Almost sure boundedness of iterates for derivative nonlinear wave equations

Abstract

We study nonlinear wave equations on ${\mathbb{R}}^{2+1}$ with quadratic derivative nonlinearities, which include in particular nonlinearities exhibiting a null form structure, with random initial data in $H_x^1\times L_x^2$. In contrast to the counterexamples of Zhou \cite{Zhou} and Foschi-Klainerman \cite{FK}, we obtain a uniform time interval $I$ on which the Picard iterates of all orders are almost surely bounded in $C_t(I;{\dot{H}}_x^1)$.

22 pages; Updated in light of an error in the previous version, statement of main result and proof have been modified