### Article

## Trajectory attractors for non-autonomous dissipative 2d Euler equations

We construct the trajectory attractor AΣ for the non-autonomous dissipative 2d Euler systems with periodic boundary conditions that contain time dependent dissipation terms −r(t)u such that 0<α≤r(t)≤β, for t≥0. External forces g(x,t),x∈T2,t≥0, also depend on time. The corresponding non-autonomous dissipative 2d Navier--Stokes systems with the same terms −r(t)u and g(x,t) and with viscosity ν>0 also have the trajectory attractor AνΣ. Such systems model large-scale geophysical processes in atmosphere and ocean. We prove that AνΣ→AΣ as viscosity ν→0+ in the corresponding metric space. Moreover, we establish the existence of the minimal limit AminΣ⊆AΣ of the trajectory attractors AνΣ as ν→0+. Every set AνΣ is connected. We prove that AminΣ is a connected invariant subset of AΣ. The problem of the connectedness of the trajectory attractor AΣ itself remains open.

We consider reaction{diusion systems with random terms that oscillate rapidly in space variables. Under the assumption that the random functions are ergodic and statistically homogeneous we prove that the random trajectory attractors of these systems tend to the deterministic trajectory attractors of the averaged reaction-diusion system whose terms are the average of the corresponding terms of the original system. Special attention is given to the case when the convergence of random trajectory attractors holds in the strong topology.

We study the limit as α → 0+ of the long-time dynamics for various approximate α-models of a viscous incompressible fluid and their connection with the trajectory attractor of the exact 3D Navier-Stokes system. The α-models under consideration are divided into two classes depending on the orthogonality properties of the nonlinear terms of the equations generating every particular α-model. We show that the attractors of α-models of class I have stronger properties of attraction for their trajectories than the attractors of α-models of class II. We prove that for both classes the bounded families of trajectories of the α-models considered here converge in the corresponding weak topology to the trajectory attractor 0 of the exact 3D Navier-Stokes system as time t tends to infinity. Furthermore, we establish that the trajectory attractor α of every α-model converges in the same topology to the attractor 0 as α → 0+. We construct the minimal limits min ⊆ 0 of the trajectory attractors α for all α-models as α → 0+. We prove that every such set min is a compact connected component of the trajectory attractor 0, and all the min are strictly invariant under the action of the translation semigroup. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.