Asymptotic behavior of the non-autonomous 3D Navier–Stokes problem with coercive force

Abstract

We construct pullback attractors to the weak solutions of the three-dimensional Dirichlet problem for the incompressible Navier-Stokes equations in the case when the external force may become unbounded as time goes to plus or minus infinity.2010 Mathematics Subject Classification. 35Q30; 35B41; 35D30. Key words and phrases. Navier-Stokes equation; weak solution; non-autonomous problem; pullback attractor; non-uniqueness.The research was partially supported by CMUC/FCT. 1 2 D.VOROTNIKOV and the initial time goes to minus infinity. The pullback mechanism appeared in much earlier works, e.g. [27] (see discussions in [29,21]), but the concept of a pullback attractor was proposed by Schmallfuss, Crauel and Flandoli (see [26,17] and the references therein) in the early 1990s, and was since then successfully applied to many systems, e.g. [6,13,25,29,34,45]. This approach turned out to be relevant in much more general situations than the one of [14] 1 . Namely, the pullback attractors characterize the behaviour of processes rather at each "finite", "present" moment than as time goes to infinity. Therefore, this notion can be used to investigate the limiting behaviour of the processes which do not have bounded complete trajectories. Such a situation can arise, for instance, for equations with coercive, i.e. unbounded as time goes to plus or minus infinity, right-hand members. The pullback attractors for the 2D Navier-Stokes system with (possibly) coercive non-autonomous body force were constructed in [10,9].The attractor theory turned out to be generalizable onto the case of the problems which lack the property of uniqueness of solutions (or where such a property remains an open problem). Obviously, such problems do not generate dynamical systems in a normal manner. One of the main motivations for the progress in this direction was the ambition to study the limiting behaviour of the weak solutions to the 3D Navier-Stokes problem. There exist several ways of construction of attractors in this case. The first one, based on the theory of multivalued semigroups, goes back to [2], and was developed in [33]. It was used for the weak solutions of the 3D Navier-Stokes problem when the righthand side is uniformly bounded in H or under an unproved hypothesis [24]. A related generalised semiflow approach was proposed in [4], and adapted to stochastic problems in [32].An alternative method employs the concept of trajectory attractor, i.e. the attractor of the translation semigroup in the space of trajectories [36,15]. The sections of the trajectory attractor coincide with a properly defined global attractor [40,16]. A similar procedure can be realized in the non-autonomous case at the presence of bounded complete trajectories, generalizing the notions of the uniform attractor and of the kernel [15,40]. The trajectory attractor technique is applicable to the weak solutions of the 3D Navier-Stokes problem [36,15,40,16]. However, it requires the uniform boundedness of the Steklov average (in time) of the square of the V * -norm of the b...

Weak solutions for a bioconvection model related to Bacillus subtilis

“…In order to simplify the presentation, we consider the autonomous case ∇φ ∈ L ∞ (independent of t). However, similar results can be obtained in the non-autonomous case via employment of the more involved theory of pullback trajectory attractors developed recently in [18]. We start with recalling some basic framework from [22,Chapter 4].…”

Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations

“…The concept of a BCTA is used to consider some further topological properties of the usual trajectory attractor (TA), where a TA means a T -compact, invariant and attracting brochette (cf. [22,31]). A T -compact and attracting brochette P is called a trajectory semi-attractor (TSA) if it is (v) positively invariant, i.e.…”

“…Let us explain the idea of a pullback trajectory attractor introduced by Vorotnikov [22]. Take another larger space E 0 such that E → E 0 and let…”

“…Then a set-valued mapping t → P t ⊂ T is said to be a pullback trajectory attractor if it is T -compact, invariant under the translation operator and it pullback attracts all trajectories in H + under the topology of C(R + ; E 0 ). Some existence conclusions of trajectory attractors were established in the nonautonomous case, see [22,24], in the autonomous case, see [2,3,20,21]. The main distinction between nonautonomous and autonomous attractor is that one is time-indexed and another is time-independent.…”

“…This phenomenon is similar to the 3D Navier-Stokes equation, the uniqueness of weak solutions remains open (cf. [11,19,21,22]). However, those weak solutions will generate a trajectory space, which consists of possibly multi-valued trajectories.…”