\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010 (2010), No. 110, pp. 1--19.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/110\hfil Effect of hyperviscosity on turbulence] {Effect of hyperviscosity on the Navier-Stokes turbulence} \author[A. Younsi\hfil EJDE-2010/110\hfilneg] {Abdelhafid Younsi} \address{Abdelhafid Younsi \newline Faculty of Mathematics USTHB, BP32 EL ALIA16111 Algiers, Algeria} \email{younsihafid@gmail.com} \thanks{Submitted December 2, 2009. Published August 9, 2010.} \subjclass[2000]{76D05, 76F20, 35B30, 35B41, 35B65, 37L30, 37K40} \keywords{Navier-Stokes equations; hyperviscosity; weak solutions; \hfill\break\indent attractor dimension; turbulence models} \begin{abstract} In this article, we modified the Navier-Stokes equations by adding a higher order artificial viscosity term to the conventional system. We first show that the solution of the regularized system converges strongly to the solution of the conventional system as the regularization parameter approaches zero, for each dimension $d\leq 4$. Then we show that the use of this artificial viscosity term leads to truncated the number of degrees of freedom in the long-time behavior of the solutions to these equations. This result suggests that the hyperviscous Navier-Stokes system is an interesting model for three-dimensional fluid turbulence. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} We regularize the Navier-Stokes equations by adding a higher-order viscosity term to the conventional system. In this paper we will restrict ourselves to periodic boundary conditions. \begin{equation} \begin{gathered} \frac{du_{\varepsilon}}{dt}+\varepsilon(-\Delta)^lu_{\varepsilon} -\nu\Delta u_{\varepsilon}+( u_{\varepsilon}.\nabla)u_{\varepsilon}+\nabla p=f(x),\quad\text{in } \Omega\times(0,\infty)\\ \operatorname{div}u_{\varepsilon}=0\quad\text{in } \Omega\times(0,\infty),\\ p(x+Le_{i},t)=p(x,t),\quad u(x+Le_{i},t)=u(x,t)\quad i=1,\dots ,d\; t\in(0,\infty)\\ u_{\varepsilon}(x,0)=u_{\varepsilon_0}(x)\quad \text{in }\Omega, \end{gathered}\label{1} \end{equation} Where $\Omega=(0,L)^{d}$ and $( e_1,\dots ,e_{d})$ is the natural basis of $\mathbb{R}^{d}$. Here $\varepsilon>0$ is the artificial dissipation parameter and $\nu>0$ is the kinematic viscosity of the fluid, $l>1$. The function $u_{\varepsilon}$ is the velocity vector field, $p$ is the pressure, and $f$ is a given force field. For $\varepsilon=0$, the model is reduced to the Navier-Stokes system. In Lions \cite{25}, the existence and uniqueness of weak solutions of the modified Navier-Stokes equations were established for all $l>0$ if $l\geq (d+2)/4$, $d$ is the space dimension. This type of regularization was proposed by Ladyzhenskaya \cite{20} and Lions \cite{26} who added the artificial hyperviscosity $(-\Delta)^{l/2}$, $l>2$ to the Navier-Stokes system. Mathematical model for such fluid motion play an important role in theoretical and computational studies of bipolar fluids \cite{7} and in the regularized Navier-Stokes equations (see \cite{7, 26, 28} and the references therein). Hyperviscosity is introduced in the works \cite{28, 30} to demonstrate global unique solvability of the Navier-Stokes equations in three dimensions. Hyperviscosity has been widely used for numerical simulations of turbulence \cite{1,3,5,6} and in computer simulations for oceanic and atmospheric flows (see \cite{4, 23}) or to control the Navier--Stokes equations \cite{31}. A well known example of such a result is the viscosity solution method for the Hamilton-Jacobi equations \cite{26}. In this paper, we will study the effect of hyperviscosity on the Navier-Stokes turbulence. First, we show that the solutions of \eqref{1} converge strongly to the corresponding solutions of the Navier--Stokes equations for $d\leq4$. This result can extend to each domain $\Omega$ with one finite size. In this result, we show that the conjecture of Lions \cite[Remarque 8.2. SecII]{25} is true, for $d\leq4$. In addition, it is an extension of a result due to Lions \cite{26} (where only the weak convergence is proved). The results in this article can be seen as an improved version of the convergence\ results announced by Yuh-Roung and Sritharan \cite{28,29}, in two different ways: On the one hand, we consider here a dimension $d\leq4$, on the other hand the order viscosity term here is $l\geq\sup(\frac{d}{2},\frac{d+2}{4})$. Next, we consider the system \eqref{1} with $l=2$; i.e., we modified the $3D$ Navier-Stokes system by adding a fourth order artificial viscosity term (Laplacian square) and we show the existence of absorbing sets. This fact implies that the system ($l=2$) possesses a global attractor $\mathfrak{A}_{\varepsilon}$. Finally, we obtain scale-invariant estimates on the Hausdorff and fractal dimensions of the global attractor $\mathfrak{A}_{\varepsilon}$ independent of $\varepsilon$ in terms of the Landau--Lifschitz theory \cite{22} of the number of degrees of freedom in turbulent flow \cite{11, 32}. In fact such an estimate that improves on the Landau-Lifschitz estimates has already been done by Avrin \cite{1} in which hyperviscous terms are spectrally added to the Navier-Stokes equations. Thus we recover the improvement on the cubic power; i.e., get a bound proportional to $G^{\frac{p}{2}}$ for $p<3$. The latter should be a possibility, as the attractor results in \cite{1} were not intended to be optimal in this direction. We would then represent an overlapping result that is new as far as we know, although readers familiar with the attractor techniques used may anticipate that such a result is possible in the hyperviscous case given the existing results in \cite{1} and the expected improvement in the Sobolev-space estimates in the fixed uniform hyperviscous case at hand. In Section 2, we present the relevant mathematical framework for the paper. In Section 3, we show the convergence of the system \eqref{1} to the conventional Navier--Stokes equations. In Section 4, we consider the hyperviscous system ($l=2$), we show the existence of a global attractor. In Section 5, we estimate the dimension of the attractor. Finally, we provide in Section 6, explicit upper bounds for the dimension of the global attractor of the