Russian Journal of Mathematical Physics最新文献

This paper studies the uniqueness of a weak solution of the incompressible Navier–Stokes Equations in the 3-dimensional case. Here the investigation is provided by using two different approaches. The first (the main) result is obtained for given functions possessing a certain smoothness, using a new approach. The other result works without additional conditions but is, in some sense, a “local” result, investigated by another approach. In addition, here the solvability and uniqueness of weak solutions to the auxiliary problems derived from the main problem are investigated.

DOI 10.1134/S1061920824030154

We study asymptotic spectral properties of the Bochner–Schrödinger operator (H_{p}=frac 1pDelta^{L^potimes E}+V) on high tensor powers of a Hermitian line bundle (L) twisted by a Hermitian vector bundle (E) on a Riemannian manifold (X) of bounded geometry under the assumption that the curvature form of (L) is nondegenerate. At an arbitrary point (x_0) of (X), the operator (H_p) can be approximated by a model operator (mathcal H^{(x_0)}), which is a Schrödinger operator with constant magnetic field. For large (p), the spectrum of (H_p) asymptotically coincides, up to order (p^{-1/4}), with the union of the spectra of the model operators (mathcal H^{(x_0)}) over (X). We show that, if the union of the spectra of (mathcal H^{(x_0)}) over the complement of a compact subset of (X) has a gap, then the spectrum of (H_{p}) in the gap is discrete, and the corresponding eigensections decay exponentially away from a compact subset.

DOI 10.1134/S1061920824030099

We obtain a uniform decomposition into Casimir eigenspaces (most of which are irreducible) of the fourth power of the adjoint representation (mathfrak{g}^{otimes 4}) for all simple Lie algebras. We present universal, in Vogel’s sense, formulas for the dimensions and split Casimir operator’s eigenvalues of all terms in this decomposition. We assume that a similar uniform decomposition into Casimir eigenspaces with universal dimension formulas exists for an arbitrary power of the adjoint representations.

DOI 10.1134/S1061920824030038

Some necessary and sufficient conditions for the strong continuity of representations of topological groups in reflexive Fréchet spaces are obtained.

DOI 10.1134/S106192082403018X

This paper studies time-periodic solutions of singularly perturbed Tikhonov systems of reaction–diffusion–advection equations with nonlinearities that include the square of the gradient of the unknown function (KPZ nonlinearities). The boundary layer asymptotics of solutions are constructed for Neumann and Dirichlet boundary conditions. The study considers both the case of quasimonotone sources and systems without the quasimonotonicity condition. The asymptotic method of differential inequalities is used to prove theorems on the existence of solutions and their Lyapunov asymptotic stability.

DOI 10.1134/S1061920824030129

We construct an appropriate restriction of the 2-dimensional Laplace operator that has compact preresolvent though the resolvent of its Friedrichs extension is not compact and, moreover, its spectrum is absolutely continuous. This result solves the Birman problem.

DOI 10.1134/S1061920824030117

We consider a difference operator acting in (l^2(mathbb Z)) by the formula (( mathcal{A} psi)_n=psi_{n+1}+psi_{n-1}+lambda e^{-2pi mathrm{i} (theta+omega n)} psi_n), (nin mathbb{Z}), where (omegain(0,1)), (lambda>0), and (thetain [0,1]) are parameters. This operator was introduced by P. Sarnak in 1982. For (omeganotin mathbb Q), the operator ( mathcal{A} ) is quasiperiodic. Previously, within the framework of a renormalization approach (monodromization method), we described the location of the spectrum of this operator. In the present work, we first establish the existence of the point spectrum for different values of parameters, and then study the eigenfunctions. To do so, using ideas of the renormalization approach, we study the difference operator on the circle obtained from the original one by the Fourier transform. This allows us, first, to obtain a new type condition guaranteeing the existence of point spectrum and, second, to describe in detail a multi-scale self-similar structure of the Fourier transforms of the eigenfunctions.

DOI 10.1134/S106192082403004X

In the paper, we consider a linear second order elliptic problem with drift in a domain perforated along the boundary. Setting homogeneous Dirichlet condition on the boundary of the cavities and homogeneous Neumann condition on the outer boundary of the domain, we prove the higher integrability of the gradient of the solution to the problem (the Boyarsky–Meyers estimate).

DOI 10.1134/S1061920824030051

A new practice of applying V.P. Maslov’s theoretical results has been implemented for analyzing fluid flow regimes that arise during their numerical modelling. In this paper, using the example of a Kolmogorov-type flow for two-dimensional motion of a viscous fluid, a rank analysis of the vorticity field and its frequency of occurrence is proposed. A similar analysis has been performed for the problem of forming columnar structures in the spatial case. It has been shown that, for the turbulent, vortex, and laminar fluid motion regimes, the rank distributions exhibit characteristics that can be used to classify the flow types.

DOI 10.1134/S1061920824030075