Автономное комплексное уравнение Гинзбурга–Ландау (Г.–Л.) и его глобальный аттрактор были изучены в ряде работ и книг (см., например, [1] – [5] и цитированную там литературу). В настоящей заметке исследуется неавтономное уравнение Г.–Л. и его траекторный аттрактор. При этом основное внимание уделяется тем случаям, когда единственность решения задачи Коши для этого уравнения к настоящему времени не установлена. 1. Рассматривается комплексное уравнение Гинзбурга–Ландау с коэффициентами и возбуждающей силой, зависящими от времени:

We study regular global attractors of dissipative dynamical semigroups with discrete or continuous time and we investigate attractors for nonautonomous perturbations of such semigroups. The main theorem states that the regularity of global attractors is preserved under small nonautonomous perturbations. Moreover, nonautonomous regular global attractors remain exponential and robust. We apply these general results to model nonautonomous reaction-diffusion systems in a bounded domain of R-3 with time-dependent external forces.Bibliography: 22 titles.

The 2D Euler equations with periodic boundary conditions and extra linear dissipative term Ru, R > 0 are considered and the existence of a strong trajectory attractor in the space L-loc(infinity)(R+,H-1) is established under the assumption that the external forces have bounded vorticity. This result is obtained by proving that any solution belonging the proper weak trajectory attractor has a bounded vorticity which implies its uniqueness (due to the Yudovich theorem) and allows to verify the validity of the energy equality on the weak attractor. The convergence to the attractor in the strong topology is then proved via the energy method. (C) 2011 Elsevier Masson SAS. All rights reserved.

We consider a reaction-diffusion system of two equations, where one equation has a small diffusion coefficient delta > 0. We construct the trajectory attractor u(delta) of such a system. We also study the limit system for delta = 0. In this system one equation is an ordinary differential equation in t, but is considered in the domain Omega x R+, where Omega is an element of R-n and R+ is the positive time axis, t. We construct the trajectory attractor u(0) of the limit system. The main result is a convergence theorem: u(delta) -> u(0) as delta -> 0(+) in the corresponding topology.

We study a reaction-diffusion system of N equations with k nonzero and N − k zero diffusion coefficients. More exactly, the first k equations of the system contain the terms a i Δu i − f j (u, v), i = 1, …, k, with the diffusion coefficient a i > 0. The right-hand sides of the other N − k equations contain only nonlinear interaction functions −h j (u, v), j = k + 1, …, N, with zero diffusion. Here u = (u 1, …, u k ) and v = (υ k+1, …, υ N ) are unknown concentration vectors. Under appropriate assumptions on the interaction functions f(·) and h(·), we construct the trajectory attractor \(\mathfrak{A}^0 \) of this reaction-diffusion system. We also find the trajectory attractors \(\mathfrak{A}^\delta \), δ = (δ 1, …, δ k ), for the analogous reaction-diffusion systems having the terms δ j Δυ j − h j (u, v), j = k + 1, …, N, with small diffusion coefficients δ j ⩾ 0 in the last N − k equations. We prove that the trajectory attractors \(\mathfrak{A}^\delta \) converge to \(\mathfrak{A}^0 \) (in an appropriate topology) as δ → 0+.

For ρ ∊ [0, 1) and ε > 0, the nonautonomous 2D Navier–Stokes equations with singularly oscillating external force are considered, together with the averaged equations formally corresponding to the limiting case ε = 0. Under suitable assumptions on the external force, the uniform boundedness of the related uniform global attractors is established, as well as the convergence of the attractors of the first system to the attractor of the second one as ε → 0+. When the Grashof number of the averaged equations is small, the convergence rate of to is controlled by Kε1−ρ.

We study a reaction-difiusion system of N equations with k nonzero and N ¡ k zero difiusion coe-cients. More exactly, the flrst k equations of the system contain the terms ai¢ui ¡ fj(u;v), i = 1;k, with the difiusion coe-cient ai > 0. The right-hand sides of the other N ¡ k equations contain only nonlinear interaction functions ¡hj(u;v), j = k + 1;N, with zero difiusion. Here u = (u1;:::;uk) and v = (vk+1;:::;vN) are unknown concen- tration vectors. Under appropriate assumptions on the interaction functions f(¢) and h(¢), we construct the trajectory attractor A0 of this reaction-difiusion system. We also flnd the trajectory attractors A-, - = (-1;:::;-k), for the analogous reaction-difiusion systems hav- ing the terms -j¢vj ¡ hj(u;v), j = k + 1;N; with small difiusion coe-cients -j > 0 in the last N ¡ k equations. We prove that the trajectory attractors A- converge to A0 (in an

A trajectory attractor \(\mathfrak{A}\) is constructed for the 2D Euler system containing an additional dissipation term −ru, r > 0, with periodic boundary conditions. The corresponding dissipative 2D Navier-Stokes system with the same term −ru and with viscosity v > 0 also has a trajectory attractor, \(\mathfrak{A}_\nu \). Such systems model large-scale geophysical processes in atmosphere and ocean (see [1]). We prove that \(\mathfrak{A}_\nu \) → \(\mathfrak{A}\) as v → 0+ in the corresponding metric space. Moreover, we establish the existence of the minimal limit \(\mathfrak{A}_{min} \) of the trajectory attractors \(\mathfrak{A}_\nu \) as v → 0+. We prove that \(\mathfrak{A}_{min} \) is a connected invariant subset of \(\mathfrak{A}\). The connectedness problem for the trajectory attractor \(\mathfrak{A}\) by itself remains open.

