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Wave-breaking phenomena for a new weakly dissipative quasilinear shallow-water waves equation

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Abstract

In this paper, we mainly study a new weakly dissipative quasilinear shallow-water waves equation, which can be formally derived from a model with the effect of underlying shear flow from the incompressible rotational two-dimensional shallow water in the moderately nonlinear regime by Wang, Kang and Liu (Appl Math Lett 124:107607, 2022). Considering the dissipative effect, the local well-posedness of the solution to this equation is first obtained by using Kato’s semigroup theory. We then establish the precise blow-up criterion by using the transport equation theory and Moser-type estimates. Moreover, some sufficient conditions which guarantee the occurrence of wave-breaking of solutions are studied according to the different real-valued intervals in which the dispersive parameter \(\theta \) being located. It is noteworthy that we need to overcome the difficulty induced by complicated nonlocal nonlinear structure and different dispersive parameter ranges to get corresponding convolution estimates.

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Acknowledgements

This work is supported by Fundamental Research Program of Shanxi Province (Grant Nos. 202203021212286 and 202203021222126).

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Authors and Affiliations

  1. School of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, Shanxi, China

    Xiaofang Dong, Xianxian Su & Kai Wang

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  1. Xiaofang Dong
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  2. Xianxian Su
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  3. Kai Wang
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Correspondence to Kai Wang.

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Dong, X., Su, X. & Wang, K. Wave-breaking phenomena for a new weakly dissipative quasilinear shallow-water waves equation. Monatsh Math (2024). https://doi.org/10.1007/s00605-024-01958-y

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