modified Navier--Stokes in terms of the relevant physical parameters. \section{Notation and preliminaries} In this section we introduce notations and the definitions of standard functional spaces that will be used throughout the paper. We denote by $H^{m}(\Omega)$, the Sobolev space of $L$-periodic functions. These spaces are endowed with the inner product \[ (u,v)= \sum_{| \beta| \leq m} (D^{\beta}u,D^{\beta}v)_{L^{2}(\Omega)} \] and the norm \[ \| u\|_{m}= \sum_{| \beta| \leq m} (\| D^{\beta}u\| _{L^{2}(\Omega)}^{2} )^{1/2}. \] $H^{-m}(\Omega)$ Denote the dual space of $H^{m}( \Omega) $. We denote by $\dot{H}^{m}(\Omega)$ the subspace of $H^{m}(\Omega)$ with, zero average \[ \dot{H}^{m}(\Omega)=\{u\in H^{m}(\Omega);\int_{\Omega} u(x)dx=0\}. \] For $m=0$, we have $\dot{H}^{m}(\Omega)=\dot{L}^{2}( \Omega) $. We introduce the following solenoidal subspaces $V_{s}$, $s\in\mathbb{R}^{+}$ which are important to our analysis \begin{gather*} V_0(\Omega) =\{u\in\dot{L}^{2}(\Omega): \operatorname{div}u=0,u.n\big|_{\Sigma_{i}}=-u.n\big|_{\Sigma_{i+3}},\; i=1,2,3\};\\ V_1(\Omega) =\{u\in\dot{H}^{1}(\Omega): \operatorname{div}u=0,\gamma_0u\big|_{\Sigma_{i}} =\gamma_0u\big|_{\Sigma_{i+3}},\;i=1,2,3\};\\ \begin{aligned} V_2(\Omega) =\{& u\in\dot{H}^{2}(\Omega): \operatorname{div}u=0,\gamma_0u\big|_{\Sigma_{i}} =\gamma_0u\big|_{\Sigma_{i+3}},\\ &\gamma_1u\big|_{\Sigma_{i}} =-\gamma_1u\big|_{\Sigma_{i+3}}, \; i=1,2,3\}; \end{aligned} \end{gather*} see \cite[Chapter III, Section 2]{32}. We refer the reader to Temam \cite{33} for details on these spaces. Here the faces of $\Omega$ are numbered as \[ \Sigma_{i}=\partial\Omega\cap\{ x_{i}=0\}\quad \text{and}\quad \Sigma_{i+3}=\partial\Omega\cap\{ x_{i}=L\} ,\; i=1,2,3. \] Here $\gamma_0$, $\gamma_1$ are the trace operators and $n$ is the unit outward normal on $\partial\Omega$. $\bullet$ The space $V_0$ is endowed with the inner product $(u,v) _{L^{2}(\Omega)}$ and norm $\| u\| = (u,u)_{L^{2}(\Omega)}^{1/2}$. $\bullet$ $V_1$ Is the Hilbert space with the norm $\| u\| _1=\| u\|_{V_1}$. The norm induced by $\dot{H} ^{1}(\Omega)$ and the norm $\| \nabla u\| $are equivalent in $V_1$. $\bullet$ $V_2$ Is the Hilbert space with the norm $\| u\| _2=\| u\|_{V_2}$. In $V_2$ the norm induced by $\dot{H}^{2}(\Omega)$ is equivalent to the norm $\| \Delta u\| $. Let $V_{s}'$ denote the dual space of $V_{s}$. We denote by $A$ the Stokes operator \[ Au=-\Delta u\text{ for }u\in D(A). \] We recall that the operator $A$ is a closed positive self-adjoint unbounded operator, with $D(A)=\{ u\in V_0,Au\in V_0\} $. We have in fact, \[ D(A)=\dot{H}^{2}(\Omega)\cap V_0=V_2. \] The eigenvalues of $A$ are $\{\lambda_{j}\}_{j=1}^{j=\infty}$, $0<\lambda_1\leq\lambda_2\leq\dots $ and the corresponding orthonormal set of eigenfunctions $\{w_{j}\}_{j=1}^{j=\infty}$ is complete in $V_0$ \[ Aw_{j}=\lambda_{j}w_{j},\quad w_{j}\in D(A_1). \] The spectral theory of $A$ allows us to define the powers $A^l$ of $A$ for $l\geq 1$, $A^l$ is an unbounded self-adjoint operator in $V_0$ with a domain $D(A^l)$ dense in $V_2\subset V_0$. We set here \[ A^lu=(-\Delta)^lu\quad \text{for }u\in D( A^l) =V_{2l}\cap V_0. \] The space $D(A^l)$ is endowed with the scalar product and the norm \[ (u,v)_{D(A^l)}=(A^lu,A^lv),\quad \| u\|_{D(A^l)}=\{(u,u)_{D(A^l)}\}^{1/2}. \] Let us now define the trilinear form $b(\cdot,\cdot,\cdot)$ associated with the inertia terms \[ b(u,v,w)=\sum_{i,j=1}^{3} \int_{\Omega} u_{i}\frac{\partial v_{j}}{\partial x_{i}}w_{j}dx. \] The continuity property of the trilinear form enables us to define (using Riesz representation Theorem) a bilinear continuous operator $B( u,v)$; $V_2\times V_2\to V_2'$ will be defined by \begin{equation} \langle B(u,v),w\rangle =b(u,v,w),\quad \forall w\in V_2. \label{2} \end{equation} Recall that for $u$ satisfying $\nabla.u=0$, we have \begin{equation} b(u,u,u)=0, \quad b(u,v,w)=-b( u,w,v). \label{3} \end{equation} Hereafter, $c_{i}$ for $i\in \mathbb{N}$, will denote a dimensionless scale invariant positive constant which might depend on the shape of the domain. Similarly, the trilinear form $b( u,v,w)$ satisfies the well-known inequalities (see, for instance, \cite[Lemma 61.1]{30} and \cite{8, 33}) \begin{equation} | b(u,v,u)| \leq c_1\| u\| ^{1/2}\| u\| _1^{3/2}\| v\|_1\quad \text{for all }u,v\in V. \label{4} \end{equation} We recall some well known inequalities that we will be using in what follows. The Ladyzhenskaya inequality (cf.\cite{19}) in $\mathbb{R}^{3}$ \begin{equation} \| u\|_{L^{\theta}(\Omega)}\leq c_2\| u\| _{L^{2}(\Omega)}^{\frac{6-\theta }{2\theta}}\| u\|_{H^{1}( \Omega)} ^{\frac{3(\theta-2)}{2\theta}} \label{5} \end{equation} for every $u\in H^{1}(\Omega)$, $2\leq\theta\leq 6$. Agmon inequality (see, e.g., \cite{8}) \begin{equation} \| u\|_{\infty}\leq c_{3}\| u\|_1 ^{1/2}\| Au\| ^{1/2}\quad \text{for all }u\in V_2 \label{6} \end{equation} Young's inequality \begin{equation} ab\leq\tfrac{\sigma}{p}a^{p}+\tfrac{1}{q\sigma^{\frac{q}{p}}}b^{q} ,\quad a,b,\sigma>0,\quad p>1,\quad q=\frac{p}{p-1}. \label{7} \end{equation} Poincar\'{e} inequality \begin{equation} \lambda_1\| u\| ^{2}\leq\| A^{1/2}u\| ^{2}\quad \text{for all }u\in V. \label{8} \end{equation} To prove uniform bounds on different norms we use the uniform Gronwall Lemma; for a proof see \cite[Lemma III 1.1]{32}. \begin{lemma}[The Uniform Gronwall Lemma] \label{lem2.1} Let $g$, $h$, $y$ be three positive locally integrable functions on $(t_{0,}+\infty)$ which satisfy \begin{gather*} \frac{dy}{dt}\leq gy+h\quad \text{for }t\geq t_0,\\ \int_{t}^{t+r}g(s)ds\leq a_1,\quad \int_{t}^{t+r}h(s) ds\leq a_2,\quad \int_{t}^{t+r}y(s)ds\leq a_{3}\quad \text{for }t\geq t_0, \end{gather*} where $a_1$, $a_2$, $a_{3}$ are positive constants. Then \[ y(t+r)\leq(\frac{a_{3}}{r}+a_2)\exp( a_1)\quad \text{for }t\geq t_{0.