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Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier–Stokes Equations in High Dimensions

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Abstract

Consider the unforced incompressible homogeneous Navier–Stokes equations on the d-torus \({\mathbb{T}^d}\) where \({d \geq 4}\) is the space dimension. It is shown that there exist nontrivial steady-state weak solutions \({u \in L^{2} (\mathbb{T}^d)}\). The result implies the nonuniqueness of finite energy weak solutions for the Navier–Stokes equations in dimensions \({d \geq 4}\); it also suggests that the uniqueness of forced stationary problem is likely to fail however smooth the given force is.

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Acknowledgements

The author is grateful to his advisor Alexey Cheskidov formany stimulating discussions and for valuable comments after reading earlier versions of the manuscript. The author is partially supported by the NSF grant DMS 1517583 through his advisor Alexey Cheskidov.

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  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Xiaoyutao Luo

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Luo, X. Stationary Solutions and Nonuniqueness of Weak Solutions for the Navier–Stokes Equations in High Dimensions. Arch Rational Mech Anal 233, 701–747 (2019). https://doi.org/10.1007/s00205-019-01366-9

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