Applied Mathematics Seminar
Oana Pocovnicu
Princeton University
Probabilistic global well-posedness of the energy-critical defocusing nonlinear wave equation on euclidean spaces
Abstract: We consider the energy-critical defocusing nonlinear wave equation
(NLW) on $\mathbb{R}^d$, $d = 3, 4, 5$. In the deterministic setting, Christ,
Colliander, and Tao showed that this equation is ill-posed below the
energy space $H^1 × L^2$.
In this talk, we take a probabilistic approach. More precisely, we
prove almost sure global existence and uniqueness for NLW with rough
random initial data below the energy space. The randomization that we
use is naturally associated with the Wiener decomposition and with
modulation spaces. The proof is based on a probabilistic perturbation
theory and on probabilistic energy bounds.
If time allows, we will briefly discuss how the above strategy also
yields a conditional almost sure global well-posedness result below
the scaling critical regularity, for the defocusing cubic nonlinear
Schrödinger equation on euclidean spaces. This talk is partially
based on joint work with Tadahiro Oh, and on joint work with Árpád
Bényi and Tadahiro Oh.
Monday March 30, 2015 at 4:00 PM in SEO 636