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

TL;DR

This paper demonstrates non-uniqueness of Leray-Hopf solutions in a dyadic Navier-Stokes shell model, revealing potential mechanisms for non-uniqueness related to symmetry breaking and weak coupling.

Contribution

It introduces a dyadic shell model illustrating non-uniqueness of solutions at critical regularity, linking nonlinear interactions to convex integration techniques.

Findings

Existence of two distinct Leray-Hopf solutions for the Obukhov model with d>2.

Solutions exhibit approximate discrete self-similarity.

Non-uniqueness arises from partial symmetry breaking.

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.