Global existence for the periodic dispersive Hunter–Saxton equation

Abstract

In this paper, we study an integrable dispersive Hunter–Saxton equation in periodic domain. Firstly, we establish the local well-posedness of the Cauchy problem of the equation in $$H^s ({\mathbb {S}}), s \ge 2,$$ by applying the Kato method. Then, based on a sign-preserve property, we obtain a global existence result for the equation. Moreover, we extend the obtained result to some periodic nonlinear partial differential equations of second order of the general form.