox (UC Dublin).
 SEAN BOHUN, Penn State
The WignerPoisson System with an External Coulomb Field

This system of equations describes the time evolution of the quantum
mechanical behaviour of a large ensemble of particles in a vacuum
where the long range interactions between the particles can be taken
into account. The model also facilitates the introduction of external
classical effects. As tunneling effects become more pronounced in
semiconductor devices, models which are able to bridge the gap between
the quantum behaviour and external classical effects become
increasingly relevant. The WP system is such a model.
Local existence is shown by a contraction mapping argument which is
then extended to a global result using macroscopic control
(conservation of probability and energy). Asymptotic behaviour of the
WP system and the underlying SP system is established with a
priori estimates on the spatial moments.
Finally, conditions on the energy are given which
(a) ensure that the solutions decay and
(b) ensure that the solutions do not decay.
 SHAOHUA CHEN, University College of Cape Breton
Boundedness and Blowup for the Solution of an
ActivatorInhibitor Model

We consider a general activatorinhibitor model

ì ï ï í
ï ï î

u_{t} = eDu  mu + 
u^{p}
v^{q}

, 

v_{t} = D Dv  nv + 
u^{r}
v^{s}






with the Neumann boundary conditions, where rq > (p1)(s+1). We
show that if r > p1 then the solutions exist long time for all
initial values and if r > p1 and q < s+1 then the solutions are
bounded for all initial values. However, if r < p1 then, for some
special initial values, the solutions will blow up.
 STEPHEN GUSTAFSON, University of British Columbia, Mathematics Department,
1984 Mathematics Rd., Vancouver, BC V6T 1Z2
Scattering for the GrossPitaevskii Equation

The GrossPitaevskii equation, a nonlinear Schroedinger equation with
nonzero boundary conditions, models superfluids and BoseEinstein
condensates. Recent mathematical work has focused on the finitetime
dynamics of vortex solutions, and existence of vortexpair traveling
waves. However, little seems to be known about the longtime
behaviour (eg. scattering theory, and the asymptotic stability of
vortices). We address the simplest such problemscattering around
the vacuum statewhich is already tricky due to the
nonselfadjointness of the linearized operator, and "longrange"
nonlinearity. In particular, our present methods are limited to
higher dimensions. This is joint work in progress with K. Nakanishi
and T.P. Tsai.
 HORST LANGE, Universitaet Köln, Weyertal 8690, 50931 Köln, Germany
Noncontrollability of the nonlinear HartreeSchrödinger and
GrossPitaevskiiSchrödinger equations

We consider the bilinear control problem for the nonlinear
HartreeSchrödinger equation [HS] (which plays a prominent
role in quantum chemistry), and for the
GrossPitaevskiiSchrödinger equation [GPS] (of the
theory of BoseEinstein condensates); for both systems we
study the case of a bilinear control term involving the position
operator or the momentum operator. A target state u_{T} Î L^{2}(R^{3}) is said to be reachable from an initial state u_{0} Î L^{2}(R^{3}) in time T > 0 if there exists a control s.t. the system
allows a solution state u(t,x) with u(0,x) = u_{0}(x), u(T,x) = u_{T}(x). We prove that, for any T > 0 and any initial datum u_{0} Î L^{2} (R^{3}) \{0}, the set of nonreachable target states
(in time T > 0) is relatively L^{2}dense in the sphere {u Î L^{2}(R^{3})  u_{L2} = u_{0}_{L2}} (for both [HS] and [GPS]).
The proof uses Fourier transform, estimates for Riesz potentials for
[HS], estimates for the Schrödinger group associated with the
Hamiltonian D+x^{2} for [GPS].
 HAILIANG LI, Department of Pure and Applied Mathematics, Osaka University, Japan
On Wellposedness and Asymptotics of Multidimensional
Quantum Hydrodynamics

In the modelling of semiconductor devices in nanosize, for instance,
MOSFET's and RTD's where quantum effects (like particle tunnelling
through potential barriers and builtup in quantum wells) take place,
the quantum hydrodynamical equations are important and dominative in
the description of the motion of electron or hole transport under the
selfconsistent electric field. These quantum hydrodynamic equations
consist of conservation laws of mass, balance laws of momentum forced
by an additional nonlinear dispersion (caused by the quantum (Bohm)
potential), and selfconsistent electric field.
In this talk, we shall review the recent progress on the
multidimensional quantum hydrodynamic equations, including the
mathematical modelings based on the moment method applied to the
WignerBoltzmann equation, rigorous analysis on the wellposedness for
general, nonconvex pressuredensity relation and regular large initial
data, long time stability of steadystate under a quantum subsonic
condition, and globalintime relaxation limit from the quantum
hydrodynamic equations to the quantum driftdiffusion equations, and
so on.
Joint with A. Jüngel, P. Marcati, and A. Matsumura.
 DONG LIANG, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3
Analysis of the SFDTD Method for ThreeDimensional Maxwell
Equations

The finitedifference timedomain (FDTD) method for Maxwell's
equations, firstly introduced by Yee, is a very popular numerical
algorithm in computational electromagnetics. However, the traditional
FDTD scheme is only conditionally stable. The computation of
threedimensional problems by the scheme will need much more computer
memory or become extremely difficult when the size of spatial steps
becomes very small. Recently, there is considerable interest in
developing efficient schemes for the problems.
In this talk, we will present a new splitting finitedifference
timedomain scheme (SFDTD) for the general threedimensional
Maxwell's equations. Unconditional stability and convergence are
proved for the scheme by using the energy method. The technique of
reducing perturbation error is further used to derive a high order
scheme. Numerical results are given to illustrate the performance of
the methods.
This research is joint work with L. P. Gao and B. Zhang.
 KIRSTEN MORRIS, University of Waterloo
Controller Design for Partial Differential Equations

Many controller design problems of practical interest involve systems
modelled by partial differential equations. Typically a numerical
approximation is used at some stage in controller design. However, not
every scheme that is suitable for simulation is suitable for
controller design. Misleading results may be obtained if care is not
taken in selecting a scheme. Sufficient conditions for a scheme to be
suitable for linear quadratic or H_{¥} controller design have
been obtained. Once a scheme is chosen, the resulting approximation
will in general be a large system of ordinary differential
equations. Standard control algorithms are only suitable for systems
with model order less than 100 and special techniques are required.
 KEITH PROMISLOW, Michigan State University
Nonlocal Models of Membrane Hydration in PEM Fuel Cells

Polymer electrolyte membrane (PEM) fuel cells are unique energy
conversion devices, effeciently generating useful electric voltage
from chemical reactants without combustion. They have recently
captured public attention for automotive applications for which they
promise high performance without the pollutants associated with
combustion.
>From a mathematical point of view the device is governed by coupled
systems of elliptic, parabolic, and degenerate parabolic equations
describing the heat, mass, and ion tranpsort through porous medias and
polymer electrolyte membranes. This talk will describe the overall
funtionality of the PEM fuel cell, presenting analysis of the slow,
nonlocal propagation of hydration fronts within the polymer
electrolyte membrane.
 TAIPENG TSAI, University of British Columbia, Vancouver
Boundary regularity criteria for suitable weak solutions of
NavierStokes equations

I will present some new regularity criteria for suitable weak
solutions of the NavierStokes equations near boundary in space
dimension 3. Partial regularity is also analyzed.
This is joint work with Stephen Gustafson and Kyungkeun Kang.

