Time decay for the bounded mean oscillation of solutions of the Schr\"odinger and wave equations

TL;DR

This paper investigates the decay properties of the BMO norm of solutions to Schr"odinger and wave equations with $L_2$ initial data, providing counterexamples to certain decay conjectures using probabilistic methods.

Contribution

It introduces counterexamples to decay conjectures for the BMO norm of solutions, employing random methods like Brownian motion, and addresses an open problem in the field.

Findings

Counterexamples to decay conjectures in BMO norm

Use of Brownian motion in PDE analysis

Addresses an open problem later solved by Keel and Tao

Abstract

Let $u(x,t)$ be the solution of the Schr\"odinger or wave equation with $L_2$ initial data. We provide counterexamples to plausible conjectures involving the decay in $t$ of the $\BMO$ norm of $u(t,\cdot)$. The proofs make use of random methods, in particular, Brownian motion. (Since this paper was written, the unsolved problem remaining in this paper has been solved by Keel and Tao.)