Existence of global attractor for the three-dimensional modified Navier-Stokes equations (Q2730680)

The author investigates the following modified Navier-Stokes equations: NEWLINE\[NEWLINE\begin{aligned} u_t+(u\nabla) u-\nabla T(\widehat u)+\nabla p=f,\text{ div }u=0,\;t\geq 0,\\ u=0\text{ on }\partial \Omega,\;u(0,x)=u_0(x),\;\Omega \subseteq \mathbb{R}^3. \tag{1}\end{aligned}NEWLINE\]NEWLINE Here \(\Omega\) is a smooth bounded domain, while NEWLINE\[NEWLINE\widehat u_{ij}=\partial_j u_i+\partial_iu_j,\;\partial_k= \partial_{x_k},\;i,j\leq 3.\tag{2}NEWLINE\]NEWLINE \(T=(T_{ij}(\widehat u))\), \(i,j\leq 3\) is a symmetric stress tensor, whose coefficients are continuous functions of the \(u_{ij}\), \(i,j\leq 3\), subject to the conditions below: NEWLINE\[NEWLINE\begin{aligned} \text{(a)}\quad & \bigl|T_{ij}(\widehat u)\bigr|\leq c\bigl(1+|\widehat u|^{2 \mu} \bigr)|\widehat u|,\;|\widehat u|^2= \sum(u_{ij})^2,\\ \text{(b)}\quad & T_{ij}(\widehat u)u_{ij}\geq \bigl(\nu_0+\nu_1 |\widehat u|^{2\mu} \bigr) |\widehat u|^2,\;i,j\leq 3,\tag{3}\\ \text{(c)}\quad & \int_\Omega \bigl(T_{ij}(\widehat v')-T_{ij} (\widehat v'')\bigr) (v_{ij}'-v_{ij}'') dx\geq\nu_2\int_\Omega \sum_{ij}(v_{ij}'-v_{ij}'')^2 dx.\end{aligned}NEWLINE\]NEWLINE A result by \textit{O. A. Ladyzhenskaya}, restated as Theorem 1.1, [Zapiski Nauchn. Semin. Leningr., Otd. Mat. Inst. Steklov 7, 1-222 (1968; Zbl 0199.10002)] states that if \(\mu\geq 0,25\) then assumptions (3) guarantee the existence of a unique global solution of (1) in a standard \(L^2\)-setting. She further proved [\textit{O. A. Ladyzhenskaya}, Philos. Trans. R. Soc. Lond., Ser. A 346, 173-190 (1994; Zbl 0807.35109)] that if \(T\) is the gradient of a scalar function \(D\), i.e. NEWLINE\[NEWLINET_{ij}(\widehat u)={\partial D(\widehat u)\over\partial u_{ij}},\tag{4}NEWLINE\]NEWLINE subject to conditions which guarantee (3) (a)--(c), then (1) has a maximal global attractor \(M\) (restated as Theorem 1.2).NEWLINENEWLINENEWLINEThe author now asks if this result remains true, if one merely assumes (3) (a)--(c) and drops (4). The answer is affirmative and given by the main result of the paper: Under the assumptions of Theorem 1.2 but with (3) (a)--(c) instead of (4), equation (1) has a global compact attractor. NEWLINENEWLINENEWLINEIn a final section the author indicates the modifications needed when \(\Omega\) is a cylindrical domain.