Russian Journal of Mathematical Physics

In the present note, we show that, for some special classes of external flows, the short-wave asymptotic behavior of the solution of linearized equations of relativistic gas dynamics can be described rather explicitly.

We consider a d-dimensional harmonic crystal, d ⩾ 1, and study the Cauchy problem with random initial data. The distribution μ_t of the solution at time t ∈ ℝ is studied. We prove the convergence of correlation functions of the measures μ_t to a limit for large times. The explicit formulas for the limiting correlation functions and for the energy current density (in the mean) are obtained in terms of the initial covariance. Furthermore, we prove the weak convergence of μ_t to a limit measure as t → ∞. We apply these results to the case when initially some infinite “parts” of the crystal have Gibbs distributions with different temperatures. In particular, we find stationary states in which there is a constant nonzero energy current flowing through the crystal. We also study the initial boundary value problem for the harmonic crystal in the half-space with zero boundary condition and obtain similar results.

For two discrete metric spaces X and Y, we consider metrics on X ⊔ Y compatible with the metrics on X and Y. As morphisms from X to Y, we consider Roe bimodules, i.e., the norm closures of bounded finite propagation operators from l²(X) to l²(Y). We study the corresponding category \(\mathcal{M}\), which is also a 2-category. We show that almost isometries determine morphisms in \(\mathcal{M}\). We also consider the case Y = X, when there is a richer algebraic structure on the set of morphisms of \(\mathcal{M}\): it is a partially ordered semigroup with the neutral element, with involution, and with a lot of idempotents. We also give a condition when a morphism is a C*-algebra.

We consider Sobolev problems (problems for an elliptic operator on a closed manifold with conditions on a closed submanifold) for the case in which these conditions are of nonlocal nature and include weighted spherical means of the unknown function over spheres of a given radius. For such problems, we establish a criterion for the Fredholm property and, in some special cases, obtain index formulas.

The stationary phase method is applied to investigate the asymptotic behavior at infinity of the Hankel transform of order zero.

A method for computing the solution to the boundary value problem for the symmetric Thomas-Fermi model of a heavy atom or ion at zero absolute temperature.