On the Cauchy problem of a periodic 2-component $\hat{I}¼$-Hunter-Saxton equation

Abstract

In this paper, we study the Cauchy problem of a periodic 2-component $\hat{I}¼$-Hunter-Saxton system. We first establish the local well-posedness for the periodic 2-component $\hat{I}¼$-Hunter-Saxton system by Kato's semigroup theory. Then, we derive precise blow-up scenarios for strong solutions to the system. Moreover, we present a blow-up result for strong solutions to the system. Finally, we give a global existence result to the system.