The Cauchy problem for higher-order modified Camassa–Holm equations on the circle

Abstract

In this paper, we investigate the Cauchy problem for the shallow water type equation ut+∂x2n+1u+12∂x(u2)+∂x(1−∂x2)−1u2+12ux2=0with low regularity data in the periodic settings. Firstly, we proved that the bilinear estimate related to the nonlinear term of the equation in space Ws (defined in page 5) is invalid with s<−n2+1. Then, the locally well-posed of the Cauchy problem for the periodic shallow water-type equation is obtained in Hs(T) with s>−n+32,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 Hs(T) with s≥−n2+1,n∈N+ with the aid of the standard Fourier restriction norm method.