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Dispersion relations for the multivelocity acoustic Peierls equations and some properties of the scalar acoustic Peierls potential. I, II. (Russian, English) Zbl 0992.35075
Sib. Mat. Zh. 42, No. 4, 771-780 (2001); 42, No. 5, 1067-1083 (2001); translation in Sib. Math. J. 42, No. 4, 648-655 (2001); 42, No. 5, 893-906 (2001).
The article is a continuation of the author’s study in [Sib. Math. J. 40, 704-725 (1998; Zbl 0934.35131)] of the mathematical questions of modeling acoustic processes in a homogeneous Maxwellian gas in the framework of kinetic theory. The main aim of the article is to obtain a more exact description for the complete acoustic field near the source of perturbation as compared to classical acoustic models. The present stage includes studying the initial step for constructing a series of available schemes providing more precise modeling of the kinetic acoustic models on the base of the linearized Boltzmann equation in the approximation of the first summator invariants. The program suggested in the above-mentioned article for realizing this stage consists in reducing the approximate equation to an equation (which can be regarded as a multivelocity analog of the Peierls equation) on a configuration space, and then applying methods of potential theory to the study of the initial approximation to a solution to the Cauchy problem for the linearized equation.
In the first part, the author formulates the basic theorem and exposes auxiliary results. In terms of special functions, the symbols of convolution kernels are calculated for the multivelocity acoustic Peierls equations and, moreover, the dispersion relations are presented (in Theorem 1). Absence of 3D-dimensional real sheets is established for the scalar dispersion relation (Theorem 2). The asymptotic of the monochromatic scalar acoustic Peierls potential is calculated at infinity in Theorem 3, and unique solvability is proven of the inverse potential problem in the class of finite distributions.
The aim of the second part is to prove the assertions exposed in Part I.
35Q35 PDEs in connection with fluid mechanics
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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