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arXiv · 2503.14699

Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data

Matei P. Coiculescu·Stan Palasek
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Abstract

We consider the Cauchy problem for the incompressible Navier-Stokes equations in dimension three and construct initial data in the critical space $BMO^{-1}$ from which there exist two distinct global solutions, both smooth for all $t>0$. One consequence of this construction is the sharpness of the celebrated small data global well-posedness result of Koch and Tataru. This appears to be the first example of non-uniqueness for the Navier-Stokes equations with data at the critical regularity. The proof is based on a non-uniqueness mechanism proposed by the second author in the context of the dyadic Navier-Stokes equations.

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Matei P. Coiculescu, Stan Palasek. 2025-03-18. Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data. https://doi.org/10.1007/s00222-025-01396-z

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