Random data Cauchy theory for nonlinear wave equations of power-type on $\mathbb{R}^3$

Jonas Luhrmann; Dana Mendelson

Random data Cauchy theory for nonlinear wave equations of power-type on $\mathbb{R}^3$

Analysis of PDEs 2014-09-16 v2

Authors: Jonas Luhrmann , Dana Mendelson

Abstract

We consider the defocusing nonlinear wave equation of power-type on $\mathbb{R}^3$ . 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

nonlinear wave equations random data Cauchy theory global existence super-critical regularity Bourgain's high-low frequency decomposition averaging effects defocusing

Cite

@article{arxiv.1309.1225,
  title  = {Random data Cauchy theory for nonlinear wave equations of power-type on $\mathbb{R}^3$},
  author = {Jonas Luhrmann and Dana Mendelson},
  journal= {arXiv preprint arXiv:1309.1225},
  year   = {2014}
}

Comments

20 pages, 1 figure, typos corrected and minor changes. To appear in Comm. PDE

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