Almost sure boundedness of iterates for derivative nonlinear wave equations

TL;DR

This paper proves that for certain derivative nonlinear wave equations with random initial data, the iterates of the solution remain almost surely bounded over a uniform time interval, highlighting stability in a probabilistic setting.

Contribution

It establishes almost sure boundedness of all Picard iterates for derivative nonlinear wave equations with null form structures, extending previous counterexamples.

Findings

Almost sure boundedness of Picard iterates in $C_t(I; \dot H_x^1)$

Uniform time interval for boundedness regardless of iteration order

Addresses stability for equations with quadratic derivative nonlinearities

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 $H_x^1\times L^2_x$. 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 $C_t(I ; \dot H_x^1)$.