On the interpolation with the potential bound for global solutions of the defocusing cubic wave equation on T2

TL;DR

This paper proves global existence of solutions for the defocusing cubic wave equation on T2 in certain Sobolev spaces by bounding the Sobolev norm using an interpolation method involving potential bounds.

Contribution

It introduces a novel interpolation approach with potential bounds to establish global solutions for the cubic wave equation on T2.

Findings

Solutions exist globally in Hs(T2) for s > 2/5.

Bounded Sobolev norms near maximal existence times.

Growth of mollified energy is controlled via interpolation.

Abstract

We prove that the solutions of the defocusing cubic wave equation on T2 exist globally in time in Hs(T2) for s > 2/5 by contradiction. Assuming that one of the maximal times of existence is finite, we prove that the Sobolev norm of each of these solutions is bounded in an open neighborhood of it by estimating the growth of a mollified energy through the interpolation with the potential bound for the low frequency part.