The Cauchy problem for a higher order shallow water type equation on the circle

In this paper, we investigate the Cauchy problem for a higher order shallow water type equationut−utxx+∂x2j+1u−∂x2j+3u+3uux−2uxuxx−uuxxx=0, where x∈T=R/2π and j∈N+. Firstly, we prove that the Cauchy problem for the shallow water type equation is locally well-posed in Hs(T) with s≥−j−22 for arbitrary initial data. By using the I-method, we prove that the Cauchy problem for the shallow water type equation is globally well-posed in Hs(T) with 2j+1−j22j+1