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

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 $\mu\downarrow 0$.