Mathematical results for some $\displaystyleα$ models of turbulence with critical and subcritical regularizations

Details Mathematical results for some $\displaystyleα$ models of turbulence with critical and subcritical regularizations Mathematical results for some $\displaystyleα$ models of turbulence with critical and subcritical regularizations

2011-05-03

arxiv.org/abs/1105.0694v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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