On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations

The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier\u2013Stokes equations perturbed by various random forces of low dimension

Qualitative features of periodic solutions of KdV

In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first order with strongly oscillating phase factors defined in terms of the KdV frequencies. The second result provides estimates for the approximation of such a solution by trigonometric polynomials of sufficiently large degree

Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation

We suggest a new derivation of a wave kinetic equation for the spectrum of the weakly nonlinear Schr\uf6dinger equation with stochastic forcing. The kinetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size of the nonlinearity and of the forcing to zero, while the time is correspondingly rescaled; then, the size of the box is sent to infinity (with a suitable rescaling of the solution). We report here the results of the first limiting procedure, analysed with full rigour in Kuksin and Maiocchi (0000), and show how the second limit leads to a kinetic equation for the spectrum, if some further hypotheses (commonly employed in the weak turbulence theory) are accepted. Finally we show how to derive from these equations the Kolmogorov-Zakharov spectra

Quasi-periodic solutions of completely resonant forced wave equations

We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.Comment: 25 pages, 1 figur

Legal regulation of mortgage lending: the experience of Russia and foreign countries

In this paper, authors carry out an analysis of regulation of legal relationships in the sphere of mortgage lending, based on studies of certain international legal acts, monographic publications and scientific publications. They analyze, in particular, the German mortgage lending system, elements of which are used in the development of the Russian mortgage system. The common features and peculiarities of mortgage lending in Germany and Russia are note

Organization of advocacy in various legal systems: comparative analysis

This article observes organization of providing legal services in different countries, such as the US, the UK, Germany, France, China and Russia. The authors describe the procedure of admitting to the legal profession and the sphere of legal activit

Peculiarities of national interests institutionalization in the North American tradition: history and modernity

This article is devoted to the analysis of characteristics of national interests’ institutionalization in the North American tradition, namely the evolution of their legal consolidation and the practice of implementation in modern condition

Passive tracer in a flow corresponding to a two dimensional stochastic Navier Stokes equations

In this paper we prove the law of large numbers and central limit theorem for trajectories of a particle carried by a two dimensional Eulerian velocity field. The field is given by a solution of a stochastic Navier--Stokes system with a non-degenerate noise. The spectral gap property, with respect to Wasserstein metric, for such a system has been shown in [9]. In the present paper we show that a similar property holds for the environment process corresponding to the Lagrangian observations of the velocity. In consequence we conclude the law of large numbers and the central limit theorem for the tracer. The proof of the central limit theorem relies on the martingale approximation of the trajectory process

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ[much less-than]1,K [much greater-than] 1, s > 1, we construct smooth initial data u 0 with ||u0||Hs , so that the corresponding time evolution u satisfies u(T)Hs[greater than]K at some time T. This growth occurs despite the Hamiltonian’s bound on ||u(t)||H1 and despite the conservation of the quantity ||u(t)||L2. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems

Radiation-induced damage and evolution of defects in Mo

The formation of defects in bcc Mo lattice as a result of 50-keV Xe bombardment is studied via atomistic simulation with an interatomic potential developed using the force-matching ab initio based approach. The defect evolution in the cascade is described. Diffusion and interaction of interstitials and vacancies are analyzed. Only small interstitial atom clusters form directly in the cascade. Larger clusters grow only via aggregation at temperatures up to 2000 K. Stable forms of clusters demonstrate one-dimensional diffusion with a very high diffusion coefficient and escape quickly to the open surface. Point vacancies have much lower diffusivity and do not aggregate. The possibility of a large prismatic vacancy loop formation near the impact surface as a result of fast recrystallization is revealed. The mobility of the vacancy dislocation loop segments is high, however, the motion of the entire loops is strongly hindered by neighbor point defects. This paper explains the existence of the large prismatic vacancy loops and the absence of the interstitial loops in the recent experiments with ion irradiation of Mo foils