The Harrington Fellows Program
Department of Mathematics, UT Austin
APRIL 29 - 30, 2011

All people are invited to attend.
However, for organizational reasons, please fill out the registration form at the link below.

Participant Registration

The conference will take place in RLM 5.124 on Friday, April 29; and in RLM 12.104 and 12.166 on Saturday, April 30.

Abstracts

Ioan Bejenaru: Near soliton evolution for Schrödinger Maps

I will discuss a stability/instability result for equivariant Schrödinger Maps with energy close to that of the ground state.

Zaher Hani: Bilinear Strichartz estimates and global well-posedness of cubic NLS on compact Riemannian manifolds

We consider the problem of global well-posedness below energy norm of the defocusing cubic nonlinear Schrödinger equation on closed (compact without boundary) Riemannian manifolds. In this context, two new main difficulties emerge:

1. inevitable loss of derivatives in the bilinear Strichartz estimates, and
2. loss of Fourier analysis due to the absence of any group structure.

To overcome the first difficulty, a sharp bilinear oscillatory integral bound on Rd is proved and used to derive (sharp at relevant scales) bilinear Strichartz estimates for variable coefficient Schrödinger parametrices. To overcome the second difficulty, new multilinear spectral analysis techniques are developed, namely bounds on the spectral localization of products of Laplacian eigenfunctions and estimates on multilinear spectral multipliers of Coifman-Meyer type. This allows to prove global well-posedness of the defocusing 2d-cubic NLS in H^s(M ) for all s > 2/3, extending without any loss of regularity a similar result on T² (Bourgain'04) where Strichartz estimates only lose a logarithmic amount of derivatives and the multi-linear analysis is classical thanks to the group structure.

Rowan Killip: Cantor blowup

We will describe the construction of solutions to semilinear wave equations that blow up on general Cantor-like sets. This is joint work with Monica Visan.

Tadahiro Oh: Birkhoff and Poincare-Dulac normal form reductions on some dispersive PDEs

We discuss applications of Birkhoff normal form reductions and Poincare-Dulac normal form reductions to some dispersive PDEs. While both normal forms are used to simplify the flow so that it is expressed in terms of (higher and higher order) resonant parts, Birkhoff normal form is used for Hamiltonian systems and Poincare-Dulac is used for (formally) analytic vector fields. Bourgain'04 combined Birkhoff normal form reductions with the I-method to establish global well-posedness of quintic NLS at low regularity. In this talk, we combine Birkhoff normal form reductions with the upside-down I-method to establish upperbounds on the growth of high Sobolev norms of solutions to NLS. Then, we use the idea from Poincare-Dulac normal form reductions to establish unconditional well-posedness of mKdV and cubic NLS on T. In particular, for cubic NLS, we establish an infinite iteration scheme to obtain an a priori energy bound in L². The first part is joint work with J. Colliander (U of Toronto) and S. Kwon (KAIST), and the second part is joint with S. Kwon (KAIST).

Fabio Pusateri: Resonances and null structures in weakly dispersive equations

We will start by describing the method of "space-time resonances" introduced by Germain, Masmoudi, and Shatah to address the question of global existence of small solutions to nonlinear dispersive equations. We will then focus on the relation between resonances and null structures, and discuss a generalization of some classical results about systems of wave equations. We will also discuss some related application to scattering for critical equations of Schrödinger type.

Robert Strain: Global solutions to a non-local diffusion equation with quadratic non-linearity

In this talk we will present our recent proof of the global in time well-posedness of the following non-local diffusion equation with α in [0, 2/3):

The initial condition is positive, radial, and non-increasing; these conditions are propagated by the equation. There is however no size restriction on the initial data. This model problem is of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation with quadratic non-linearity. This is a joint work with Joachim Krieger.

Xin Yu: The Strauss conjecture on asymptotically Euclidean manifolds

We verify the Strauss conjecture for semilinear wave equations on asymptotically Euclidean manifolds when n = 3, 4. We also give an almost sharp lifespan for the subcritical case 2 <= p < p_c when n = 3. The main ingredients include a Keel-Smith-Sogge type estimate with 0 < μ < 1/2 and weighted Strichartz estimates of order two. This is joint work with Chengbo Wang.

Xiaoyi Zhang: Global wellposedness and scattering for the 3D-energy critical NLS on the domain exterior to a ball

I will discuss our recent work on the 3D defocusing nonlinear Schrödinger equation on the domain exterior to a ball. For the initial boundary value problem with Dirichlet boundary condition, we prove global wellposedness and scattering for radial solutions in the critical space. Similar results are also shown to be true in the case of Neumann boundary conditions.