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Global Existence and Blow-Up Phenomena for the Hunter-Saxton Equation on the Line

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Abstract

In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS) equation on the line. Firstly, we establish the local well-posedness for the integral form of the (HS) equation in some special spaces \(E^s_{p,r}\), which mix Lebesgue spaces and homogeneous Besov spaces. Then we present a global existence result and provide a sufficient condition for strong solutions to blow up in finite time for the equation. The global existence is a new result comparing to the circle case in Yin (SIAM J Math Anal 36:272–283, 2004), since Yin (SIAM J Math Anal 36:272–283, 2004) proved that all solutions of (HS) equation with initial data that are not constant functions blow up in finite time. Finally, we give the ill-posedness and the unique continuation of the (HS) equation.

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Acknowledgements

This work was partially supported by NNSFC (No. 11671407), FDCT (No. 098/2013/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A03031104).

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Authors and Affiliations

  1. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Weikui Ye & Zhaoyang Yin

  2. Faculty of Information Technology, Macau University of Science and Technology, Macau, China

    Zhaoyang Yin

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  1. Weikui Ye
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  2. Zhaoyang Yin
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Communicated by A. Constantin.

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Ye, W., Yin, Z. Global Existence and Blow-Up Phenomena for the Hunter-Saxton Equation on the Line. J. Math. Fluid Mech. 24, 25 (2022). https://doi.org/10.1007/s00021-021-00630-x

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  • DOI: https://doi.org/10.1007/s00021-021-00630-x

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