Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models

Abstract

In this paper we present analytical studies of three-dimensional viscous and inviscid simplified Bardina turbulence models with periodic boundary conditions. The global existence and uniqueness of weak solutions to the viscous model has already been established by Layton and Lewandowski. However, we prove here the global well-posedness of this model for weaker initial conditions. We also establish an upper bound to the dimension of its global attractor and identify this dimension with the number of degrees of freedom for this model. We show that the number of degrees of freedom of the long-time dynamics of the solution is of the order of $(L/l_d)^{12/5}$, where $L$ is the size of the periodic box and $l_d$ is the dissipation length scale-- believed and defined to be the smallest length scale actively participating in the dynamics of the flow. This upper bound estimate is smaller than those established for Navier-Stokes-$\alpha$, Clark-$\alpha$ and Modified-Leray-$\alpha$ turbulence models which are of the order $(L/l_d)^{3}$. Finally, we establish the global existence and uniqueness of weak solutions to the inviscid model. This result has an important application in computational fluid dynamics when the inviscid simplified Bardina model is considered as a regularizing model of the three-dimensional Euler equations.