  The generalized Hunter-Saxton system comprises several well-known models from
fluid dynamics and serves as a tool for the study of fluid convection and
stretching in one-dimensional evolution equations. In this work, we examine the
global regularity of periodic smooth solutions of this system in $L^p$, $p \in
[1,\infty)$, spaces for nonzero real parameters $(\lambda,\kappa)$. Our results
significantly improve/extend those by Wunsch et al. [27-29] and Sarria [21].
Furthermore, we study the effects that different boundary conditions have on
the global regularity of solutions by replacing periodicity with a homogeneous
three-point boundary condition and establish finite-time blowup of a
local-in-time solution of the resulting system for particular values of the
parameters.
