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Hyperbolic regularizations of conservation laws

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Abstract

One of the problems of the kinetics of nonequilibrium processes is related to the lack of information concerning most of the nonequilibrium variables, namely, those which have no intuitive physical meaning, i.e., cannot be defined from the experiment. Moreover, the number of nonequilibrium variables is so large that a reasonable amount (from the physical point of view) of boundary conditions is insufficient for posing the mixed problem. What do the initial data for the Cauchy problem and the boundary conditions for the mixed problem mean in this case? In fact, we must assume that the initial-boundary data for most of the nonequilibrium variables (the higher-order momenta) are arbitrary! The British physicists Chapman and Enskog conjectured that, for “physically correct” models of continuum mechanics, the influence of the higher-order momenta is “inessential.” There are some postulates of physical correctness, but we do not dwell on them. For us it is of importance to understand what the fact that the influence of the higher-order momenta is “inessential” means from the mathematical point of view. The paper is devoted to this very topic.

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To the memory of Leonid Romanovich Volevich

The research was financially supported by the Russian Foundation for Basic Research (under grant no. 06-01-00095) and by the DFG Project 436 RUS 113/895/0-1.

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Palin, V.V., Radkevich, E.V. Hyperbolic regularizations of conservation laws. Russ. J. Math. Phys. 15, 343–363 (2008). https://doi.org/10.1134/S1061920808030051

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