The Cauchy problem for quadratic and cubic Ostrovsky equation with negative dispersion. (October 2018) - Wang, JunFang; Yan, Wei
Abstract: In this paper, we consider the Cauchy problem for the quadratic and cubic Ostrovsky equation with negative dispersion ∂ x u t − β ∂ x 3 u + 1 k ∂ x ( u k ) − γ u = 0, β < 0, γ > 0, ( k = 2, 3 ) . Firstly, by using the Strichartz estimates instead of the Cauchy–Schwarz inequalities, we give an alternative proof of Lemma 1.2 of Isaza and Mejía (2006). Secondly, by using the Strichartz estimates instead of the Cauchy–Schwarz inequalities, we give an alternative proof of Lemma 1.3 of Isaza and Mejía (2007). Thirdly, we prove that the Cauchy problem for the cubic Ostrovsky equation is locally well-posed in H s ( R ) with s ≥ 1 4 . Finally, we prove that the Cauchy problem for the cubic Ostrovsky equation is not well-posed in H s ( R ) with s < 1 4 .