} \] \end{lemma} Denoting by $G$ the dimensionless Grashoff number \cite{10}, this number measures the relative strength of the forcing and viscosity. \section{Strong convergence for the hyperviscous system} In this Section, we give a new Theorem which ensures the strong convergence of the solutions of the system \eqref{1} to the corresponding solutions of the Navier--Stokes equations for $d\leq4$. This result can extend to each domain $\Omega$ with one finite size. Moreover, we show that $u_{\varepsilon }\in C(0,T;V_0)$. Using the operators defined above, we can write the modified system \eqref{1} in the evolution form \begin{gather} \frac{du_{\varepsilon}}{dt}+\varepsilon A^lu_{\varepsilon}+\nu Au_{\varepsilon}+B(u_{\varepsilon},u_{\varepsilon}) =f(x),\quad \text{in }\Omega\times(0,\infty) \label{9}\\ u_{\varepsilon_0}(x) =u_{\varepsilon_0}, \quad \text{in } \Omega. \label{10} \end{gather} The existence and uniqueness results for initial value problem \eqref{1} can be found in \cite{25}, \cite[Chap.1, Remarque 6.11]{26}. The following theorem collects the main result in this work \begin{theorem} \label{thm1} For $l\geq (d+2)/4$, $d$ is the space dimension, for $\varepsilon>0$ fixed, $f\in L^{2}(0,T;V_0')$ and $u_{\varepsilon_0}\in V_0$ be given. There exists a unique weak solution of \eqref{1} which satisfies \[ u_{\varepsilon}\in L^{2}(0,T;V_l)\cap L^{\infty}( 0,T;V_0),\quad \forall T>0. \] \end{theorem} Note that the conventional Navier-Stokes system can be written in the evolution form \begin{gather} \frac{du}{dt}+\nu Au+\hat{B}(u,u) =f(x) \quad \text{in }\Omega\times(0,\infty)\label{11}\\ u(0) =u_0\quad \text{in }\Omega. \label{12} \end{gather} \begin{theorem} \label{thm3.2} For $d\leq4$, for $f\in L^{2}(0,T;V_0)$ and $u_0\in V_0$ be given. There exists a weak solution of \eqref{11}-\eqref{12} which satisfies $u\in L^{\infty}(0,T;V_0)\cap L^{2}(0,T;V_1)$, for $T>0$. For $d=2$, $u$ is unique (Lions \cite{25}). \end{theorem} We will establish various estimates uniform in $\varepsilon$ for the solutions of the modified Navier Stokes. These bounds will be used to establish the limit of these solutions to the conventional Navier Stokes equations. \begin{proposition}\label{prop1} For $d\leq4$ and for $\varepsilon>0$ fixed, $f\in L^{2}(0,T;V_0)$ and $u_{\varepsilon_0}\in V_0$. The weak solution $u_{\varepsilon}(t)$ of the modified Navier-Stokes equations satisfy \begin{itemize} \item[(i)] $u_{\varepsilon}$ is uniformly bounded in $L^{\infty}( 0,T;V_0) $, \item[(ii)] $u_{\varepsilon}$ is uniformly bounded in $L^{2}(0,T;V_1)$. \end{itemize} \end{proposition} We need the following Lemma proved in Temam \cite[Lemma 4.1.ChIII,Sec4]{33}. \begin{lemma} \label{lem1} The form $b$ is trilinear continuous on $V\times V\times V_{s}$ if $s\geq d/2$ and \[ \| b(u,v,w)\|\leq c_{4}\| u\| \| v\|_1\| w\|_{s}. \] \end{lemma} Applying\ Lemma \ref{lem1}, we obtain the following result. \begin{lemma} \label{lem2} Let $u_{\varepsilon}(t)$ be a weak solution of the modified Navier-Stokes system. Then $B( u_{\varepsilon})$ belongs to $L^{2}( 0,T;V_l')$ for $l\geq d/2$. \end{lemma} \begin{proof} By the definition of the operator $B$ and the above Lemma, we obtain \[ | \langle B(u(t),v)\rangle | =| b(u(t),u(t),v)| \leq c_{4}\| u(t)\|\| u(t)\|_1\| v\|_{V_l'},\quad \forall v\in V_l. \] Thus, \[ \| B(u(t))\|_{V_l'}\leq c_{4}\| u( t) \|\| u(t)\|_1\quad \text{for }0\leq t\leq T. \] \end{proof} \begin{lemma} \label{lem3} If $f\in L^{2}(0,T;V_1')$, then, for any solution $u_{\varepsilon}(t)$ of problem \eqref{1} the time derivative $\frac{du_{\varepsilon}}{dt}$ is uniformly bounded in $L^{2}( 0,T;V_l')$. \end{lemma} \begin{proof} Due to Lemma \ref{lem2} $B(u_{\varepsilon})$ belongs to $L^{2}(0,T;V_l')$, since $f-\varepsilon A^lu_{\varepsilon}-\nu Au_{\varepsilon}$ belongs to $L^{2}(0,T;V_l')$, this implies that $\frac{du_{\varepsilon}} {dt}\ $belongs to $L^{2}(0,T;V_l')$. \end{proof} \begin{lemma} \label{lem3.7} The function $u_{\varepsilon}$ is almost everywhere equal to a continuous function from $[0,T]$ to the space $V_0$. \end{lemma} \begin{proof} Since $u_{\varepsilon}\in L^{2}(0,T;V_1)\cap L^{\infty }(0,T;V_0)$ and $\frac{du_{\varepsilon}}{dt}\in L^{2}(0,T;V_l')$, the weak continuity in $V_0$ is a direct consequence of \cite[Lemma 1.4.ChIII,Sec1]{33}. Similarly, it follows that $u_{\varepsilon}(0)$ converges to $u(0)$ in $V_0$, and since $u_{\varepsilon_0}$ converges to $u_0$ in $V_l'$, we conclude that $u(0)=u_0$. \end{proof} Now we prove the strong convergence. It follows from (ii) of Proposition \ref{prop1} and from Lemma \ref{lem3}, that \[ u_{\varepsilon_{n}}\in\mathcal{X=}\{u_{\varepsilon_{n}}\in L^{2}(0,T;V_1),\; \frac{du_{\varepsilon_{n}}}{dt}\in L^{2}( 0,T;V_l')\} \] with bounds independent of $\varepsilon_{n}$. Hence (i) $u_{\varepsilon_{n}}\to u$ in $L^{2}(0,T;V_l)$ weakly; and (ii) $\frac{du_{\varepsilon_{n}}} {dt}\to\frac{du}{dt}$ in $L^{2}(0,T;V_l')$ weakly; These two properties allow us to establish the strong convergence result. The proof of the following theorem can be found in Temam \cite[Theorem 2.1, Chapter III, Sec\ 2]{33}. \begin{theorem} \label{thm3} The injection of $\mathcal{X}=\{u\in L^{2}(0,T;V_1)$, $\frac{du_{\varepsilon}}{dt}\in L^{2}( 0,T;V_l')\}$ into $\mathcal{Y}=\{ u\in L^{2}(0,T;V_0)\} $ is compact. \end{theorem} By virtue of the above estimates and the compactness Theorem \ref{thm3}. We can now state our first result. \begin{theorem}\label{thm4} For $l\geq\sup(\frac{d}{2},\frac{d+2}{4})$ and for $d\leq4$, the weak solution $u_{\varepsilon}$ of the modified Navier-Stokes equations \eqref{1} given by Theorem \ref{thm1} converges strongly in $L^{2}(0,T;V_0)$ as $\varepsilon\to0$ to $u$ the weak solution of the system \eqref{9}-\eqref{10}. \end{theorem} \begin{proof} Theorem \ref{thm1} and Lemma \ref{lem1} are satisfied for $l\geq \sup(\frac{d}{2},\frac{d+2}{4})$. We use part $ii)$ of Proposition \ref{prop1} and Lemma \ref{lem3} we can deduce that the weak solutions $u_{\varepsilon_{n}}\in\mathcal{X=}\{u_{\varepsilon_{n}}\in L^{2}(0,T;V_1)$, $\frac{du_{\varepsilon_{n}}}{dt}\in L^{2}( 0,T;V_l')\}$. Hence, the compactness Theorem \ref{thm3} implies the strong convergence in $L^{2}(0,T;V_0)$. \end{proof} The following proposition is a consequence of Proposition \ref{prop1}. \begin{proposition}\label{prop2} For all $w\in L^{2}(0,T;V_1)$, $\forall\frac{dw}{dt}\in L^{2}(0,T;V_1')$ \begin{itemize} \item[(a)] $\lim_{n\to\infty}\int_0^{T} (\frac{du_{\varepsilon_{n}}(t)}{dt},w)dt=\int_0^{T}(\frac {du(t)}{dt},w(t))dt$, \item[(b)] $\lim_{n\to\infty}\int_0^{T}(\nabla u_{\varepsilon_{n} }(t),\nabla w(t))dt=\int_0^{T}( \nabla u(t),\nabla