Global well-posedness for the Benjamin equation in low regularity

Abstract

In this paper we consider the initial value problem of the Benjamin equation $${\partial }_{t}u+\nu \stackrel{˝}{(}{\partial }_x^{2}u)+\mu {\partial }_x^{3}u+{\partial }_x{u}^2=0,$$ where $u:\mathbb{R}\times [0,T]\mapsto \mathbb{R}$, and the constants $\nu ,\mu \in \mathbb{R},\mu \ne 0$. We use the I-method to show that it is globally well-posed in Sobolev spaces $H^s(\mathbb{R})$ for $s>-3/4$. Moreover, we use some argument to obtain a good estimative for the lifetime of the local solution, and employ some multiplier decomposition argument to construct the almost conserved quantities.

29 pages