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.

Keywords:

• Almost sure global well-posedness
• Nonlinear wave equation
• Random initial data

2010 Mathematics Subject Classfication:

• 35L05
• 35R60
• 35Q55

Acknowledgments

The authors would like to sincerely thank Gigliola Staffilani for all of her help. They are grateful to Michael Eichmair for his encouragement and support and to Andrea Nahmod for stimulating discussions. The first author wishes to thank his advisor Michael Struwe for his support and guidance.