### Local well-posedness and a priori bounds for the modified Benjamin-Ono equation

Zihua Guo

#### Abstract

We prove that the complex-valued modified Benjamin-Ono (mBO) equation is analytically locally well posed if the initial data $\phi$ belongs to $H^s$ for $s\geq 1/2$ with $\| {\phi} \| _{L^2}$ sufficiently small, without performing a gauge transformation. The key ingredient is that the logarithmic divergence in the high-low frequency interaction can be overcome by a combination of $X^{s,b}$ structure and smoothing effect structure. We also prove that the real-valued $H^\infty$ solutions to the mBO equation satisfy a priori local-in-time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s>1/4$.

#### Article information

Source
Adv. Differential Equations, Volume 16, Number 11/12 (2011), 1087-1137.

Dates
First available in Project Euclid: 17 December 2012