Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces

2008/12/10 by Zihua Guo, Guo, Zihua · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.0812.1825

Abstract

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation ∂_t u+|∂_x|^1+α∂_x u+uu_x=0, u(x,0)=u₀(x), is locally well-posed in the Sobolev spaces H^s for s>1-α if 0≤ α≤ 1. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru \citeIKT to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkovin \citeMST. Moreover, as a bi-product we prove that if 0

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