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Macrodynamic equations and description of large scale system. (English) Zbl 0718.93002
The aggregated models for determining a small number of macroquantities, characterizing the system as a whole, are often used in order to describe systems, consisting of a large number of elements. Analysis of the aggregated dynamical models from physics, chemistry, biology and economics showed that many of them are described by the equations of macrodynamics. The class of macrodynamic equations was introduced and studied by the author and L. I. Rozonoer [e.g., J.Franklin, Inst. 318, 283-314, 315-347 (1984; Zbl 0554.93004, Zbl 0554.93005, res.)], the author [U.S.S.R. Comput. Math. Math. Phys. 25, No.4, 125-133 (1986); translation from Zh. Vychisl. Mat.Mat. Fiz. 25, No.4, 521-534 (1985)] and elsewhere.
In this work main definitions, connected with macrodynamic equations, are given. As example the equations of chemical kinetics, the equations of gase dynamics and the Boltzmann equation are considered. The main results on the properties of macrodynamic equations, obtained earlier by L. I. Rozonoer [“Irreversible thermodynamics far from equilibrium”, in: I. Lamprecht and A. I. Zotin (eds.), “Thermodynamics and kinetics of biological processes” (Berlin 1983), pp. 219-238], are discussed. In particular, the questions of existence, uniqueness and stability of stationary states, special variational principle, singularly perturbed macrodynamics equations are considered.
93A15 Large-scale systems
93C20 Control/observation systems governed by partial differential equations