w(t) ) dt$, \item[(c)] $\lim_{n\to\infty}\int_0^{T}b( u_{\varepsilon_{n}}(t),u_{\varepsilon_{n}}(t),w(t)) dt=\int_0^{T}b(u(t),u( t),w(t))dt$. \end{itemize} \end{proposition} Let us now establish the limit of the equations \eqref{9} as $\varepsilon_{n}\to0$. Taking the inner product of \eqref{9} with a test function $\varphi\in\mathcal{D}(0,T;\mathcal{D}(A^{l/2}))$ then integrate by parts and using the convergence Proposition \ref{prop2} we can pass to the limit as $\varepsilon_{n}\to0$, we get $-\int_0^{T}( u,\varphi')dt+\nu\int_0 ^{T}(\nabla u,\nabla\varphi)dt+\int_0^{T}b(u,u,\varphi) dt=\int_0^{T}\langle f,\varphi\rangle dt$. Here the term $\varepsilon_{n}\int_0^{T}(A^{l/2}u_{\varepsilon_{n} }(t),A^{l/2}\varphi(t))dt$ approaches $0$ as $\varepsilon_{n}\to 0$. Since the weak solution $u_{\varepsilon_{n} }$ is in $L^{2}(0,T;V_1)$ with a uniform bound in $\varepsilon_{n}$ and we obtain \[ \varepsilon_{n}\int_0^{T}| (A^{l/2}u_{\varepsilon_{n}},A^{l/2}\varphi)| dt \leq\varepsilon_{n}\int_0^{T}| (u_{\varepsilon_{n}},A^l \varphi)| dt \leq c\varepsilon_{n}. \] Since $u\in L^{2}(0,T;V_1 )\cap L^{\infty}(0,T;V_0)$, we can conclude that $u$ is indeed the weak solution for the conventional Navier-Stokes equations. \section{The hyperviscous Navier-Stokes system and attractors} Now, we consider modifications of the 3D Navier-Stokes system by adding a fourth order artificial viscosity term (Laplacian square) depending on a small parameter $\varepsilon$ to the conventional system. \begin{equation} \begin{gathered} \frac{du_{\varepsilon}}{dt}+\varepsilon A^{2}u_{\varepsilon}+\nu Au_{\varepsilon}+B( u_{\varepsilon},u_{\varepsilon})=f( x),\quad\text{in }\Omega\times(0,\infty)\\ \operatorname{div}u_{\varepsilon}=0,\quad\text{in } \Omega\times(0,\infty),u_{\varepsilon}( x,0) =u_{\varepsilon_0}(x) \quad\text{in }\Omega,\\ p(x+Le_{i},t)=p(x,t),\quad u(x+Le_{i},t)=u(x,t)\quad i=1,2,3.\; t\in (0,\infty) \end{gathered} \label{13} \end{equation} where $\Omega=(0,L)^{3}$. In this section we will show the existence of the compact global attractor $\mathfrak{A}_{\varepsilon}$ associated with the semigroup $S_{\varepsilon}(t)$ generated by the problem \eqref{13}. For the theory of global attractors see \cite{2,8,14,18,27,30,32}. For $\varepsilon=0$ weak solutions of problem are known to exist by a basic result by Leray from 1934 \cite{24}, only the uniqueness of weak solutions remains as an open problem. Then the known theory of global attractors of infinite dimensional dynamical systems is not applicable to the 3D Navier--Stokes system. The theory of trajectory attractors for evolution partial differential equations was developed in \cite{30}, which the uniqueness theorem of solutions of the corresponding initial-value problem is not proved yet, e.g. for the 3D Navier--Stokes system (see, for instance,\cite{14,30}). Such trajectory attractor is a classical global attractor but in the space of weak solutions. The problem of upper semicontinuity of global attractors for the 2D with periodic boundary conditions was discussed by Yuh-Roung Ou and S. S. Sritharan\ in \cite{28}. For related results which use the theory has been introduced by Foias, Sell, and Temam in \cite{12,32} to show that the system \eqref{1} possesses an inertial manifold (see \cite{1,29,32}). The existence and uniqueness results for initial value problem \eqref{13} are consequence of Theorem \ref{thm4} for $l=2$ and $d=3$. \begin{theorem} \label{thm5}Let $\Omega\subset\mathbb{R}^{3}$, and let $f\in L^{2}(0,T;V_2')$ and $u_{\varepsilon_0}\in V_0$ be given. Then there exists a unique weak solution of \eqref{13} which satisfies $u_{\varepsilon}\in C([ 0,T];V_0)\cap L^{2}(0,T;V_2)$, for all $T>0$. Then as $\varepsilon\to0$, the solution $u_{\varepsilon}$ converges to the weak solution of the Navier-Stokes equations. \end{theorem} Now, we show that the semigroup $S_{\varepsilon}(t)$ has an absorbing ball in $V_0$ and an absorbing ball in $V_1$. Then we show that $S_{\varepsilon}(t)$ admits a compact attractor in $V_0$ for each $\varepsilon\geq0$. We take the inner product of \eqref{13} with $u_{\varepsilon}$, we obtain the energy equality \[ \frac{d}{dt}\| u_{\varepsilon}\| ^{2}+2\varepsilon\| Au_{\varepsilon}\| ^{2}+2\nu\| \nabla u_{\varepsilon }\| ^{2}=2(f,u_{\varepsilon}). \] Here we have used the fact that $b(u_{\varepsilon},u_{\varepsilon },u_{\varepsilon})=0$. By applying Young's inequality and the Poincar\'{e} Lemma, we get \begin{equation} \frac{d}{dt}\| u_{\varepsilon}\| ^{2}+2\varepsilon\| Au_{\varepsilon}\| ^{2}+\nu\| \nabla u_{\varepsilon }\| ^{2}\leq\frac{\| f\| ^{2}}{\nu\lambda_1}, \label{14} \end{equation} we drop the term $2\varepsilon\| Au_{\varepsilon}\| ^{2}$, we obtain \[ \frac{d}{dt}\| u_{\varepsilon}\| ^{2}+\nu\lambda_1\| u_{\varepsilon}\| ^{2}\leq\frac{\| f\| ^{2}}{\nu\lambda_1}, \] by integrating the above inequality from $0$ to $t$,we get \begin{equation} \| u_{\varepsilon}(t)\| ^{2}\leq\| u_{\varepsilon_0}\| ^{2}e^{-\nu\lambda_1t}+\rho_0^{2}( 1-e^{-\nu\lambda_1t}),\text{ }t>0, \label{15} \end{equation} where $\rho_0=\frac{1}{\nu\lambda_1}\| f\| $. Hence for any ball $B_{R_0}=\{ u_{\varepsilon_0}\in V_0;\| u_{\varepsilon_0}\| \leq R_0\} $ there is a ball $B(0,\delta_0)$ in $V_0$ centered at origin with radius $\delta_0>\rho_0$ $(R_0>\delta_0)$ such that \begin{equation} S_{\varepsilon}(t)B_{R_0}\subset B_{r_0}\quad \text{for }t\geq t_0( B_{R_0}) =\frac{1}{\nu\lambda_1}\log\frac{R_0^{2}-\rho_0^{2} }{\delta_0^{2}-\rho_0^{2}}. \label{16} \end{equation} The ball $B_{\delta_0}$ is said to be absorbing and invariant under the action of $S_{\varepsilon}(t)$. Taking the limit in \eqref{15} we obtain \begin{equation} \limsup_{t\to\infty}\| u_{\varepsilon}(t) \| \leq\rho_0. \label{17} \end{equation} We integrate \eqref{14} from $t$ to $t+r$, we obtain for $u_{\varepsilon 0}\in B_{R_0}$, \begin{equation} \int_{t}^{t+r}\| u_{\varepsilon}\|_1^{2}ds\leq\frac{1} {\nu}(\frac{r\| f\| ^{2}}{\nu\lambda_1}+\| u_{\varepsilon}(t) \| ^{2}),\forall r>0,\text{ }\forall t\geq t_0(B_{R_0}). \label{18} \end{equation} With the use of \eqref{17} we conclude that \begin{equation} \limsup_{t\to\infty}\int_{t}^{t+r}\| u_{\varepsilon }\| _1^{2}ds\leq\frac{r}{\nu^{2}\lambda_1}\| f\| ^{2}+\frac{\| f\| ^{2}}{\nu^{3}\lambda_1^{2}}, \label{19} \end{equation} from