### Abstracts

November 27 & December 3, 2002

**Jim Colliander**, University of Toronto

*Well-posedness for quasilinear (uniformly) parabolic PDE*

This talk will describe a proof of well-posedness for the initial-boundary
value problem for quasilinear parabolic PDE. The discussion will combine
ideas presented in the books "Linear and Quasilinear Equations
of Parabolic Type" by Ladyzhenskaya - Solonnikov - Uralceva and
"Elliptic PDE of 2nd Order" by Gilbarg-Trudinger.

January 29, 2003

**Fridolin Ting**, University of Toronto.

Asymptotic behavior (near finite extinction time) for the fast diffusion
equation with exponent m=(N-2)/(N+2), N >2

I will summarize the results obtained by Del Pino and Saez (2001). They
proved that for continuous, positive and sufficiently nice decaying
initial data, the solution goes asymptotically (as t approaches vanishing
time) to a self similar solution parameterized by \lambda > 0 and
x in R^N. Techniques involved are similar to those used by Ye (1984)
on global existence and convergence of Yamabe flow.

Supporting papers:

M. Del Pino & M. Saez, "On the Extinction Profile for Solutions
of u_t = \Laplace u^(N-2)/(N+2)", Indiana University Mathematics
Journal, Vol. 50, No. 1 (2001)

R. Ye, "Global existence and convergence of Yamabe Flow",
Journal of Differential Geometry, 39, (1994), 35-50

February 5 & 12, 2003

**Jim Colliander**, University of Toronto.

Variations of a theme by Morawetz

The identification of monotone-in-time quantities underpins some of
the basic insights into the long-time behavior on nonlinear Schrodinger
evolutions. For example, in the focusing setting, the variance identity
implies a monotone behavior implying the existence of blow-up solutions.
In the defocusing case, Morawetz identities provide spacetime norm bounds
implying scattering behavior. This talk describes a unified approach
to obtaining monotone-in-time quantities for certain NLS evolutions,
generalizing these two classic examples. A scattering result for the
3d cubic defocusing case will also be discussed. This talk describes
joint work with M. Keel, G. Staffilani, H. Takaoka and T. Tao

February 26, 2003

**Robert Jerrard**, University of Toronto

*Dynamics of Ginzburg-Landau vortices: general background*

In this talk I will describe some results from the calculus of variations
that describe the structure and stability proposerties of Ginzburg-Landau
vortices. These results are useful for studying questions about dynamics.
This talk will be aimed at non-experts.

Reference:

The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differential
Equations 14 (2002), no. 2, 151--191.

March 5, 2003

**Robert Jerrard**, University of Toronto

*Dynamics of Ginzburg-Landau vortices* *II*

In this talk I will sketch the derivation of dynamical laws for Ginzburg-Landau
vortices for several different types of evolution equation, and I will
discuss general stability results that are useful in actual proofs of
all these results.

March 12, 2003

**Robert Jerrard**, University of Toronto

*Long time asymptotics for Ginzburg-Landau heat flow*

In this talk i will go over a paper that describes the long-time limit
of finite-energy solutions of the Ginzburg-Landau heat flow on the plane.

References:

(1) Bauman, Patricia(1-PURD); Chen, Chao-Nien(1-IN); Phillips, Daniel(1-PURD);
Sternberg, Peter(1-IN) Vortex annihilation in nonlinear heat flow for
Ginzburg-Landau systems. (English. English summary) European J. Appl.
Math. 6 (1995), no. 2, 115--126. 35Q99 (82D55)

(2) Kalantarov, V. K.; Lady\v zenskaja, O. A. Stabilization of the solutions
of a certain class of quasilinear parabolic equations as $t\rightarrow
\infty $. (Russian) Sibirsk. Mat. Zh. 19 (1978), no. 5, 1043--1052,
1214. 35K60 (35B40)

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