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# Communications in Analysis and Geometry

## Volume 28 (2020)

### Number 4

### The First of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

### Almost sure boundedness of iterates for derivative nonlinear wave equations

Pages: 943 – 977

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n4.a5

#### Authors

#### 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 $H^1_x \times L^2_x$. In contrast to the counterexamples of Zhou [73] and Foschi–Klainerman [23], we obtain a uniform time interval $I$ on which the Picard iterates of all orders are almost surely bounded in $C_t (I ; \dot{H}^1_x)$.

S. Chanillo is funded in part by NSF DMS-1201474.

M. Czubak is funded in part by the Simons Foundation #246255.

D. Mendelso was funded in part by NSF DMS-1128155 during the completion of this work.

A. Nahmod is funded in part by NSF DMS-1201443 and DMS-1463714.

G. Staffilani is funded in part by NSF DMS-1362509, DMS-1462401, by the John Simon Guggenheim Foundation, and by the Simons Foundation.

Received 21 May 2018

Accepted 1 April 2019

Published 1 October 2020