which we obtain \begin{equation} \limsup_{t\to\infty}\frac{1}{t}\int_0^{t}\| u_{\varepsilon }\|_1^{2}ds\leq\frac{\| f\| ^{2}}{\nu^{2} \lambda_1}, \label{20} \end{equation} this verifies that the left-hand side is finite. To show that the semigroup $S_{\varepsilon}(t)$ has an absorbing set in $V_1$, we consider the strong solutions and take the inner product of \eqref{13} with\ $Au_{\varepsilon}$, we obtain \begin{equation} \frac{1}{2}\frac{d}{dt}\| A^{1/2}u_{\varepsilon}\| ^{2}+\varepsilon\| A^{3/2}u_{\varepsilon}\|^{2}+\nu\| Au_{\varepsilon}\|^{2}=-b(u_{\varepsilon},u_{\varepsilon},Au_{\varepsilon })+(f,Au_{\varepsilon}). \label{21} \end{equation} By applying Young's inequality, we obtain \[ (f,Au_{\varepsilon})\leq\| f\| \| Au_{\varepsilon }\| \leq\frac{\nu}{4}\| Au_{\varepsilon}\| ^{2}+\frac{1}{\nu}\| f\| ^{2}. \] By using the Agmon's inequality \eqref{6} and Young's inequality we can estimate the last term in the left-hand side of \eqref{21} as follows \begin{align*} | b(u_{\varepsilon},u_{\varepsilon},Au_{\varepsilon})| & \leq\| u_{\varepsilon}\|_{\infty}\| u_{\varepsilon }\|_1\| Au_{\varepsilon}\| \\ & \leq c_{4}\| u_{\varepsilon}\|_1^{3/2}\| Au_{\varepsilon}\| ^{3/2}\\ & \leq\frac{\nu}{4}\| Au_{\varepsilon}\| ^{2}+c_{4} \| u_{\varepsilon}\|_1^{6}. \end{align*} Hence we obtain from \eqref{21} \[ \frac{d}{dt}\| u_{\varepsilon}\|_1^{2}+2\varepsilon\| A^{3/2}u_{\varepsilon}\|^{2}+\nu\| Au_{\varepsilon }\| ^{2}\leq\frac{2}{\nu}\| f\| ^{2}+2c_{5} \| u_{\varepsilon}\| _1^{6}. \] Dropping the positive terms associated with $\varepsilon$ we have \begin{equation} \frac{d}{dt}\| u_{\varepsilon}\|_1^{2}+\nu\| A_1u_{\varepsilon}\| ^{2}\leq\frac{2\| f\| ^{2} }{\nu}+2c_{4}\| u_{\varepsilon}\|_1^{6} \label{22} \end{equation} we apply the uniform Gronwall Lemma to \eqref{22} with \[ g=2c_{4}\| u_{\varepsilon}\|_1^{4},\quad h=\frac {2\| f\| ^{2}}{\nu},\quad y=\| u_{\varepsilon }\|_1^{2}. \] Thanks to \eqref{15}-\eqref{19} we estimate the quantities $a_1$, $a_2$, $a_{3}$ in Gronwall Lemma by \[ a_1=2c_{4}a_{3}^{2},\quad a_2=\frac{2r\| f\| ^{2}}{\nu},\quad a_{3}=\frac{r\| f\| ^{2}}{\nu^{2}\lambda_1} +\frac{\| f\| ^{2}}{\nu^{3}\lambda_1^{2}} \] and we obtain \[ \| u_{\varepsilon}(t)\|_1^{2}\leq (\frac{a_{3}}{r}+a_2)\exp(a_1)=R_1^{2}\quad \text{for }t\geq t_0,\; t_0\text{ as in }(\text{\ref{16}}). \] Hence, for any ball $B_{R_1}$, there exists a ball $B_{\delta_1}$, in $V_1$ centered at origin with radius $R_1>\delta_1>\rho_1$ such that \[ S_{\varepsilon}(t)B_{R_1}\subset B_{\delta_1}\quad \text{for }t\geq t_1(B_{R_0})=t_0(B_{R_0})+1+\frac{1} {\nu\lambda_1}\log\frac{R_1^{2}-\rho_1^{2}}{\delta_1^{2}-\rho_1^{2} }. \] The ball $B_{\delta_1}$ is said to be absorbing and invariant for the semigroup $S_{\varepsilon}(t)$. Furthermore, if $B$ is any bounded set of $V_0$, then $S_{\varepsilon }(t)B\subset B_{\delta_1}$ for $t\geq t_1(B,R_0)$, this shows the existence of an absorbing set in $V_1$. Since the embedding of $V_1$ in $V_0$ is compact, we deduce that $S_{\varepsilon}(t)$ maps a bounded set in $V_0$ into a compact set in $V_0$. In addition, the operators $S_{\varepsilon}(t)$ are uniformly compact for $t\geq t_1( B,R_0)$. That is, \[ \cup_{t\geq t_1} S_{\varepsilon}(t,0,B_{R_0}) \] is relatively compact in $V_0$. Due to a the standard procedure (cf., for example, \cite[Theorem I.1.1]{32} for details), one can prove that there is a global compact attractor $\mathfrak{A}_{\varepsilon}$ for the operators $S_{\varepsilon}(t)$ for $\varepsilon\geq0$. Note that the global attractor $\mathfrak{A}_{\varepsilon}$ must be contained in the absorbing balls $V_0$ and $V_1$ \begin{equation} \mathfrak{A}_{\varepsilon}= \cap_{t_1\geq0}\overline{ \cup_{t\geq t_1} B_{\delta_1}(t)}\subset B_{\delta_0}\cap B_{\delta_1}. \label{23} \end{equation} Notice that all the above bounds are independent of $\varepsilon$. \section{Estimates of Dimensions of the Global Attractor} Our aim in this section is to study the finite dimensionality of the global attractor. In the first part we will prove the differentiability property of $S_{\varepsilon}(t)$ and in the second part we will provide estimates of the fractal and Hausdorff dimensions of their global attractors $\mathfrak{A}_{\varepsilon}$. Using the trace formula \cite[Chapters V and VI]{32}, we estimate the Hausdorff and the fractal dimensions of the global attractor $\mathfrak{A}_{\varepsilon}$ in $V$. For a solution $u_{\varepsilon}(t)=S_{\varepsilon}( t) u_{\varepsilon_0}$, $t\geq0$, lying on the attractor $u_{\varepsilon_0}\in\mathfrak{A}_{\varepsilon}$, we see from \eqref{13} that the linearized flow around $u_{\varepsilon}$ is given by the equation \begin{equation} \begin{gathered} U_{\varepsilon}'+\varepsilon A^{2}U_{\varepsilon}+\nu AU_{\varepsilon }+B( u_{\varepsilon},U_{\varepsilon})+B(U_{\varepsilon },u_{\varepsilon})=0,\quad\text{in }V'\\ U_{\varepsilon}(0)=\xi,\quad\text{in }V. \end{gathered} \label{24} \end{equation} We show the differentiability of the semigroup $S_{\varepsilon}$ with respect to the initial data in the space $V$. \begin{theorem}\label{thm6} For any $t>0$, the function $u_{\varepsilon_0}\to u_{\varepsilon}(t) =S_{\varepsilon}(t)u_{\varepsilon_0}$ is Fr\'{e}chet differentiable on the attractor $\mathfrak{A}_{\varepsilon}$. Its differential is the linear operator \[ D(S_{\varepsilon}(t)u_{\varepsilon_0})=L( t,u_{\varepsilon_0}) :\xi\in V\to U_{\varepsilon}(t)\in V\text{,\ }t\in[0,T] , \] where $U_{\varepsilon}(t)$ is the solution of \eqref{24}. \end{theorem} \begin{proof} Let \[ \eta(t)=v_{\varepsilon}(t)-u_{\varepsilon}(t)-U_{\varepsilon}(t),\quad U_{\varepsilon}(0) =\xi=v_{\varepsilon_0}-u_{\varepsilon_0}. \] Clearly, $\eta$ satisfies \[ \eta_{t}+\varepsilon A^{2}\eta+\nu A\eta+B(\eta,v_{\varepsilon} )+B(v_{\varepsilon},\eta)-B(w_{\varepsilon},w_{\varepsilon})=0,\quad \eta(0)=0 \] where $w_{\varepsilon}=v_{\varepsilon}-u_{\varepsilon}$. Taking the inner product of the last equation with $\eta$ and using the identity $B(v_{\varepsilon},\eta,\eta)=0$ we obtain \begin{equation} \frac{d\| \eta\| ^{2}}{dt}+2\varepsilon\| A\eta\| ^{2}+2\nu\| \eta\|_1^{2}=2b(\eta ,v_{\varepsilon},\eta)-2b(w_{\varepsilon},w_{\varepsilon},\eta). \label{25} \end{equation} By \eqref{4} the first term in