1 research outputs found
On a higher dimensional version of the Benjamin--Ono equation
We consider a higher dimensional version of the Benjamin--Ono equation,
, where
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 -Sobolev
spaces , with if and if . We also provide ill-posedness results.Comment: We also show that in dimension 2 our results are shar