Global well-posedness for the Benjamin equation in low regularity

TL;DR

This paper proves the global well-posedness of the Benjamin equation in low regularity Sobolev spaces using the I-method, multiplier decomposition, and almost conserved quantities, extending understanding of its solution behavior.

Contribution

It establishes global well-posedness for the Benjamin equation in Sobolev spaces with regularity above -3/4, employing novel analytical techniques like the I-method and multiplier decomposition.

Findings

Global well-posedness in H^s for s > -3/4

Development of almost conserved quantities for the Benjamin equation

Improved estimates for the lifetime of local solutions

Abstract

In this paper we consider the initial value problem of the Benjamin equation $$ \partial_{t}u+\nu \H(\partial^2_xu) +\mu\partial_{x}^{3}u+\partial_xu^2=0, $$ where $u:\R\times [0,T]\mapsto \R$, and the constants $\nu,\mu\in \R,\mu\neq0$. We use the I-method to show that it is globally well-posed in Sobolev spaces $H^s(\R)$ 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.