Mathematical results for some $\displaystyle{\alpha}$ models of turbulence with critical and subcritical regularizations

TL;DR

This paper proves the existence and uniqueness of solutions for a family of turbulence models with critical and subcritical regularizations, providing bounds on singularities and extending previous regularity results.

Contribution

It establishes the existence of unique weak solutions for a broad class of $oldsymbol{ extalpha}$ turbulence models, including critical and subcritical cases, with bounds on singularities.

Findings

Unique regular weak solutions for critical models

Existence of weak solutions for subcritical models

Upper bounds on Hausdorff dimension of singular set

Abstract

In this paper, we establish the existence of a unique "regular" weak solution to turbulent flows governed by a general family of $\alpha$ models with critical regularizations. In particular this family contains the simplified Bardina model and the modified Leray-$\alpha$ model. When the regularizations are subcritical, we prove the existence of weak solutions and we establish an upper bound on the Hausdorff dimension of the time singular set of those weak solutions. The result is an interpolation between the bound proved by Scheffer for the Navier-Stokes equations and the regularity result in the critical case.