Almost optimal local well-posedness for modified Boussinesq equations

Abstract

In this article, we investigate a class of modified Boussinesq equations, for which we provide first an alternate proof of local well-posedness in the space $(H^s\cap L^\infty)\times (H^s\cap L^\infty)(\mathbb{R})$ ($s\geq 0$) to the one obtained by Constantin and Molinet [7]. Secondly, we show that the associated flow map is not smooth when considered from $H^s\times H^s(\mathbb{R})$ into $H^s(\mathbb{R})$ for s<0, thus providing a threshold for the regularity needed to perform a Picard iteration for these equations.

Keywords

• modified boussinesq equation
• well-posedness
• ill-posedness