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

Abstract

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 ∈ [1, ∞), spaces for nonzero real parameters (λ, κ). 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-intime solution of the resulting system for particular values of the parameters.