The author assumes that the relationship between the stress tensor and the deformation velocity tensor is given by: \[ \tau_{ij}=- p\delta_{ij}+ {1\over 2} \Biggl({\partial\over \partial x_i} \varphi_j(u)+ {\partial\over \partial x_j} \varphi_i(u)\Biggr),\qquad i, j= 1, 2, 3,\tag{1} \] where \(p\) denotes the pressure of the fluid and \(\delta_{ij}\) is the Kronecker’s delta, \(\varphi: \mathbb{R}^3\to \mathbb{R}^3\) is a function of \(u\) given by: \(\varphi(u)= \sigma(|u|)u\), where \(\sigma\) satisfies some assumptions. In the case when \(\varphi(u)= \mu u\), the relationship (1) reduces to the classical linear law. Introducing (1) into the general equations of conservation of momentum he obtains the following modified Navier-Stokes equations for incompressible fluids: \[ {\partial\over \partial t} u- \Delta \varphi(u)+ (u\cdot \nabla) u+ \nabla p- \nabla (\nabla \cdot \varphi(u))= f\quad\text{in }\Omega,\qquad \nabla\cdot u= 0.\tag{2} \] The author assumes that \(\Omega\subset \mathbb{R}^3\) is an open bounded set with boundary of class \(C^{1,1}\). The main result in this paper states the following:
The dynamical system defined by equations (2) possesses a (weak) attractor \(A\) and it is possible to estimate the number of determining modes for the corresponding semigroup \(\{S_t\}_{t\geq 0}\) of operators. Furthermore, if the external force \(f\) is “small”, then the unique stationary solution \(u\) of equations (2) is stable and \(A\equiv \{u\}\).