Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

Abstract

<p style='text-indent:20px;'>The three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt (Kelvin-Voight) fluids in bounded domains is considered in this work. We investigate the long-term dynamics of such viscoelastic fluid flow equations with "fading memory" (non-autonomous). We first establish the existence of an absorbing ball in appropriate spaces for the semigroup defined for the Kelvin-Voigt fluid flow equations of order one with "fading memory" (transformed autonomous coupled system). Then, we prove that the semigroup is asymptotically compact, and hence we establish the existence of a global attractor for the semigroup. We provide estimates for the number of determining modes for both asymptotic as well as for trajectories on the global attractor. Once the differentiability of the semigroup with respect to initial data is established, we show that the global attractor has finite Hausdorff as well as fractal dimensions. We also show the existence of an exponential attractor for the semigroup associated with the transformed (equivalent) autonomous Kelvin-Voigt fluid flow equations with "fading memory". Finally, we show that the semigroup has Ladyzhenskaya's squeezing property and hence is quasi-stable, which also implies the existence of global as well as exponential attractor having finite fractal dimension.</p>