Abstract
The purpose of this paper is to give an exact division for the well-posedness and ill-posedness (non-existence) of the Hunter–Saxton equation on the line. Since the force term is not bounded in the classical Besov spaces (even in the classical Sobolev spaces), a new mixed space \(\mathcal {B}\) is constructed to overcome this difficulty. More precisely, if the initial data \(u_0\in \mathcal {B}\cap \dot{H}^{1}(\mathbb {R}),\) the local well-posedness of the Cauchy problem for the Hunter–Saxton equation is established in this space. Contrariwise, if \(u_0\in \mathcal {B}\) but \(u_0\notin \dot{H}^{1}(\mathbb {R}),\) the norm inflation and hence the ill-posedness is presented. It’s worth noting that this norm inflation occurs in the low frequency part, which exactly leads to a non-existence result. Moreover, the above result clarifies a corollary with physical significance such that all the smooth solutions in \(L^{\infty }(0,T;L^{\infty }(\mathbb {R}))\) must have the \(\dot{H}^1\) norm.
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Acknowledgements
Guo was supported by the National Natural Science Foundation of China (No. 12301298, No. 12161004). Ye was supported by the National Natural Science Foundation of China (No. 12271051, No. 12371095), the general project of NSF of Guangdong province (No. 2021A1515010296). Yin was partially supported by the National Natural Science Foundation of China (No. 12171493), the Macao Science and Technology Development Fund (No. 0091/2018/A3), Guangdong Province of China Special Support Program (No. 8-2015) and the key project of the Natural Science Foundation of Guangdong Province (No. 2016A030311004).
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Guo, Y., Ye, W. & Yin, Z. Sharp ill-posedness for the Hunter–Saxton equation on the line. J. Evol. Equ. 24, 31 (2024). https://doi.org/10.1007/s00028-024-00962-x
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DOI: https://doi.org/10.1007/s00028-024-00962-x