| Abstract: |
| We consider an incompressible, non-Newtonian fluid of power-law type confined to a bounded domain in either $\mathbb{R}^2$ or $\mathbb{R}^3$. On its boundary, we consider the so-called dynamic slip boundary condition, i.e.
$$ \partial_t \boldsymbol u + s(\boldsymbol u) + [\boldsymbol S(D \boldsymbol u) \boldsymbol n]_\tau = 0, $$
where $s$ has some non-linear, possibly even implicit, relation to the velocity field $\boldsymbol u$ and $\boldsymbol S$ is the Cauchy stress. Our primary goal is to study if such a model possesses a finite-dimensional attractor.
Firstly, we use an iteration scheme in Nikolskii-Bochner spaces to obtain additional fractional time regularity, provided that the power-law exponent is in the supercritical range $r \geq 11/5$. Thanks to this regularity, we can use a method of trajectories to obtain the attractor. Moreover, its dimension is finite, and it is even the exponential attractor, provided that $s$ satisfies some reasonable growth condition.
In the two-dimensional setting, we are able to achieve also a better space regularity, namely $L^\infty_{\text{loc}}(0, T; W^{2, q} (\Omega))$. With such regularity, it is now possible to show the differentiability of the solution operator. This opens the way to apply a method of Lyapunov exponents, which gives us a much better dimension estimate. |
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