This work is a review with proofs of a group of results on the stochastic Burgers equation with small viscosity, obtained during the last two decades. These results jointly show that the equation makes a surprisingly good model of hydrodynamical turbulence. The model provides natural and rigorously justified analogies of a number of key predictions of the theory of turbulence, including the main assertions of the Kolmogorov approach to turbulence, known as the K41 theory.

УСПЕХИ МАТЕМАТИЧЕСКИХ НАУК Марко Иосифович Вишик 23 июня 2012 г. ушел из жизни Марко Иосифович Вишик, выдающийся математик, крупнейший специалист и один из основоположников теории дифференциальных уравнений с частными производными.М. И. Вишик родился 19 октября 1921 г. во Львове.В 8 лет потерял отца.Он очень нежно вспоминал мать, которая смогла дать ему обра

We prove that for any given homotopic C-1-maps u, v : G -> M in a nontrivial homotopy class from a metric graph into a closed manifold of negative sectional curvature, the distance between u and v can be bounded by 3 (length(u) + length(v)) + C(kappa, rho/20), where rho > 0 is a lower bound of the injectivity radius and -kappa < 0 an upper bound for the sectional curvature of M. The constant C(kappa, epsilon) is given byC(kappa, epsilon) = 8 sh(kappa)(-1)(1) + 8 sh(kappa)(-1) (1/sh(kappa)(epsilon))with sh(kappa)(t) = sinh(root kappa t). Various applications are given.

We consider random perturbations of 2D Navier–Stokes equations. Under some natural conditions on random forces, we study asymptotic properties of solutions and stationary measures. 0 Introduction We consider the 2D Navier–Stokes (NS) system perturbed by a random force: u− ν∆u+ (u,∇)u+∇p = η(t, x), div u = 0. (0.1) The space variable x belongs either to a bounded domain, and then the Dirichlet boundary condition is imposed, or to the two-dimensional torus T = R/2πZ, and then we assume that ∫ T2 u dx ≡ ∫ T2 η dx ≡ 0. Denoting by H the corresponding L-space of divergence-free vector fields with the natural norm | · | and by Π the orthogonal projections to H , we write (0.1) as a random system in H : u(t) + νLu(t) +B(u(t), u(t)) = η(t). (0.2) Here L = −Π∆, B(u, u) = Π(u,∇)u, and η(t) = Πη(t, ·) (see [CF88]). We denote by {ej} the Hilbert basis in H formed by the eigenfunctions of the operator L with eigenvalues 0 0: η = ηe(t, x) = √ e ∞ ∑ k=−∞ ηk(x)δ(t − ke), ηk(x) = ∞ ∑ j=1 bjξjkej(x), (0.3) Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK; Steklov Institute of Mathematics, 8 Gubkina St., 117966 Moscow, Russia; E-mail: S.B.Kuksin@ma.hw.ac.uk. Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK; E-mail: A.Shirikyan@ma.hw.ac.uk.

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We study quasilinear Hamiltonian partial differential equations with one-dimensional space variable in a segment of real line. We assume that the equation has a family of n-frequency time-quasiperiodic solutions, depending on an n-dimensional amplitude vector, and prove that most of these solutions persist under Hamiltonian perturbations of the equation by a nonlinear term which contains less derivatives than the linear part of the unperturbed equation. The result is similar to one proved in (K) for perturbations which contain no derivatives.

*This paper was written while both authors were guests of the Forschungsinstitut fur Mathematik at the ETH Zurich, and we thank the institute for its hospitality, pleasant working atmosphere and helpful staff. In particular, we thank Jiirgen Moser for numerous stimulating discussions on the subject. We also benefitted from a remark by Sigurd Angenent concerning the analyticity of the solutions. The second author also thanks the Deutsche Forschungsgemeinschaft for their financial support through a Heisenberg grant.

We study the Hilbert manifold formed by all pairs (almost complex structure on the tents T-4, pseudoholomorphic 2-torus in T-4) (the homotopy type of the two-tori is fixed). We prove that for a typical structure the number of the pseudoholomorphic two-tori is finite and even. The situations when these numbers are zero and non-zero both happen for open non-empty sets of almost complex structures.

We study partial differential equations of hamiltonian form and treat them as infinite-dimensional hamiltonian systems in a functional phase-space ofx-dependent functions. In this phase space we construct an invariant symplectic capacity and prove a version of Gromov's (non)squeezing theorem. We give an interpretation of the theorem in terms of the “energy transition to high frequencies” problem.

We prove that linear Schrödinger operators -tΔ + q on a torus or ona bounded smooth domain in Rd considered with Dirichlet boundary conditions, have a strongly nonresonant spectrum for any potential q of generic type (generic in the sense of Kolmogorov measure). As a consequence, a Krylov-Bogolubov averaging theorem holds for nonlinear perturbations of the corresponding Schrödinger evolution equations.

We treat the nonlinear Klein-Gordon (NKG) equation as the Sine-Gordon (SG) equation, perturbed by a higher order term. It is proved that most small-amplitude finite-gap solutions of the SG equation, which satisfy either Dirichlet or Neumann boundary conditions, persist in the NKG equation and jointly form partial central manifolds, which are ''Lipschitz manifolds with holes''. Our proof is based on an analysis of the finite-gap solutions of the boundary problems for SG equation by means of the Schottky uniformization approach, and an application of an infinite-dimensional KAM-theory.