# The IVP for the Kuramoto-Sivashinsky equation in low regularity Sobolev spaces

## TL;DR

This paper establishes local and global well-posedness of the Kuramoto-Sivashinsky equation in Sobolev spaces with regularity above 1/2, and analyzes the flow map and solution behavior as a parameter approaches zero.

## Contribution

It proves well-posedness in low regularity Sobolev spaces and demonstrates the sharpness of the regularity threshold for the flow map.

## Key findings

- Well-posedness for s > 1/2 in Sobolev spaces
- Flow map is not C^2 for s < 1/2
- Solution behavior analyzed as parameter μ approaches zero

## Abstract

In this work, we study the initial-value problem associated with the Kuramoto-Sivashinsky equation. We show that the associated initial value problem is locally and globally well-posed in Sobolev spaces $H^s(\mathbb{R})$, where $s>1/2$. We also show that our result is sharp, in the sense that the flow-map data-solution is not $C^2$ at origin, for $s<1/2$. Furthermore, we study the behavior of the solutions when $\mu\downarrow 0$.