Improved local well-posedness for the periodic "good" Boussinesq equation

TL;DR

This paper establishes improved local well-posedness results for the periodic 'good' Boussinesq equation in lower regularity spaces using a normal form approach, revealing smoothing effects even for rough initial data.

Contribution

It extends well-posedness results to lower regularity spaces for the periodic 'good' Boussinesq equation using normal form techniques.

Findings

Well-posedness holds for s > -3/8 in H^s × H^{s-2}.

The solution exhibits smoothing effects for mean-zero initial data.

The approach improves upon previous results for s > -1/4.

Abstract

We prove that the "good" Boussinesq model with the periodic boundary condition is locally well-posed in the space $H^{s}\times H^{s-2}$ for $s > -3/8$. In the proof, we employ the normal form approach, which allows us to explicitly extract the rougher part of the solution. This also leads to the conclusion that the remainder is in a smoother space $C([0,T], H^{s+a}), where $0 <= a < \min (2s+1, 1/2)$. If we have a mean-zero initial data, this implies a smoothing effect of this order for the non-linearity. This is new even in the previously considered cases $s > -1/4$.