Adapted Linear-Nonlinear Decomposition And Global Well-posedness For Solutions To The Defocusing Cubic Wave Equation On $\mathbb{R}^{3}$

TL;DR

This paper establishes global well-posedness for the defocusing cubic wave equation on a0a0a0 with initial data in certain Sobolev spaces, by controlling the variation of an almost conserved quantity over long time intervals.

Contribution

It introduces a novel decomposition approach to estimate the variation of an almost conserved quantity, extending well-posedness results to lower regularity data.

Findings

Proves global well-posedness for data in H^s d7 H^{s-1} with 13/18 < s < 1.

Develops a method to control the variation of an almost conserved quantity over long time intervals.

Decomposes solutions into linear and nonlinear parts to facilitate estimates.

Abstract

We prove global well-posedness for the defocusing cubic wave equation with data in $H^{s} \times H^{s-1}$, $1>s>{13/18}$. The main task is to estimate the variation of an almost conserved quantity on an arbitrary long time interval. We divide it into subintervals. On each of these subintervals we write the solution as the sum of its linear part adapted to the subinterval and its corresponding npnlinear part. Some terms resulting from this decomposition have a controlled global variation and other terms have a slow local variation.