Abstract:
This article explores a one-dimensional system of equations for the discrete model of a gas (Carleman system of equations). The Carleman system is the Boltzmann kinetic equation of a model one-dimensional gas consisting of two particles. For this model, momentum and energy are not retained. On the example of the Carleman model, the essence of the Boltzmann equation can be clearly seen. It describes a mixture of “competing” processes: relaxation and free movement. We prove the existence of a global solution of the Cauchy problem for the perturbation of the equilibrium state with periodic initial data. For the first time we calculate the stabilization speed to the equilibrium state (exponential stabilization).
Citation:
S. A. Dukhnovskii, “On a speed of solutions stabilization of the Cauchy problem for the Carleman equation with periodic initial data”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 21:1 (2017), 7–41
\Bibitem{Duk17}
\by S.~A.~Dukhnovskii
\paper On a speed of solutions stabilization of the Cauchy problem for the Carleman equation with periodic initial data
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2017
\vol 21
\issue 1
\pages 7--41
\mathnet{http://mi.mathnet.ru/vsgtu1529}
\crossref{https://doi.org/10.14498/vsgtu1529}
\elib{https://elibrary.ru/item.asp?id=29245095}
Linking options:
https://www.mathnet.ru/eng/vsgtu1529
https://www.mathnet.ru/eng/vsgtu/v221/i1/p7
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S. A. Dukhnovskii, “Painlevé Test and a Self-Similar Solution of the Kinetic Model”, J Math Sci, 275:5 (2023), 613
Evgeny Radkevich, Olga Vasil'eva, Georgiy Filippov, Lecture Notes in Civil Engineering, 282, Proceedings of FORM 2022, 2023, 65
S. A. Dukhnovsky, “A self–similar solution and the tanh–function method for the kinetic Carleman system”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2022, no. 1, 99–110
S. A. Dukhnovskii, “Global existence theorems of a solution of the Cauchy problem for systems of the kinetic Carleman and Godunov–Sultangazin equations”, Eurasian Math. J., 12:1 (2021), 97–102
S. A. Dukhnovskii, “On the Rate of Stabilization of Solutions to the Cauchy Problem for the Godunov–Sultangazin System with Periodic Initial Data”, J Math Sci, 259:3 (2021), 349
S. A. Dukhnovskii, “Resheniya sistemy Karlemana cherez razlozhenie Penleve”, Vladikavk. matem. zhurn., 22:4 (2020), 58–67
S. A. Dukhnovskii, “Painlevé test and a self-similar solution of the kinetic model”, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 176 (2020), 91–94
S. A. Dukhnovskii, “O skorosti stabilizatsii reshenii zadachi Koshi dlya sistemy uravnenii Godunova—Sultangazina s periodicheskimi nachalnymi dannymi”, Materialy IV Mezhdunarodnoi nauchnoi konferentsii “Aktualnye problemy
prikladnoi matematiki”. Kabardino-Balkarskaya respublika, Nalchik, Prielbruse, 22–26 maya 2018 g. Chast I, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 165, VINITI RAN, M., 2019, 88–113
S. A. Dukhnovskii, “Asymptotic stability of equilibrium states for Carleman and Godunov–Sultangazin systems of equations”, Moscow University Mathematics Bulletin, 74:6 (2019), 246–248
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