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

2019/01/18 by Bustamante, Eddye, Urrea, José Jiménez, Mejía, Jorge
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.06329

Abstract

In this work we prove that the initial value problem (IVP) associated to the two-dimensional Benjamin-Ono equation . u_t+\mathcal H Δu +uu_x · amp;\hspace-2mm=0, (x,y)∈\mathbb T², t∈\mathbb R,
u(x,y,0) · amp;\hspace-2mm=u₀(x,y), \ , where \mathcal H denotes the Hilbert transform with respect to the variable x and Δ is the Laplacian with respect to the spatial variables x and y, is locally well-posed in the periodic Sobolev space H^s(\mathbb T²), with s>7/4.

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