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Construction of a Family of Stable Finite-Time Blowup Solutions for the Viscous Boussinesq System

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Abstract

This paper concerns with the local dynamical behavior near explicit finite-time blowup solutions with the smooth initial data for the three-dimensional viscous Boussinesq system. More precisely, we show that there exists a family of explicit blowup solutions with the smooth initial data and infinite energy in whole space \({\mathbb {R}}^3\). Meanwhile, we employ a suitable Nash–Moser iteration scheme by Yan (J Differ Equ 261:1973–2005, 2016) to prove Lyapunov nonlinear stability of those explicit finite-time blowup solutions for three-dimensional viscous Boussinesq system in a smooth moving domain with the timelike boundary condition. This means that we find a family of stable finite-time blowup solutions for the three-dimensional viscous Boussinesq system in a smooth bounded domain.

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Acknowledgements

The author expresses his sincere thanks to the anonymous referees for the comment and suggestion on this paper. This work is supported by Guangxi Natural Science Foundation No 2021JJG110002, NSF. No 12161006 and the special foundation for Guangxi Ba Gui Scholars.

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  1. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, People’s Republic of China

    Weiping Yan

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Correspondence to Weiping Yan.

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Communicated by Nader Masmoudi.

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Yan, W. Construction of a Family of Stable Finite-Time Blowup Solutions for the Viscous Boussinesq System. Ann. Henri Poincaré 24, 1971–2003 (2023). https://doi.org/10.1007/s00023-023-01267-4

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