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

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  • 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 &\hspace{-2mm} = 0, \qquad\qquad (x, y)\in\mathbb R^2, \; t\in\mathbb R, \\ u(x, y, 0)&\hspace{-2mm} = u_0(x, y), \end{array} \right\}\, , $

    where $0 < \alpha\leq1$, $D_x^{\alpha}$ denotes the operator defined through the Fourier transform by

    $(D_x^{\alpha}f)\widehat{\;\;}(\xi, \eta): = |\xi|^{\alpha}\widehat{f}(\xi, \eta)\, , ~~~~~~~~~~~~~~~~~~~~~~~~(0.1)$

    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 $s>\dfrac32+\dfrac14(1-\alpha)$.

    Mathematics Subject Classification: 35Q53.


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