Attractors of weak solutions to a regularized system of motion equations for fluids with memory (Q647899)

The authors prove the existence of a trajectory and global attractor for a semi-dynamical system generated by weak solutions to the Navier-Stokes system of equations with memory: \[ \partial_t v + (v\cdot\nabla) v - \mu_1 \text{Div }\int_0^t e^{-(t-s)/\lambda} {\mathcal E}(v)(s,Z(s,t,x))ds -\mu_0 \Delta v + \nabla p = \varphi, \] \(\text{div } v =0\), subject to initial and Dirichlet boundary conditions. Here, \({\mathcal E}(v)\) is the rate of deformation tensor, and \(Z\) denotes the regularized trajectory evaluated at the time \(t\) and starting from the point \(x\) at time \(s\leq t\). The main result establishes the existence of a minimal trajectory attractor under the condition \(\mu_0>\mu_1 \lambda\).