The focusing energy-critical nonlinear wave equation with random initial data

Abstract

We consider the focusing energy-critical quintic nonlinear wave equation in three dimensional Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in ${\dot{H}}_x^s({\mathbb{R}}^3)\times H_x^{s-1}({\mathbb{R}}^3)$, for any $s>1/2$. By randomizing radial initial data in ${\dot{H}}_x^s({\mathbb{R}}^3)\times H_x^{s-1}({\mathbb{R}}^3)$ for $s>5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton which give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the first long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.

75 pages, minor typos corrected and updated references