In this paper, we examine a one-dimensional system of equations for a discrete gas model (the Godunov-Sultangazin system). The Godunov-Sultangazin system is the Boltzmann kinetic equation for a model one-dimensional gas consisting of three groups of particles. In this model, the momentum is preserved whereas the energy is not. We prove the existence of a unique global solution to the Cauchy problem for a perturbation of the equilibrium state with periodic initial data. For the first time, we find the rate of stabilization to the equilibrium state (exponential stabilization).
In this article, we consider the one--dimensional kinetic system of Carleman equations. The Carleman system is the kinetic Boltzmann equation. This system describes a monatomic rarefied gas consisting of two groups of particles. One particle from the first group, interacting with a particle of the first group, transforms into two particles of the second group. Similarly, two particles of the second group, interacting with themselves, transform into two particles of the first group, respectively. We found traveling wave solutions by using the tanh--function method for nonlinear partial differential system. The results of the work can be useful for mathematical modeling in various fields of science and technology: kinetic theory of gases, gas dynamics, autocatalysis. The obtained exact solutions are new.
Исследуется одномерная система уравнений для дискретной модели газа (система уравнений McKean). Система McKean является кинетическим уравнением Больцмана модельного одномерного газа, состоящего из двух групп частиц. Система проходит тест Пенлеве при определенных условиях на многообразие особенностей. Кроме того, кинетическая система допускает редукцию (автомодельное решение), которая позволяет свести данную систему к системе обыкновенных дифференциальных уравнений, для которой тест Пенлеве выполняется и возможно найти решение.
In this article we discuss the kinetic system of Carleman equations. Research of technological processes for the production of building materials is based on autocatalysis reactions. The Carleman system is a special case of the discrete Boltzmann equation. In many problems of the kinetic theory of gases, gas dynamics, autocatalysis chemistry, and other fields of science and technology, Cauchy problems of hyperbolic equations arise that describe various processes. In the one-dimensional case, the system Carleman describes autocatalysis. A theorem on the stabilization of solutions of the Carleman system is formulated.
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