# Almost optimal local well-posedness for improved modified Boussinesq equations

## TL;DR

This paper establishes near-optimal local well-posedness results for improved modified Boussinesq equations, providing an alternative proof and identifying regularity thresholds for the flow map smoothness.

## Contribution

It offers a new proof of local well-posedness in a specific function space and determines the regularity threshold where the flow map ceases to be smooth.

## Key findings

- Alternative proof of local well-posedness in $(H^s\cap L^\infty)\times (H^s\cap L^\infty)$ for $s\geq 0$.
- Flow map is not smooth from $H^s\times H^s$ to $H^s$ for $s<0$.
- Identifies the regularity threshold for Picard iteration applicability.

## Abstract

In this article, we investigate a class of improved 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. 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.