the right-hand side of \eqref{25} has the estimate \begin{align*} | 2b(\eta,v_{\varepsilon},\eta)| & \leq2c_1\| \eta\| ^{1/2}\| \eta\|_1^{\frac{3}{2} }\| v_{\varepsilon}\|_1\\ & \leq2c_1R_1\| \eta\| ^{1/2}\| \eta\|_1^{3/2}\\ & \leq\frac{c_1^{4}R_1^{4}}{\nu^{3}}\| \eta\| ^{2} +\frac{3\nu}{4}\| \eta\|_1^{2}. \end{align*} Employing the inequalities \eqref{4} we estimate the second term in the right hand side of \eqref{25} as follows \[ 2b(w_{\varepsilon},w_{\varepsilon},\eta) \leq 2c_2\| \eta\|_1\| w_{\varepsilon}\|_1^{2} \leq\frac{2c_2^{2}}{\nu}\| w_{\varepsilon}\|_1 ^{4}+\frac{\nu}{2}\| \eta\|_1^{2}. \] Hence, we obtain from \eqref{25} \[ \frac{d\| \eta\| ^{2}}{dt}+2\varepsilon\| A\eta\| ^{2}+\frac{3\nu}{4}\| \eta\|_1^{2} \leq\frac{c_1^{4}R_1^{4}}{\nu^{3}}\| \eta\| ^{2} +\frac{2c_1^{2}}{\nu}\| w_{\varepsilon}\|_1^{4} \] we drop the positive terms $2\varepsilon\| A\eta\| ^{2}$ and $\frac{3\nu}{4}\| \eta\|_1^{2}$ we get \begin{equation} \frac{d\| \eta\| ^{2}}{dt}\leq\frac{c_1^{4}R_1^{4}} {\nu^{3}}\| \eta\| ^{2}+\frac{2c_1^{2}}{\nu}\| w_{\varepsilon}\| _1^{4}. \label{26} \end{equation} From the classical Gronwall Lemma (see \cite{33}), \eqref{26} gives \[ \| \eta\| ^{2}\leq\frac{2c_1^{2}}{\nu}\int_0 ^{t}\| w_{\varepsilon}\|_1^{4}\exp(\int_{s}^{t}\frac {c_1^{4}R_1^{4}}{\nu^{3}}d\tau)ds. \] Thus \begin{equation} \| \eta\| ^{2}\leq C_{o}\int_0^{t}\| w_{\varepsilon }\| _1^{4}ds,\quad C_{o}=\frac{2c_1^{2}}{\nu}\exp(\frac {Tc_1^{4}R_1^{4}}{\nu^{3}}). \label{27} \end{equation} The difference \[ w_{\varepsilon}(t)=v_{\varepsilon}(t) -u_{\varepsilon}(t) =S_{\varepsilon}(t) v_{\varepsilon_0}-S_{\varepsilon}(t)u_{\varepsilon_0} \] satisfies the equation \begin{gather*} \frac{dw_{\varepsilon}}{dt}+\varepsilon A^{2}w_{\varepsilon}+\nu Aw_{\varepsilon}+B(w_{\varepsilon},v_{\varepsilon})+B(v_{\varepsilon },w_{\varepsilon})-B(w_{\varepsilon},w_{\varepsilon})=0,\\ w_{\varepsilon }(0)=v_{\varepsilon_0}-u_{\varepsilon_0}=w_{\varepsilon_0}. \end{gather*} Taking the inner product of the last equation with $w_{\varepsilon}$, we obtain \begin{equation} \frac{d}{dt}\| w_{\varepsilon}\| ^{2}+2\varepsilon\| Aw_{\varepsilon}\| ^{2}+2\nu\| w_{\varepsilon}\| _1^{2}=2b(w_{\varepsilon},w_{\varepsilon},v_{\varepsilon}). \label{28} \end{equation} By using inequalities \eqref{4}, and Young's inequality we obtain \[ | 2b(w_{\varepsilon},v_{\varepsilon},w_{\varepsilon})| \leq2c_1\| v_{\varepsilon}\| \| w_{\varepsilon }\| _1^{3/2}\| w_{\varepsilon}\|^{1/2} \leq\frac{c_1^{4}R^{4}}{\nu^{3}}\| w_{\varepsilon}\| ^{2}+\frac{3\nu}{4}\| w_{\varepsilon}\|_1^{2}. \] Substituting the above result into \eqref{28}, we obtain \begin{equation} \frac{d}{dt}\| w_{\varepsilon}\| ^{2}+2\varepsilon\| Aw_{\varepsilon}\| ^{2}+\frac{5\nu}{4}\| w_{\varepsilon }\| _1^{2}\leq\frac{c_1^{4}R^{4}}{\nu^{3}}\| w_{\varepsilon}\| ^{2}. \label{29} \end{equation} We drop the positive terms $2\varepsilon\| Aw_{\varepsilon}\| ^{2}$ and $\frac{5\nu}{4}\| w_{\varepsilon}\|_1^{2}$ to obtain the following differential inequality \begin{equation} \frac{d}{dt}\| w_{\varepsilon}\| ^{2}\leq\frac{c_1^{4} R^{4}}{\nu^{3}}\| w_{\varepsilon}\| ^{2}. \label{30} \end{equation} Using the classical Gronwall Lemma we deduce from \eqref{30} that \begin{equation} \| w_{\varepsilon}\| ^{2}\leq\| w_{\varepsilon }(0)\| ^{2}\exp(\frac{Tc_1^{4}R^{4}}{\nu^{3} }). \label{31} \end{equation} From \eqref{31} we deduce that \begin{equation} \int_0^{t}\| u_{\varepsilon}(t)-v_{\varepsilon }(t)\| ^{2}dt\leq C_1\| u_{\varepsilon 0}-v_{\varepsilon_0}\| ^{2},\quad C_1=T\exp(\frac{Tc_1^{4}R^{4} }{\nu^{3}}), \label{32} \end{equation} with \eqref{27} we conclude that \[ \| \eta\| ^{2}\leq C_{o}C_1^{2}\| u_{\varepsilon 0}-v_{\varepsilon_0}\| ^{4}, \] then we deduce from \eqref{27} and \eqref{32} that \begin{equation} \| \eta\| ^{2}\leq C_2\| w_{\varepsilon}( 0)\| ^{4},\quad \text{where }C_2=\frac{2c_1^{2}T^{2}}{\nu }\exp(\frac{Tc_1^{4}(2R^{4}+R_1^{4})}{\nu^{3}}) \label{33} \end{equation} this shows that \[ \frac{\| v_{\varepsilon}(t)-u_{\varepsilon}(t)-U_{\varepsilon }(t)\| ^{2}}{\| v_{\varepsilon_0}-u_{\varepsilon_0}\| ^{2}}\leq C_2\| v_{\varepsilon_0}-u_{\varepsilon_0}\| ^{2}\to0\quad \text{as }\| v_{\varepsilon_0}-u_{\varepsilon 0}\| _1\to0 \text{ on }\mathfrak{A}_{\varepsilon}. \] The differentiability of $S_{\varepsilon}(t)$ is proved. \end{proof} From Theorem \ref{thm6}\ the function $S_{\varepsilon}(t)$ is Fr\'{e}chet differentiable on $\mathfrak{A}_{\varepsilon}$ for $t>0$. For $\xi\in V_0$, there exists a unique solution $U_{\varepsilon}$ of \eqref{24} satisfies \[ U_{\varepsilon}\in C([0,T] ;V_0)\cap L^{2}(0,T;V_2)\text{ \ }\forall T>0. \] With the differentiability ensured in Theorem \ref{thm5} we can then define a linear map $L(t;u_{\varepsilon_0}):\xi\in V_0\to U_{\varepsilon}(t)\in V_0$ where $U_{\varepsilon}$ is the solution of \eqref{24}. We can apply the trace formula (see \cite{8} and \cite[Section V. 3]{32}) to find a bound on the dimension of the global attractor $\mathfrak{A}_{\varepsilon}$. We consider the trace $TrF'(u_{\varepsilon })$ of the linear operator $F'(u_{\varepsilon})$ and for $m\in \mathbb{N}$, the number \[ q_{m}=\limsup_{t\to\infty}\sup_{u_{\varepsilon _0}\in \mathfrak{A}_{\varepsilon}} \sup_{\substack{\xi_1\in V_0,\, |\xi_1| \leq 1\\ i=1,\dots ,m}} \frac{1}{t}\int_0^{t}TrF'(S_{\varepsilon}( \tau)u_{\varepsilon_0})\circ Q_{m}(\tau)d\tau \] where $Q_{m}(\tau)=Q_{m}(\tau,u_{\varepsilon_0};\xi _1,\dots ,\xi_{m})$ is the orthogonal projector in $V_0$ onto the space spanned by $U_{\varepsilon}^{1}(\tau) ,\dots ,U_{\varepsilon }^{m}(\tau)$. where $U_{\varepsilon}^{j}(\tau)$ $=L( \tau,u_{\varepsilon_0}).