Singularity \& Regularity Issues for Simplified Models of Turbulence

TL;DR

This paper investigates the singularity and regularity properties of simplified turbulence models, specifically Leray-α models, providing bounds on the complexity of their singular sets and connecting these to known results for Navier-Stokes equations.

Contribution

It establishes an upper bound on the Hausdorff dimension of singularities in Leray-α models for subcritical parameters, bridging previous results for Navier-Stokes and regularity.

Findings

Bound on the Hausdorff dimension of singular set for Leray-α models

Interpolation between Scheffer's Navier-Stokes bound and known regularity results

Extension of regularity analysis to a family of turbulence models

Abstract

We consider a family of Leray-$\alpha$ models with periodic boundary conditions in three space dimensions. Such models are a regularization, with respect to a parameter $\theta$, of the Navier-Stokes equations. In particular, they share with the original equation (NS) the property of existence of global weak solutions. We establish an upper bound on the Hausdorff dimension of the time singular set of those weak solutions when $\theta$ is subcritical. The result is an interpolation between the bound proved by Scheffer for the Navier-Stokes equations and the regularity result proved in \cite{A01}.