


Nonlinear Partial Differential Equations
Org: Walter Craig (McMaster), Robert McCann (Toronto) and Catherine Sulem (Toronto) [PDF]
 ANDREI BIRYUK, McMaster University
Infinite energy solutions of multidimensional Burgerstype
equations
[PDF] 
We consider the Burgerstype equation
u_{t} + (f(u) ·Ñ) u = nDu 

in R^{n} with L^{¥} (not necessary potential) initial
vector field. We show that there is a unique global smooth solution,
satisfying the initial condition in a weak sense.
 MARINA CHUGUNOVA, McMaster University, Hamilton, ON
Diagonalization of the Hamiltonian Coupledmode System in the
Presence of Symmetry
[PDF] 
We consider the Hamiltonian coupledmode system derived in nonlinear
optics, photonics, and atomic physics. Since the Dirac energy is not
bounded, the standard stability analysis based on Lyapunov functionals
does not work for gap solitons. Spectral stability of gap solitons
depends on eigenvalues of the linearized coupledmode equations, which
are equivalent to a fourbyfour Dirac system with signindefinite
metric. In the special class of symmetric nonlinear potentials, we
construct a blockdiagonal representation of the linearized
couplemode equations when the spectral problem reduces to two coupled
twobytwo Dirac systems.
 GALIA DAFNI, Concordia University, 1455 de Maisonneuve Blvd. West,
Montreal, Quebec H3G 1M8
Divcurl lemmas for local Hardy spaces and BMO
[PDF] 
Nonhomogeneous versions of the divcurl lemma of Coifman, Lions, Meyer
and Semmes are given in the context of local Hardy spaces (in the
sense of Goldberg), on R^{n} and on domains. A version for
BMO on domains (joint work with Chang and Sadosky) will also be
discussed.
 PHILIPPE GUYENNE, McMaster University, 1280 Main St. W, Hamilton, ON
Solitary wave interactions
[PDF] 
We study numerically the interactions of solitary water waves for the
full Euler equations. Both collisions of co and counterpropagating
solitary waves are considered. We show that the collisions are
inelastic and generate small residual waves. Comparisons with KdV
predictions and lab experiments will be discussed.
 SLIM IBRAHIM, McMaster
On the global and illposedness for a 2D NLS with exponential
type nonlinearity
[PDF] 
In this work, we define a new criticality notion for solutions of a 2D
NLS equation with exponential type nonlinearity. We prove global
wellposedness in the subcritical and critical cases, while
illposedness is shown in the supercritical case.
 REINHARD ILLNER, University of Victoria
Jeffery's equation: The motion of ellipsoidal bodies in
incompressible viscous flows
[PDF] 
In 1922 Jeffery derived an ODE for the motion of the principal axis of
a rotationally symmetric, elongated ellipsoid in a Stokes flow. The
equation is
p¢ = 
1
2

(curlu_{0} Ùp + l(S[u_{0}] p (p^{T} S[u_{0}] p) p) 

where S[u_{0}] is the symmetric part of the velocity Jacobian, and
l is a parameter depending on the geometry of the ellipsoid.
Jeffery's equation finds applications in flow problems arising in the
manufacture of artifacts with immersed objects (glass, metal or
plastic "sticks") to modify the elastoplastic behaviour of the
product. In this talk, I will discuss some of these applications, and
I will outline a new derivation of the Jeffery equation via asymptotic
analysis from an exterior boundary value problem for the
NavierStokes equations.
This is joint work with Michael Junk, Konstanz.
 ROBERT JERRARD, University of Toronto
Refined Jacobian estimates and vortex dynamics for the
GrossPitaevsky equation
[PDF] 
We study dynamics of vortices in solutions of the GrossPitaevsky
equations on 2dimensional domains. These equations model certain
superfluids, and they contain a dimensionless parameter e.
Results characterizing vortex dynamics in the limit e® 0
have been known since the late '90s. Our results improve on this
earlier work in several ways: they are valid for fixed small
e rather than only for a sequence of solutions with
e tending to zero; and they are valid for larger numbers
of vortices and longer time scales than previous work. The refined
Jacobian estimates mentioned in the title of the talk play a crucial
technical role in the proof and are possibly of independent interest.
This is joint work with D. Spirn.
 NIKY KAMRAN, McGill University
Decay of scalar waves in Kerr geometry
[PDF] 
We consider the Cauchy problem for the scalar wave equation in the
Kerr geometry for a rotating black hole in equilibrium, and prove that
the solutions corresponding to initial data compactly supported
outside the event horizon end to zero in L^{¥}_{loc} as t ®¥ in the BoyerLindquist slicing. The analysis is more
difficult that in the case of the Dirac equation, owing to the
presence of an ergosphere, that is a region of spacetime outside the
event horizon in which the Killing vector corresponding to time
translations becomes spacelike. This is joint work with Felix
Finster, Joel Smoller and ShingTung Yau.
 JOSE ALBERTO MONTERO, McMaster University
Stable Critical points to the Ginzburg Landau equations
[PDF] 
In this talk we describe some stable solutions to the Ginzburg Landau
equation that develop singularities as a small parameter goes to zero.
Our main tools are the weak jacobians of Jerarrd and Soner, together
with some standard gammaconvergence techniques.
 VLADISLAV PANFEROV, McMaster University, Hamilton, Ontario
Regular solutions of the Boltzmann equation in 1D in space
[PDF] 
I will discuss the problem of regularity of weak solutions of the
nonlinear Boltzmann equation with onedimensional (planewave)
symmetry. By imposing certain truncations on possible configurations
of collisions of particles we are able to prove that the initialvalue
problem has global weak solutions in L^{¥} provided that the
initial data are essentially bounded and satisfy a mild additional
condition meaning that the velocity averages obtained by free
streaming remain bounded for all times. The obtained regularity is
enough to guarantee uniqueness and propagation of bounds for
derivatives.
 FRIDOLIN TING, Lakehead University, 955 Oliver Road, Thunder Bay, ON
P7B 5E1
(In)stability of pinned fundamental vortices
[PDF] 
We study the (in)stability of pinned ±1 vortex solutions to the
GinzburgLandau equations with external potentials in R^{2}. For
smooth and sufficiently small external potentials, there exists a
perturbed vortex solution centered near critical points of the
external potential (called pinned vortices). We show that pinned
vortices centered near maxima (resp. minima) of the potential are
orbitally stable (resp. unstable) w.r.t. gradient and Hamiltonian
flow. This is joint work with I. M. Sigal.
 AGNES TOURIN, McMaster University, HH, 1280 Main Street West, Toronto, ON
L8S 4K1
A particular fully nonlinear degenerate parabolic equation
arising in Finance
[PDF] 
I will present a monotone and stable approximation of the fully
nonlinear degenerate parabolic equation derived recently by Cheridito,
Soner and Touzi from the stochastic control problem of
superreplicating a contingent claim under gamma constraints. I will
show some numerical results that are of practical interest. This
equation is nonstandard but the theory of viscosity solutions still
provides good methods for solving this type of problem in a robust
manner and the convergence of the numerical scheme can be proved.
 MAXIM TROKHIMTCHOUK, California  Berkeley


