Pullback attractors for a non-autonomous incompressible non-Newtonian fluid

Abstract

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors A V in space V (has H 2 -regularity, see notation in Section 2) and A H in space H (has L 2 -regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing A V = A H , which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.

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

“…In [59], the authors of the present paper proved the existence and regularity of the cocycle attractor for the non-Newtonian fluid without delays, by establishing that the associated cocycle satisfies the pullback condition (PC) (see Lemma 2.3 in [59]) and by using the regularity of solutions and embedding theorems. Since C H and C W are not uniform convex Banach spaces, the argument used in Section 3 of [59] seems difficult to apply. Also, the method of energy equality is not applicable because C H and C W are not Hilbert spaces.…”

“…Compared with the work of [59], the new problem encountered in this paper is that C H and C W are neither Hilbert spaces nor uniform convex Banach spaces. In [59], the authors of the present paper proved the existence and regularity of the cocycle attractor for the non-Newtonian fluid without delays, by establishing that the associated cocycle satisfies the pullback condition (PC) (see Lemma 2.3 in [59]) and by using the regularity of solutions and embedding theorems.…”

“…Also, the method of energy equality is not applicable because C H and C W are not Hilbert spaces. At the same time, if the spatial domain is unbounded, the embedding W → H is no longer compact and the arguments used both in Section 3 and Section 4 of [59] are inadequate. We note that the approach presented in this paper is still valid for the case when the spatial domain is unbounded; see Remark 6.2 for details.…”

“…Zhao and Zhou [59] proved the existence and regularity of the pullback attractors (in the version of cocycle attractor) for the non-Newtonian fluid without delays. Schmalfuss [51] gave conditions for the existence of a pullback attractors for cocycles based on the so-called pullback convergence and proved the unique existence of pullback (or cocycle) attractor for the generalized non-autonomous Navier-Stokes equations.…”

“…One of the advantages of the theory of pullback attractors is that it allows one to handle more general nonautonomous terms and it works under random environments as well (see [13]). In fact, the theory of pullback attractors has been widely used and there are many works concerning this subject; one can refer to [10], [13]- [16], [20,23,33,34,38,39,48,55,59].…”