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arXiv · 2407.06179

Non-uniqueness in the Leray-Hopf class for a dyadic Navier-Stokes model

Stan Palasek
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Abstract

The uniqueness of Leray-Hopf solutions to the incompressible Navier-Stokes equations remains a significant open question in fluid mechanics. This paper proposes a potential mechanism for non-uniqueness, illustrated in a natural dyadic shell model. We show that, for the Obukhov model with $d>2$, there exist initial data at the critical regularity that give rise to two distinct Leray-Hopf solutions. These solutions exhibit an approximately discretely self-similar structure, with non-uniqueness resulting from a partial breaking of the scaling symmetry. The fundamental observation is that, in a certain scenario, the dynamics reduce to a sequence of weakly coupled finite-dimensional systems. Moreover, the predominant nonlinear interactions are identical to those arising in convex integration, suggesting the possibility of a similar construction in the full PDE setting.

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Stan Palasek. 2024-07-08. Non-uniqueness in the Leray-Hopf class for a dyadic Navier-Stokes model. https://arxiv.org/abs/2407.06179

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