On finite-dimensional attractor to 2D incompressible fluid with stress-diffusion

Abstract

We consider incompressible visco-elastic rate-type fluid flow, with the Cauchy stress defined in terms of positive-definite symmetric tensor $B$. The tensor $B$ obeys an evolution equation, combining Oldroyd-B and Giesekus model, and, crucially, a diffusion term. Restricting ourselves to the 2D periodic problem, we show existence of a finite-dimensional global attractor. We also estimate its dimension explicitely in terms of the equation parameters.