# SCIENTIFIC PROGRAMS AND ACTIVITIES

April 16, 2014

## Working Group on Nonlinear Evolution Equations

### 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)