On the completeness of eigenfunctions system in solving problems for polyatomic gases

V. V. Sushkov

Taurida Journal of Computer Science Theory and Mathematics, no. 1 (2024), pp. 102-115

Résumé

The basic equation in considering problems of rarefied gas behavior is the well-known Boltzmann equation, which comprises partial derivatives of the distribution function with respect to coordinates and time. Hence, to obtain a solution, it is necessary to additionally boundary conditions, that is, to formulate an initial-boundary value problem. The complex structure of the collision integral of the Boltzmann equation has led to a focus on using collision integral models (therefore, this equation is also called a model). The most famous of these is called the Bhatnagar-Gross-Krook (BGK) model, although Landau proposed it approximately at the same time. The main advantage of using the BGK operator is the possibility of reducing any problem to a linear system of integro-differential equations, which subsequently implies either reducing the problem to a one-dimensional case or developing a method for solving vector-matrix integro-differential equations. The developed method can not only serve as a tool for solving specific problems in mathematical physics but also construct a more general theory of boundary value problem solutions for an entire class of integro-differential equations. Let's illustrate the methodology with an example of an equation to which the consideration of boundary value problems for the model Boltzmann equation with the BGK collision integral reduces for polyatomic gases. it is shown that the eigenvalues of the characteristic equation can be divided into eigenvalues of the continuous spectrum (all finite points on the real axis) and the discrete spectrum (the point at infinity) and have constructed the corresponding eigenfunctions. At the same time, the eigenfunctions of the continuous spectrum are vector distributions, while the eigenfunctions of the discrete spectrum are vector functions with polynomial components. The central property characterizing the constructed set of eigenfunctions, can be considered their completeness, understood as the ability to expand any holder vector function into the eigenfunctions of the equation. This article considers the properties of the spectrum of the characteristic equation arising in the solution of the BGK equation for polyatomic gases. It is proved that it consists of the point at infinity, which is a quadruple point of the discrete spectrum, and points on the real axis, forming the continuous spectrum. These correspond to the eigenfunctions of the discrete spectrum, which are polynomial vectors, and the eigenfunctions of the continuous spectrum, which are generalized functions. The correctness of using eigenfunctions of this type is demonstrated. The completeness of the set of eigenfunctions in the class of functions satisfying the Holder condition on the closed real half-axis is proven.

Détail

Keywords: BGK equation, polyatomic gases, eigenfunctions, discrete spectrum, continuous spectrum, completeness theorem.