Differentiability of the solution operator and the dimension of the attractor for certain power–law fluids

We study the dynamics of a two-dimensional homogeneous incompressible fluid of power–law type, with the viscosity behaving like (1+|Du|)p−2, p⩾2. Here Du is the symmetric velocity gradient. Thanks to the recent regularity results of Kaplický, Málek and Stará, we prove that the solution operator is differentiable. This enables us to use the Lyapunov exponents to estimate the dimension of the exponential attractor. In the Dirichlet setting, the obtained estimates are better than in the case of the Navier–Stokes system.