On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on ℝ³

2015/06/03 by Luhrmann, Jonas, Mendelson, Dana
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1506.01250

Abstract

We consider energy sub-critical defocusing nonlinear wave equations on ℝ³ and establish the existence of unique global solutions almost surely with respect to a unit-scale randomization of the initial data on Euclidean space. In particular, we provide examples of initial data at super-critical regularities which lead to unique global solutions. The proof is based on probabilistic growth estimates for a new modified energy functional. This work improves upon the authors' previous results in [25] by significantly lowering the regularity threshold and strengthening the notion of uniqueness.

Related

• A remark on almost sure global well-posedness of the energy-critical defocusing nonlinear wave equations in the periodic setting
• Probabilistic global well-posedness of the energy-critical defocusing\n quintic nonlinear wave equation on \ℝ³
• Random data Cauchy theory for nonlinear wave equations of power-type on ℝ³
• Almost sure global well-posedness for the energy-critical defocusing\n nonlinear wave equation on \ℝ^d, d=4 and 5
• Global well-posedness of the energy subcritical nonlinear wave equation with initial data in a critical space