CAMP/Nonlinear PDEs Seminar

Wednesdays at 4pm in Eckhart 202.

Spring 2009 Schedule

April 1
Monica Visan, University of Chicago
The focusing energy-critical nonlinear Schrodinger equation

We introduce the defocusing and focusing energy-critical nonlinear Schrodinger equations. We then present joint work with Rowan Killip on the focusing problem.

April 22 at 3pm in E202
Sijue Wu, University of Michigan
Almost global well-posedness of the 2-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period $[0, T/\epsilon]$ for initial data of type $\epsilon\Phi$, where $T$ depends only on $\Phi$. We show that for such data there exists a unique solution for a time period $[0, e^{T/\epsilon}]$. This is achieved by better understandings of the nature of the nonlinearity of the water wave equation.

May 6
Ovidiu Savin, Columbia University
Minimizers of convex functionals arising in random surfaces

We discuss $C^1$ regularity of minimizers to $\int F(\nabla u)dx$ in two dimensions for certain classes of non-smooth convex functionals $F$. In particular our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of Kenyon, Okounkov and others. This is a joint work with D. De Silva.

May 13
Alexander Kiselev, University of Wisconsin
Nonolocal maximum principles for active scalars

I will discuss nonlocal maximum principles for a family of dissipative active scalar equations. This technique gives proofs of global regularity for the critical surface quasi-geostrophic equation, Burgers equation and some other related models. The talk is based on works joint with Fedya Nazarov, Roman Shterenberg and Sasha Volberg.

June 3
John Gibson, Georgia Institute of Technology
Equilibria, traveling waves, and periodic orbits in plane Couette flow

Equilibrium, traveling wave, and periodic orbit solutions of pipe and plane Couette flow can now be computed precisely at Reynolds numbers above the onset of turbulence. These invariant solutions capture the complex dynamics of coherent roll-streak structures in wall-bounded flows and provide a framework for connecting wall-bounded turbulence to dynamical systems theory. We present a number of newly computed solutions of plane Couette flow and observe how they are visited by the turbulent flow. What emerges is a picture of low-Reynolds turbulence as a walk among a set of weakly unstable invariant solutions.

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