On the well-posedness for the Chen-Lee equation in periodic Sobolev spaces

We prove that the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation u_t + uu_x + βHu_xx + (Hu_x - u_xx) = 0, where x ∈ T, t > 0, η > 0 and H denotes the usual Hilbert transform, is locally and globally well-posed in the Sobolev spaces H^s(T) for any s > - ½. We also prove some ill-posedness issues when s < -1.

DOI: https://doi.org/10.15446/recolma.v50n1.62187

Sobre el buen planteamiento de la ecuación de Chen-Lee en espacios de Sobolev periódicos

Ricardo Pastrán¹, Oscar Riaño¹

¹ Universidad Nacional de Colombia, Bogotá, Colombia. rapastranr@unal.edu.co, ogrianoc@unal.edu.co

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Abstract

We prove that the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation u_t + uu_x + β H u_xx + η (H u_x - u_xx) = 0, where x ∈ T, t > 0, η > 0 and H denotes the usual Hilbert transform, is locally and globally well-posed in the Sobolev spaces H^s(T) for any s > -½. We also prove some ill-posedness issues when s < -1.

Keywords: Cauchy problem, local and global well-posedness, Benjamin-Ono equation.

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2010 Mathematics Subject Classification: 34A12, 35Q35.

Resumen

Probamos que el problema de valor inicial asociado a una perturbación de la ecuación de Benjamín-Ono o ecuación de Chen-Lee u_t + uu_x + β H u_xx + η (H u_x - u_xx) = 0, donde x ∈ T, t > 0, η > 0 y H denota la transformada de Hilbert usual, es localmente y globalmente bien planteado en espacios de Sobolev H^s(T) para cualquier s > -½. También probamos un tipo de mal planteamiento cuando s < -1.

Palabras claves: Problema de Cauchy, buen planteamiento local y global, ecuación de Benjamín-Ono.

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(Recibido: julio de 2015 Aceptado: enero de 2016)