The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations

Abstract

In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equation $$\left. \begin{array}{rl} u_t+D_x^{\alpha} u_x +\mathcal Hu_{yy} +uu_x &=0,\qquad\qquad (x,y)\in\mathbb R^2,\; t\in\mathbb R, u(x,y,0)&=u_0(x,y), \end{array} \right\}\,,$$ where $0<\alpha\leq1$, $D_x^{\alpha}$ denotes the operator defined through the Fourier transform by \begin{align} (D_x^{\alpha}f)\widehat{\;}(\xi,\eta):=|\xi|^{\alpha}\widehat{f}(\xi,\eta)\,, \end{align} and $\mathcal H$ denotes the Hilbert transform with respect to the variable $x$, is locally well posed in the Sobolev space $H^s(\mathbb R^2)$ with $s>\dfrac32+\dfrac14(1-\alpha)$.