On a higher dimensional version of the Benjamin--Ono equation

Abstract

We consider a higher dimensional version of the Benjamin--Ono equation, ${\partial }_{t}u-{\mathcal{R}}_1\Delta u+u{\partial }_{x_1}u=0$, where ${\mathcal{R}}_1$ denotes the Riesz transform with respect to the first coordinate. We first establish sharp space--time estimates for the associated linear equation. These estimates enable us to show that the initial value problem for the nonlinear equation is locally well-posed in $L^2$-Sobolev spaces $H^s({\mathbb{R}}^d)$, with $s>5/3$ if $d=2$ and $s>d/2+1/2$ if $d\ge 3$. We also provide ill-posedness results.

We also show that in dimension 2 our results are sharp