Random Data Cauchy Theory for Nonlinear Wave Equations of Power-Type on ℝ³

Abstract

We consider the defocusing nonlinear wave equation of power-type on ℝ3. We establish an almost sure global existence result with respect to a suitable randomization of the initial data. In particular, this provides examples of initial data of super-critical regularity which lead to global solutions. The proof is based upon Bourgain's high-low frequency decomposition and improved averaging effects for the free evolution of the randomized initial data.