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

Jaeho Choi; Nitin Krishna; Nicole Magill; Alejandro Sarria

doi:10.3934/dcdsb.2019142

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2019, Volume 24, Issue 12: 6349-6365. Doi: 10.3934/dcdsb.2019142

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

1.
Department of Mathematics, University of Maine, Orono, ME 04469, USA
2.
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
3.
Department of Mathematics, Cornell University, Ithaca, NY 14850, USA
4.
Department of Mathematics, University of North Georgia, Dahlonega, GA 30533, USA

Received: September 2018

Revised: December 2018

Early access: July 2019

Published: September 2019

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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 $ L^p $, $ p \in [1,\infty) $, spaces for nonzero real parameters $ (\lambda,\kappa) $. 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.

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Mathematics Subject Classification: 35B44, 35B10, 35B65, 35Q35, 35B40.

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References

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