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arXiv · 2006.11861

Remarks on the non-uniqueness in law of the Navier-Stokes equations up to the J.-L. Lions' exponent

Kazuo Yamazaki
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Abstract

Lions (1959, Bull. Soc. Math. France, \textbf{87}, 245--273) introduced the Navier-Stokes equations with a viscous diffusion in the form of a fractional Laplacian; subsequently, he (1969, Dunod, Gauthiers-Villars, Paris) claimed the uniqueness of its solution when its exponent is not less than five quarters in case the spatial dimension is three. Following the work of Hofmanov$\acute{\mathrm{a}}$, Zhu and Zhu (2019, arXiv:1912.11841 [math.PR]), we prove the non-uniqueness in law for the three-dimensional stochastic Navier-Stokes equations with the viscous diffusion in the form of a fractional Laplacian with its exponent less than five quarters.

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Kazuo Yamazaki. 2020-06-21. Remarks on the non-uniqueness in law of the Navier-Stokes equations up to the J.-L. Lions' exponent. https://doi.org/10.1016/j.spa.2022.01.016

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