Local Equilibrium of the Carleman Equation

The Bloch Principle for L2(R) Stabilization of Solutions to the Cauchy Problem for the Carleman Equation

“…The first step of stabilization of the localized solution of the kinetic Carleman equation to the local equilibrium state consists in distributing (up to the equilibrium energy state) the energy between solitons (as in the case of periodic initial data [2]). …”

“…Section 3), in the nonsingular zone N Z == {(ζ, k) | ζ ∈ Z, ζ = 0, −1, k ∈ [0, 2π]}, the zeros of σ ζ,k (p) lie in the left half-plane Re p −εμ for some μ = O(1) > 0. As in the case of periodic initial data [2,3], a priori estimate holds for the truncated system (5.1) in the weighted space…”

“…For example, for w(x, t) = ζ∈Z w ζ (t)e iζx the linearized operator on the Fourier coefficients (cf. [2,3]) is expressed by The solution has the form…”

The investigation of evolution of the harmonic perturbation the stationary solution of the boundary value problem for a system of the Carleman equations

“…However the system has the main properties more complete system. It explains the fact that the investigation of the Carleman system is very interested now [1][2][3][4][5][6]. One of the important properties of the Carleman system is the stability of the stationary solutions.…”

Numerical investigation of some stationary solutions of the Carleman system

“…This system can be used for mathematical modelling problems of the kinetic theory of gasses, chemistry and in particular, for research of building materials. So, the investigation of the Carleman system is very interested now [1][2][3][4][5]. One of the important properties of the initial boundary problem for Carleman system is the stability of the stationary solutions.…”