\xi_{j}$, $j=1,\dots ,m$, $t\geq0$, $\ $are $m$ solutions of \eqref{24}, corresponding to $\xi=\xi_1 ,\dots ,\xi_{m}\in V_1$. Let $\varphi_{j}( \tau)$, $j=1,\dots ,m$, $\tau\geq0$, be an orthonormal basis of for $\tilde{Q}_{m}(\tau)V_0=$span $\{ U_{\varepsilon}^{1}(\tau),\dots ,U_{\varepsilon}^{m}( \tau) \} $, $\varphi_{j}(t)\in V_1$ for $j=1,\dots ,m$, since $U_{\varepsilon}^{1}(\tau) ,\dots ,U_{\varepsilon}^{m}(\tau) \in V_1$, $\tau\in \mathbb{R}^{+}$. From the general result in \cite[Section V.3.41]{32}, we have that if $q_{m}<0$ for some $m\in N$ then the global attractor has finite Hausdorff and fractal dimensions estimated respectively as \begin{gather} \dim_{H}(\mathfrak{A}_{\varepsilon}) \leq m,\label{34}\\ \dim_{F}(\mathfrak{A}_{\varepsilon}) \leq m(1+\max_{1\leq j\leq m-1}\frac{(q_{j})_{+}}{\| q_{m}\| }). \label{35} \end{gather} Then we have \begin{align*} TrF'(S_{\varepsilon}(\tau)u_{\varepsilon 0})\circ Q_{m}(\tau) & = \sum_{j=1}^{\infty} (TrF'(u_{\varepsilon}(\tau))\circ Q_{m}(\tau) \varphi_{j}(\tau),\varphi_{j}(\tau))\\ & = \sum_{j=1}^{m} (F'(u_{\varepsilon}(\tau))\varphi_{j}( \tau) ,\varphi_{j}(\tau)), \end{align*} Recall that $(\cdot,\cdot)$ denotes the scalar product in $V_0$, we write using \eqref{2} and \eqref{3}, \begin{align*} &Tr(F'(u_{\varepsilon}(\tau))\varphi_{j}(\tau) ,\varphi_{j}(\tau)) \\ & = \sum_{j=1}^{m} (-\varepsilon A^{2}\varphi_{j}-\nu A\varphi_{j}-B(\varphi _{j},u_{\varepsilon})-B(u_{\varepsilon},\varphi_{j}),\varphi_{j})\\ & = \sum_{j=1}^{m} (-\varepsilon\| A\varphi_{j}\| ^{2}-\nu\| A^{\frac{1}{2} }\varphi_{j}\|^{2}-b( u_{\varepsilon},\varphi_{j},\varphi_{j})-b( \varphi_{j},u_{\varepsilon},\varphi_{j})); \end{align*} thus \begin{equation} Tr(F'(u_{\varepsilon}(\tau))\varphi_{j}(\tau) ,\varphi_{j}(\tau))= \sum_{j=1}^{m} (-\varepsilon\| \varphi_{j}\|_2^{2}-\nu\| \varphi_{j}\| _1^{2}-b(\varphi_{j},u_{\varepsilon} ,\varphi_{j})). \label{36} \end{equation} We estimate the nonlinear term as follows \[ |\sum_{j=1}^m b(\varphi_{j},u,\varphi_{j})| =|\sum_{j=1}^m \int_{\Omega} \sum_{k,l=1}^3 \varphi_{jk}\frac{\partial u_{_l}}{\partial x_{k}}(x) \varphi_{jl}dx| \] whence for almost every $x\in\Omega$ we have \[ |\sum_{j=1}^m \sum_{k,l=1}^3 \varphi_{jk}\frac{\partial u_{_l}}{\partial x_{k}}(x) \varphi_{jl}dx| \leq\| u\|_1\| \rho\| \] where \[ \| u(x)\|_1=( \sum_{k,l=1}^3 \| D_{i}u_{k}(x)\| ^{2})^{\frac{1}{2} }\text{ and }\rho(x)= \sum_{j=1}^m \sum_{i=1}^3 (\varphi_{ji}(x))^{2}. \] Therefore, \begin{equation} \big|\sum_{j=1}^m b(\varphi_{j},u,\varphi_{j}) \big| \leq\int_{\Omega}\rho(x) \| u(x)\|_1dx \label{37} \end{equation} with the Schwarz inequality \begin{equation} |\sum_{j=1}^m b(\varphi_{j},u,\varphi_{j})|\leq\| u( x) \|_1\| \rho(x)\| . \label{38} \end{equation} Applying the weighted Sobolev-Lieb-Thirring inequality \cite[Theorem A.3.1]{32}, there exists $c_{5}$ independent of the family $\varphi_{j}$, $m$ and of $\varepsilon$ such that \begin{equation} \| \rho\| ^{2}\leq c_{5} \sum_{j=1}^m \| \varphi_{j}(x)\|_1^{2}. \label{39} \end{equation} Insert \eqref{39} into \eqref{38} to find \[ |\sum_{j=1}^m b(\varphi_{j},u,\varphi_{j})|\leq\| u\|_1(c_{5} \sum_{j=1}^m \| \varphi_{j}(x)\|_1^{2})^{^{1/2} }, \] using the Young inequality we obtain \[ |\sum_{j=1}^m b(\varphi_{j},u,\varphi_{j})|\leq\frac{\nu}{2} \sum_{j=1}^m \| \varphi_{j}(x)\|_1^{2}+\frac{c_{5}} {2\nu}\| u\|_1^{2}. \] By using the Sobolev embedding Theorem $V_2\subset V_1$, we have \begin{equation} c_{6}\| \varphi_{j}\|_1^{2}\leq\| \varphi_{j}(x)\|_2 \label{40} \end{equation} for an absolute constant $c_{6}$. Using the inequalities above \eqref{36} gives \[ TrF'(u_{\varepsilon}(\tau))\circ Q_{m}(\tau) \leq-\varepsilon c_{6} \sum_{j=1}^m \| \varphi_{j}\|_1^{2}-\frac{\nu}{2} \sum_{j=1}^m \| \varphi_{j}(x)\|_1^{2}+\frac{c_{5}} {2\nu}\| u\|_1^{2}. \] We now use the estimate for $\rho$. In fact it is $\lambda_{m}\sim c\lambda_1m^{2/3}$ in 3D, which can be found for example in \cite{11} or \cite[Lemma VI.2.1]{32}, there exists a constant $c_{7}$ such that \[ \sum_{j=1}^m \| \varphi_{j}(x)\|_1^{2}\geq\lambda_1+\dots +\lambda_{m}\geq c_{7}\lambda_1m^{5/3}, \] use \eqref{40} to estimate $TrF'(u_{\varepsilon}( \tau))\circ Q_{m}(\tau)$ as follows \begin{equation} TrF'(u_{\varepsilon}(\tau))\circ Q_{m}(\tau) \leq-(\varepsilon c_{7}+\frac{\nu}{2})c_{7} \lambda_1m^{5/3}+\frac{c_{5}}{2\nu}\| u_{\varepsilon }\| _1^{2}. \label{41} \end{equation} Kolmogorov's mean rate of dissipation of energy in turbulent flow (see e.g. \cite[VI.(3.20)]{11,16,32}) is defined as \begin{equation} \epsilon=\lambda_1^{3/2}\nu\limsup_{t\to\infty} \sup_{u_{\varepsilon_0}\in\mathfrak{A}_{\varepsilon}}\frac{1}{t}\int_0 ^{t}\| u_{\varepsilon}(\tau)\|_1^{2} d\tau\label{42} \end{equation} the maximal mean rate of dissipation of energy on the attractor, which is finite thanks to \eqref{20}. Hence \[ \frac{1}{t}\int_0^{t}Tr(F'(S_{\varepsilon}( \tau) u_{\varepsilon_0})\circ Q_{m}(\tau))d\tau\leq-(\varepsilon c_{7}+\frac{\nu}{2})c_{7}\lambda_1m^{\frac{5}{3} }+\frac{c_{5}}{2\nu}\frac{1}{t}\int_0^{t}\| u_{\varepsilon}(\tau)\|_1^{2}d\tau. \] Using \eqref{42} we can estimate the quantities $q_{m}$ \[ q_{m}\leq-\kappa_1m^{5/3}+\kappa_2, \] with \[ \kappa_1=(\varepsilon c_{6}+\frac{\nu}{2})c_{7}\lambda_1\quad \text{and}\quad \kappa_2=\frac{c_{5}}{2\nu^{2}\lambda_1^{3/2}}\epsilon. \] Therefore, if $m'\in \mathbb{N}$ is defined by \[ m'-1<(\frac{2\kappa_2}{\kappa_1})^{3/5} =(\frac{2c_{5} }{\nu^{2}\lambda_1^{5/2}(2\varepsilon c_{6}+\nu) c_{7}})^{3/5}\epsilon^{3/5}\leq m', \] then $q_{m'}\leq0$, setting $c_{8}^{\varepsilon}=(\frac{2c_{5}} {\nu^{2}\lambda_1^{5/2}(2\varepsilon c_{6}+\nu) c_{7} })^{3/5}$so that from \eqref{34}-\eqref{35} this $m'$ is an upper bound for the dimension of the global attractor, \[ \dim_{H}(\mathfrak{A}_{\varepsilon}) \leq\dim_{F}(\mathfrak{A}_{\varepsilon}) \leq c_{8}^{\varepsilon}\epsilon^{3/5}. \] Using \eqref{20} we can estimate the energy dissipation flux $\epsilon$ by \begin{equation} \epsilon\leq\frac{\lambda_1^{1/2}\| f\| ^{2}}{\nu}. \label{43} \end{equation} To make the dimension estimate more explicit, we can estimate the energy dissipation flux $\epsilon$ in terms of $G$ by \begin{equation} \epsilon\leq\lambda_1^{2}\nu^{3}G^{2}. \label{44} \end{equation} Therefore, using \eqref{44} we prove the following Proposition. \begin{proposition} The global attractor $\mathfrak{A}_{\varepsilon}$ of the regularized 3D Navier-Stokes \eqref{13}, is finite dimensional, in $V_0$ has finite Hausdorff and fractal dimensions, which can be estimated in terms of the Grashoff number by \[ \dim_{H}(\mathfrak{A}_{\varepsilon})\leq\dim_{F}( \mathfrak{A}_{\varepsilon})\leq c_{9}G^{6/5} \] where $c_{9}=c_{8}^{\varepsilon}\nu^{9/5}\lambda_1^{6/5}$. \end{proposition} We can estimate $c_{8}^{\varepsilon}$ as follow \[ c_{8}^{\varepsilon}\leq(\frac{2c_{5}}{\nu^{3}\lambda_1^{5/2}c_{7} })^{3/5}=c_{8}^{0}. \] Then there