- Probabilistic scattering for the 4D energy-critical defocusing nonlinear wave equation
- 01/25/2018
- 11:00 AM - 12:00 PM
- C304 Wells Hall
- Jonas Lührmann
We consider the Cauchy problem for the energy-critical defocusing
nonlinear wave equation in four space dimensions. It is known that for
initial data at energy regularity, the solutions exist globally in time
and scatter to free waves. However, the problem is ill-posed for initial
data at super-critical regularity, i.e. for regularities below the
energy regularity.
In this talk we study the super-critical data regime for this Cauchy
problem from a probabilistic point of view, using a randomization
procedure that is based on a unit-scale decomposition of frequency
space. We will present an almost sure global existence and scattering
result for randomized radially symmetric initial data of super-critical
regularity. This is the first almost sure scattering result for an
energy-critical dispersive or hyperbolic equation for scaling
super-critical initial data.
The main novelties of our proof are the introduction of an approximate
Morawetz estimate to the random data setting and new large deviation
estimates for the free wave evolution of randomized radially symmetric data.
This is joint work with Ben Dodson and Dana Mendelson.