Some problems of the phenomenological equation of spin combustion

A. A. Grebeneva ; V. A. Lukyanenko

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2022), pp. 7-29

Résumé

The model of spin combustion, generalization of the Ja. B. Zeldovich model with a pseudo-differential operator $\left(-\Delta\right)^{\alpha}$, $0\alpha1$ is investigated. The equation is a singularly perturbed nonlinear parabolic equation of the Van-der-Pol's type. Spin combustion modes were discovered as new non-stationary effects by Zeldovich, who developed the theory of burning condensed systems with solid-phase products. In auto-oscillating combustion, the front of the reaction remains flat and moves at an oscillating rate about the average. In the spin regimes on the surface of the burning sample, there are rotating reactions along the front. The article reviews the basic models of the theory of combustion of condensed systems. The mathematical model of the phenomenological equation of spin combustion was proposed by Zeldovich together with A. P. Aldushin and B. A. Malomed. Spatial non-stationary effects of burning hollow cylinder with radius $R$ were investigated by them. The research in the field of theoretical foundations of mathematical modeling of combustion modes belongs to Ivleva and Merzhanov. Their model consists of the equation of thermal conductivity and kinetic equation. The results of experimental, numerical and analytical studies of the burning surface spins are presented in the works by B. V. Novozhilov. The article considers a model of spin burning on a real axis. The integral representation of the problem by Fourier transformation is constructed. For comparison, a model of gas-free spin combustion on the entire plane and corresponding to it an integral representation of the problem with Neumann conditions are given. In the study of the combustion model on a real axis with periodic conditions on a spatial variable and its spectral problem, the consistency with the combustion model in a circumference, which corresponds to the spin modes of combustion of a thin-walled cylinder, has been established. By using discrete Fourier transform, the problem in the form of a nonlinear integral equation of convolution type is obtained. For bounded areas, the operator methods of mathematical physics have been used to estimate non-linear cubic components. The operator form of the problem of spin combustion is presented, it is proved that its operator is a generator of the holomorphic semigroup. The type of the semigroup can be specified by studying the spectrum of the operator. The local solvability of the problem for bounded areas has been proven. Of interest is the model of spin combustion on the real axis with a delay on the spatial variable or time.

Détail

Keywords: nonlinear equation, nonstationary effects, rotating waves, bifurcation analysis, integral representation.