Probabilistic global well-posedness and Gibbs measure evolution for radial nonlinear Schr\"odinger and wave equations on the unit ball.
- Series
- CDSNS Colloquium
- Time
- Monday, February 24, 2014 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Aynur Bulut – Univ. of Michigan
In this talk we will discuss recent work, obtained in collaboration with
Jean Bourgain, on new global well-posedness results along Gibbs measure
evolutions for the radial nonlinear wave and Schr\"odinger equations posed
on the unit ball in two and three dimensional Euclidean space, with
Dirichlet boundary conditions.
We consider initial data chosen according to a Gaussian random process
associated to the Gibbs measures which arise from the Hamiltonian structure
of the equations, and results are obtained
almost surely with respect to these probability measures. In particular,
this renders the initial value problem supercritical in the sense that
there is no suitable local well-posedness theory for
the corresponding deterministic problem, and our results therefore rely
essentially on the probabilistic structure of the problem.
Our analysis is based on the study of convergence properties of solutions.
Essential ingredients include probabilistic a priori bounds, delicate
estimates on fine frequency interactions, as well as the use of invariance
properties of the Gibbs measure to extend the relevant bounds to
arbitrarily long time intervals.