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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...

Abstract. We consider the initial-boundary value problem for the coupled Navier-Stokes-Keller-Segel-Fisher-Kolmogorov-Petrovskii-Piskunov system in two-and three-dimensional domains. The problem describes oxytaxis and growth of Bacillus subtilis in moving water. We prove existence of global weak solutions to the problem. We distinguish between two cases determined by the cell diffusion term and the space dimension, which are referred as the supercritical and subcritical ones. At the first case, the choice of the growth function enjoys wide range of possibilities: in particular, it can be zero. Our results are new even at the absence of the growth term. At the second case, the restrictions on the growth function are less relaxed: for instance, it cannot be zero but can be Fisher-like. In the case of linear cell diffusion, the solution is regular and unique provided the domain is the whole plane. In addition, we study the long-time behaviour of the problem, find dissipative estimates, and construct attractors.

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This paper deals with the initial-boundary value problem for the system of motion equations of an incompressible viscoelastic medium with Jeffreys constitutive law in an arbitrary domain of two-dimensional or three-dimensional space. The existence of weak solutions of this problem is obtained.