We consider for rho is an element of [0, 1] and epsilon > 0 small, the nonautonomous weakly damped wave equation With a Singularly oscillating external forcepartial derivative(2)(t)u - Delta u + gamma partial derivative(t)u = - f(u) + g(0)(t) + epsilon(-rho) g(1)(t/epsilon),together with the averaged equationpartial derivative(2)(t)u - Delta u + gamma partial derivative(t)u = -f(u) + g(0)(t).Under suitable assumptions oil the nonlinearity and the external force, we prove the uniform (with respect to epsilon) boundedness of the attractors A(epsilon) in the weak energy space. If rho < 1, we establish the convergence of the attractor A(epsilon) of the first equation to the attractor A(0) ofthe second one, as epsilon -> 0(+). On the other hand, if rho = 1, this convergence may fail. When A(0) is exponential, then the convergence rate of A(epsilon) to A(0) is controlled by M epsilon(eta), for some M >= 0 and some eta = eta(rho) is an element of (0, 1). (c) 2008 Elsevier Masson SAS. All rights reserved.

Vladimir Gilelevich Maz’ya, the prominent mathematician and author of numerous publications and fundamental results in various fields of analysis and mathematical physics, celebrated his 70th birthday on 31 December 2007. V. G. Maz’ya was born in Leningrad in 1937. His father was killed at the front in 1941, and both his grandfathers and grandmothers died during the siege of Leningrad. His mother raised Vladimir all by herself. The two lived together on her meagre salary of an accountant, sharing a nine-square-meter room in a large communal flat. Vladimir finished secondary school with a gold medal, and in his last school years he was a repeated winner of city olympiads in mathematics and physics. In 1955 Maz’ya entered the Faculty of Mathematics and Mechanics of Leningrad State University (LSU). His first papers were published quite early: the first (on the Dirichlet problem for second-order elliptic equations) appeared in Doklady Akad. Nauk SSSR in 1959 when he was a fourth-year student. In the same year he gave two talks at the seminar of V. I. Smirnov, on necessary and sufficient conditions for the validity of integral inequalities of Sobolev type. The results were published in Doklady in 1960. For this work he became the first winner of the Prize for Young Mathematicians, established in 1962 by the Leningrad Mathematical Society. After graduating from the university, Maz’ya obtained a position of junior research fellow at the Research Institute of Mathematics and Mechanics of LSU. In 1961 he organized a mathematical school for high school students at the Faculty of Mathematics and Mechanics and became its first director. He defended his Ph.D. thesis entitled “Classes of sets and embedding theorems for function spaces” in 1962 at Moscow State University. It was based on ideas from his talks at Smirnov’s seminar. In their reviews, the opponents and the external reviewer noted that the level of the work far exceeded the requirements of the Higher Certification Commission for Ph.D. theses, and his work was recognized as outstanding at the thesis defence in the Academic Council of Moscow State University. Maz’ya had no formal research advisor either for his master’s thesis or for his Ph.D. thesis: he himself chose his research topics, and he solved the problems by

General methods for constructing and studying global attractors of nonautonomous evolution partial differential equations are presented. The nonautonomous 2D Navier-Stokes system with time-dependent external force serves as the main example. The Kolmogorov epsilon-entropy and fractal dimension of global attractors are considered for this system and other important equations in mathematical psychics. The convergence of global attractors of nonautotnomous equations with singularly oscillating terms to attractors of the corresponding "limit" equations is also established. Bibliography: 136 titles.

On 26 April 2007 Leonid Romanovich Volevich passed away after a severe illness. He was born in Moscow on 11 July 1934. His father was a well-known neuropathologist, and his mother was a translator and teacher of foreign literature. In 1952 he entered the Faculty of Mechanics and Mathematics of Moscow State University, where in his fourth year he began participating in O. A. Oleinik’s seminar. In 1957 he graduated and became a Ph.D. student of K. I. Babenko in the Division of Applied Mathematics of the Steklov Mathematical Institute. In 1960 he defended his Ph.D. thesis “Local properties of solutions of systems of partial differential equations” and stayed to work in the department led by Babenko at the Institute of Applied Mathematics organized by M. V. Keldysh. There he worked his whole life, in the position of principal researcher from 1997. In 1971 he defended his D.Sc. thesis “Investigations of non-homogeneous pseudodifferential operators (regularity of solutions and the Cauchy problem)”. Volevich was a permanent and active participant in M. I. Vishik’s university seminar from its very beginning. In 1965 the Executive Committee of the Moscow Mathematical Society appointed Oleinik as editor of the journal Trudy Moskovskogo Matematicheskogo Obshchestva (translated as Transactions of the Moscow Mathematical Society) and Volevich as deputy editor. After Oleinik’s death he became the editor of the journal by the decision of the Executive Committee. Fifty-three volumes of the journal were published with his direct assistance. He was also a member of the editorial board of the journal Mathematische Nachrichten. Volevich contributed greatly to the general theory of partial differential equations. The foundations of this theory, which goes back to investigations of the classical mathematicians of the 18th and 19th centuries, were laid by J. Hadamard, S. N. Bernstein, I. G. Petrovskii, J. Leray, and S. L. Sobolev. The theory was later further developed and extended in work of Vishik, O. A. Ladyzhenskaya, L. Nirenberg, Oleinik, L. Hörmander, and others.