exists a constant $c_{10}=c_{8}^{0}\nu^{9/5}\lambda _1^{6/5}$ independent of $\varepsilon$. Hence \begin{theorem} \label{thm5.3} The Hausdorff and fractal dimensions of the global attractor $\mathfrak{A}_{\varepsilon}$ of the regularized 3D Navier-Stokes \eqref{13}, $\dim_{F}( \mathfrak{A}_{\varepsilon})$ and $\dim_{H}(\mathfrak{A}_{\varepsilon})$ respectively, satisfy \[ \dim_{H}(\mathfrak{A}_{\varepsilon})\leq\dim_{F}( \mathfrak{A}_{\varepsilon})\leq c_{10}G^{6/5}. \] \end{theorem} \section{Numbers of degrees of freedom in turbulent flows} In this Section, we estimate the effects of hyperviscosity on the turbulent flow. An argument from the classical theory of turbulence (see, L. Landau and Lifshitz \cite{22}) suggests that there are finitely many degrees of freedom in turbulent flows. Heuristic physical arguments are used to justify this assertion and to provide an estimate for this number of degrees of freedom by dividing a typical length scale of the flow, $l_0=\lambda_1^{-1/2}$, by the Kolmogorov dissipation length scale $l_{\epsilon}$; i.e., $l_{\epsilon}=\frac{\nu^{3}}{\epsilon}$ where $\epsilon$ is Kolmogorov's mean rate of dissipation of energy in turbulent flow and taking the third power in 3D. We will express our primary attractor results in terms of the Kolmogorov length-scale $l_{\epsilon}$ and the Landau-Lifschitz estimates \cite{22} of the number of degrees of freedom in turbulent flow \cite{11, 32} and we can easily observe such compatibility that exists between these estimates and the number of degrees of freedom in turbulence (see also \cite{22}). Such estimates will give us useful information about the capability of \eqref{13} to approximate Navier-Stokes equations dynamics. We will show that the corresponding number of degrees of freedom is proportional to the dimension of the global attractor. By Holder's inequality the right hand side of \eqref{37} can be estimated as follow \begin{align*} \int_{\Omega}\| u(x)\|_1\| \rho(x)\| dx &\leq\| \rho(x)\|_{L^{5/3}(\Omega)}\| A^{1/2}u_{\varepsilon}(x) \|_{L^{5/2}(\Omega)}\\ &\leq(c_{5} \sum_{j=1}^m \| \varphi_{j}\|_1^{2})^{3/5}\| A^{1/2}u_{\varepsilon}(x) \|_{L^{5/2}(\Omega).} \end{align*} By Young's inequality we obtain \begin{equation} \int_{\Omega}\| u(x)\|_1\| \rho(x)\| dx\leq\frac{\nu}{2} \sum_{j=1}^m \| \varphi_{j}\|_1^{2}+\frac{c_{5}}{\nu^{3/2}} \| A^{1/2}u_{\varepsilon}(x)\| _{L^{5/2}(\Omega)}^{5/2}. \label{45} \end{equation} Using \eqref{40}, \eqref{45} we can majorize $TrF'( u_{\varepsilon}(\tau)) \circ\tilde{Q}_{m}(\tau)$ as follows \begin{equation} \begin{aligned} &TrF'(u_{\varepsilon}(\tau))\circ \tilde{Q}_{m}(\tau)\\ &\leq-\varepsilon c_{6} \sum_{j=1}^m \| \varphi_{j}(\tau)\|_1^{2}-\frac{\nu}{2} \sum_{j=1}^m \| \varphi_{j}(x)\|_1^{2}+\frac{c_{5}} {\nu^{3/2}}\| A^{1/2}u_{\varepsilon}( x) \|_{L^{5/2}(\Omega)}^{5/2}. \end{aligned}\label{46} \end{equation} Note that in the 3D case we have $\lambda_{j}\geq c_{11}L^{-2}j^{\frac{2}{3}}$ for some positive universal constant (see, for example \cite[Lemma VI 2.1]{32}). Therefore, \begin{equation} \sum_{j=1}^m \| \varphi_{j}(x)\|_1^{2}\geq\lambda_1+\dots +\lambda_{m}\geq c_{12}\lambda_1m^{5/3}. \label{47} \end{equation} Taking into account \eqref{46} and \eqref{51}\ then yields \begin{align*} &TrF'(u_{\varepsilon}(\tau))\circ Q_{m}(\tau)d\tau \\ &\leq-\varepsilon c_{6}c_{12}\lambda_1 ^{2}m^{5/3}-\frac{\nu}{2}c_{12}\lambda_1m^{5/3}+\frac{c_{5} }{\nu^{3/2}}\| A^{1/2}u_{\varepsilon}( x) \|_{L^{5/2}(\Omega)}^{5/2}.\\ & \leq(-\varepsilon c_{6}-\frac{\nu}{2})c_{12}\lambda_1m^{5/3}+\frac{c_{5}}{\nu^{3/2}}\| A^{1/2}u_{\varepsilon}( x)\|_{L^{5/2}(\Omega)}^{5/2}.\\ & \leq-(\varepsilon c_{6}+\frac{\nu}{2})c_{12}\lambda_1m^{5/3}+\frac{c_{5}}{\nu^{3/2}}\| A^{1/2}u_{\varepsilon}(x) \|_{L^{5/2}(\Omega)}^{5/2}. \end{align*} Thanks to \eqref{5} with $\theta=5/2$, we have \[ \| A^{1/2}u_{\varepsilon}(x) \|_{L^{5/2}(\Omega)} \leq c_2\| A^{1/2}u_{\varepsilon}(x) \|^{1/2}\| A^{1/2}u_{\varepsilon}(x) \|_1^{1/5} \] and hence \begin{equation} \| A^{1/2}u_{\varepsilon}(x) \|_{L^{\frac{5}{2} }(\Omega)}^{5/2}\leq c_2^{5/2}\| A^{1/2}u_{\varepsilon}( x)\|^{\frac{5}{4}}\| A^{1/2}u_{\varepsilon}( x)\|_1^{1/2}. \label{48} \end{equation} In fact, the norm $\| A_1^{1/2}u_{\varepsilon}\|_1$ is equivalent to the norm $\| u_{\varepsilon}\|_2$ in $V_2 $. This means \begin{equation} \| Au_{\varepsilon}(x)\| d_2\leq\| A^{1/2} u_{\varepsilon}(x)\|_1\leq d_1\| Au_{\varepsilon}(x)\| . \label{49} \end{equation} Notice that $d_1$ and $d_2$ do not depend on $\varepsilon$. Then, from the above and using H\"{o}lder's inequality\ we obtain \begin{equation} \begin{aligned} &\limsup_{t\to\infty}\sup_{u_{\varepsilon_0}\in\mathfrak{A} _{\varepsilon}}\frac{1}{t}\int_0^{t}\| A^{1/2}u_{\varepsilon }(\tau,x) \|_{L^{5/2}(\Omega)} ^{5/2}d\tau\\ &\leq C_{3}\limsup_{t\to\infty}\sup_{u_{\varepsilon 0}\in\mathfrak{A}_{\varepsilon}}\frac{1}{t}\int_0^{t}\| A^{\frac{1}{2} }u_{\varepsilon}(x) \|^{5/4}d\tau\label{50} \end{aligned} \end{equation} where $C_{3}=c_2^{5/2}d_1M^{1/2}$ and \begin{equation} M=\sup_{t\in[0,T]}\sup_{u_{\varepsilon_0}\in\mathfrak{A}_{\varepsilon}\cap D( A)}\| Au_{\varepsilon}(x)\|\label{51} \end{equation} it is clear that $M$ is finite. On the other hand, using \eqref{42} we have \begin{equation} \begin{aligned} &\sup_{u_{\varepsilon_0}\in\mathfrak{A}_{\varepsilon}}(\limsup_{t\to \infty}\frac{1}{t}\int_0^{t}\| A^{1/2}u_{\varepsilon}(x)\|^{5/4}d\tau)\\ &\leq\sup_{u_{\varepsilon_0} \in\mathfrak{A}_{\varepsilon}}(\limsup_{t\to\infty}\frac{1}{t} \int_0^{t}\| A^{1/2}u_{\varepsilon}(x)\| ^{2}d\tau)^{5/8} \leq \big(\frac{\epsilon}{\lambda_1^{3/2}\nu}\big)^{5/8}. \end{aligned} \label{52} \end{equation} For $u_{\varepsilon_0}\in\mathfrak{A}_{\varepsilon}$, we can estimate the quantities $q_{m}(t)$, $q_{m}$ \[ q_{m}=\limsup_{t\to\infty}q_{m}(t)\leq-\kappa _1m^{5/3}+\kappa_2, \] where \[ \kappa_1=(\varepsilon c_{6}+\frac{\nu}{2})c_{12}\lambda_1, \quad \kappa_2=C_{3}\frac{c_{5}}{\nu^{3/2}}(\frac{\epsilon} {\lambda_1^{3/2}\nu})^{5/8}. \] Therefore, if $m'\in \mathbb{N}$ is defined by \begin{equation} m'-1<\big(\frac{2\kappa_2}{\kappa_1}\big)^{3/5} =\Big(\frac{4C_{3} c_{5}}{(2\varepsilon c_{6}+\nu)\lambda_1^{\frac{31}{16}}\nu^{\frac{17}{8} }c_{12}}\Big)^{3/5}\epsilon^{3/8}