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

Eddye Bustamante; José Jiménez Urrea; Jorge Mejía

doi:10.3934/cpaa.2019057

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2019, Volume 18, Issue 3: 1177-1203. Doi: 10.3934/cpaa.2019057

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

Departamento de Matemáticas, Universidad Nacional de Colombia, A. A. 3840 Medellín, Colombia

Received: February 28, 2018

Revised: July 31, 2018

Published: May 2019

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

Keywords:

Benjamin Ono equation.

Mathematics Subject Classification: 35Q53.

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