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Asymptotic stability of explicit infinite energy blowup solutions of the 3D incompressible Navier-Stokes equations

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Abstract

In this paper, we study the dynamical stability of a family of explicit blowup solutions of the three-dimensional (3D) incompressible Navier-Stokes (NS) equations with smooth initial values, which is constructed in Guo et al. (2008). This family of solutions has finite energy in any bounded domain of ℝ3, but unbounded energy in ℝ3. Based on similarity coordinates, energy estimates and the Nash-Moser-Hörmander iteration scheme, we show that these solutions are asymptotically stable in the backward light-cone of the singularity. Furthermore, the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing data. Finally, the result also shows that in the absence of physical boundaries, the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 12231016 and 12071391) and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A1515010860).

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Fangyu Han & Zhong Tan

  2. Fujian Provincial Key Laboratory on Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, China

    Zhong Tan

  3. Shenzhen Research Institute of Xiamen University, Shenzhen, 518057, China

    Zhong Tan

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  1. Fangyu Han
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Han, F., Tan, Z. Asymptotic stability of explicit infinite energy blowup solutions of the 3D incompressible Navier-Stokes equations. Sci. China Math. 66, 2523–2544 (2023). https://doi.org/10.1007/s11425-022-2059-1

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