Sharp local well-posedness for the "good" Boussinesq equation

TL;DR

This paper establishes the precise regularity threshold for local well-posedness of the 'good' Boussinesq equation on both periodic and nonperiodic domains, improving previous results through advanced bilinear estimates.

Contribution

It proves the sharp local well-posedness in $H^{-1/2}$ and ill-posedness below this regularity, refining earlier thresholds and employing novel bilinear estimates in $X^{s,b}$ spaces.

Findings

Well-posedness in $H^{-1/2}( ext{domain})$

Ill-posedness below $H^{-1/2}$

Improved regularity thresholds over previous work

Abstract

In the present article, we prove the sharp local well-posedness and ill-posedness results for the "good" Boussinesq equation on $\mathbb{T}$; the initial value problem is locally well-posed in $H^{-1/2}(\mathbb{T})$ and ill-posed in $H^s(\mathbb{T})$ for $s<-1/2$. Well-posedness result is obtained from reduction of the problem into a quadratic nonlinear Schr\"odinger equation and the contraction argument in suitably modified $X^{s,b}$ spaces. The proof of the crucial bilinear estimates in these spaces, especially in the lowest regularity, rely on some bilinear estimates for one dimensional periodic functions in $X^{s,b}$ spaces, which are generalization of the bilinear refinement of the $L^4$ Strichartz estimate on $\mathbb{R}$. Our result improves the known local well-posedness in $H^s(\mathbb{T})$ with $s>-3/8$ given by Oh and Stefanov (2012) to the regularity threshold $H^{-1/2}(\mathbb{T})$. Similar ideas also establish the sharp local well-posedness in $H^{-1/2}(\mathbb{R})$ and ill-posedness below $H^{-1/2}$ for the nonperiodic case, which improves the result of Tsugawa and the author (2010) in $H^s(\mathbb{R})$ with $s>-1/2$ to the limiting regularity.