Periodic Cauchy Problem for one Two-dimensional Generalization of the Benjamin-Ono Equation in Sobolev Spaces of Low Regularity

Abstract

In this work we prove that the initial value problem (IVP) associated to the two-dimensional Benjamin-Ono equation $$\begin{array}{rl}{u}_t+\mathcal{H}\Delta u+u{u}_x& =0,(x,y)\in {\mathbb{T}}^2,t\in \mathbb{R},\\ u(x,y,0)& =u_0(x,y),\end{array}\},$$ where $\mathcal{H}$ denotes the Hilbert transform with respect to the variable $x$ and $\Delta$ is the Laplacian with respect to the spatial variables $x$ and $y$, is locally well-posed in the periodic Sobolev space $H^s({\mathbb{T}}^2)$, with $s>7/4$.