The Cauchy problem for higher-order modified Camassa–Holm equations on the circle. (October 2019) - Yan, Wei; Li, Yongsheng; Zhai, Xiaoping; Zhang, Yimin
Abstract: In this paper, we investigate the Cauchy problem for the shallow water type equation u t + ∂ x 2 n + 1 u + 1 2 ∂ x ( u 2 ) + ∂ x ( 1 − ∂ x 2 ) − 1 u 2 + 1 2 u x 2 = 0 with low regularity data in the periodic settings. Firstly, we proved that the bilinear estimate related to the nonlinear term of the equation in space W s (defined in page 5) is invalid with s < − n 2 + 1 . Then, the locally well-posed of the Cauchy problem for the periodic shallow water-type equation is obtained in H s ( T ) with s > − n + 3 2, n ≥ 2 for arbitrary initial data. Thus, our result improves the result of Himonas and Misiolek (Commun. Partial Differ. Equ, 23(1998), 123–139.), where they have proved that the problem is locally well-posed for small initial data in H s ( T ) with s ≥ − n 2 + 1, n ∈ N + with the aid of the standard Fourier restriction norm method.