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



Sagun Chanillo (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Magdalena Czubak (Department of Mathematics, University of Colorado, Boulder, Col., U.S.A.)

Dana Mendelso (Department of Mathematics, University of Chicago, Illinois, U.S.A.)

Andrea Nahmod (Department of Mathematics, University of Massachusetts, Amherst, Mass., U.S.A.)

Gigliola Staffilani (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)


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