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

## TL;DR

This paper proves local well-posedness for a two-dimensional generalization of the Benjamin-Ono equation in periodic Sobolev spaces with regularity above 7/4, expanding understanding of its mathematical properties.

## Contribution

It establishes the local well-posedness of the 2D Benjamin-Ono equation in Sobolev spaces of low regularity, a novel result for this class of equations.

## Key findings

- Well-posedness holds for s > 7/4 in H^s(𝕋^2).
- The analysis extends the theory of Benjamin-Ono equations to two dimensions.
- The results provide a foundation for further study of solutions in low regularity spaces.

## Abstract

In this work we prove that the initial value problem (IVP) associated to the two-dimensional Benjamin-Ono equation $$\left. \begin{array}{rl} u_t+\mathcal H \Delta u +uu_x &\hspace{-2mm}=0,\qquad\qquad (x,y)\in\mathbb T^2,\; t\in\mathbb R,\\ u(x,y,0)&\hspace{-2mm}=u_0(x,y), \end{array} \right\}\,,$$ 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 $s>7/4$.