The Cauchy problem for the Ostrovsky equation with positive dispersion

2015/05/22 by Wei Yan, Yan, Wei, Yongsheng Li +5
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #math.AP

paper · pdf · doi:10.48550/arxiv.1505.05995

31. arXiv admin note: substantial text overlap with arXiv:1411.0890

openalex publication_date 2015/05/22 · arxiv created 2017/06/15 · arxiv updated 2017/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is devoted to studying the Cauchy problem for the Ostrovsky equation ∂_x(u_t-β∂_x³u +(1)/(2)∂_x(u²)) -γu=0, with positive β and γ. This equation describes the propagation of surface waves in a rotating oceanic flow. We first prove that the problem is locally well-posed in H^-(3)/(4)(\R). Then we reestablish the bilinear estimate, by means of the Strichartz estimates instead of calculus inequalities and Cauchy-Schwartz inequalities. As a byproduct, this bilinear estimate leads to the proof of the local well-posedness of the problem in H^s(\R) for s>-(3)/(4), with help of a fixed point argument.

Citations

• The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity

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