On the $ L^p $ regularity of solutions to the generalized Hunter-Saxton system

Abstract

The generalized Hunter-Saxton system comprises several well-kno-wn 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 \begin{document}$ L^p $\end{document} , \begin{document}$ p \in [1,\infty) $\end{document} , spaces for nonzero real parameters \begin{document}$ (\lambda,\kappa) $\end{document} . Our results significantly improve and extend those by Wunsch et al. [ 29 , 30 , 31 ] and Sarria [ 23 ]. 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.