Existence and regularity of pullback attractors for an incompressible non-Newtonian fluid with delays

Abstract

This paper studies an incompressible non-Newtonian fluid with delays in two-dimensional bounded domains. We first prove the existence and uniqueness of solutions. Then we establish the existence of pullback attractors { A C H ( t ) } t ∈ R \{\mathscr {A}_{_{\mathcal {C}_{_H}}}(t)\}_{t\in \mathbb {R}} (has L 2 L^2 -regularity), { A C W ( t ) } t ∈ R \{\mathscr {A}_{_{\mathcal {C}_{_W}}}(t)\}_{t\in \mathbb {R}} (has H 2 H^2 -regularity), and { A E H 2 ( t ) } t ∈ R \{\mathscr {A}_{_{E^2_{_H}}}(t)\}_{t\in \mathbb {R}} (has L 2 L^2 -regularity), { A E W 2 ( t ) } t ∈ R \{\mathscr {A}_{_{E^2_{_W}}}(t)\}_{t\in \mathbb {R}} (has H 2 H^2 -regularity) corresponding to two different processes associated to the fluid, respectively. Meanwhile, we verify the regularity of the pullback attractors by proving \[ A C H ( t ) = A C W ( t ) , A E H 2 ( t ) = A E W 2 ( t ) , ∀ t ∈ R , \mathscr {A}_{_{\mathcal {C}_{_H}}}(t) =\mathscr {A}_{_{\mathcal {C}_{_W}}}(t),\quad \mathscr {A}_{_{E^2_{_H}}}(t)=\mathscr {A}_{_{E^2_{_W}}}(t), \quad \forall \,t\in \mathbb {R}, \] and \[ A E H 2 ( t ) = J ( A C H ( t ) ) = J ( A C W ( t ) ) = A E W 2 ( t ) , ∀ t ∈ R , \mathscr {A}_{_{E^2_{_H}}}(t) =J(\mathscr {A}_{_{\mathcal {C}_{_H}}}(t)) =J(\mathscr {A}_{_{\mathcal {C}_{_W}}}(t)) =\mathscr {A}_{_{E^2_{_W}}}(t),\quad \forall \,t\in \mathbb {R}, \] where J J is a linear operator. By the regularity we reveal the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data. This effect implies, in the case of delays, that the regularity of the fluid in its history state does not play an important role on the regularity of its eventual state. Finally, we give some remarks.