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September  2013, 6(3): 533-543. doi: 10.3934/krm.2013.6.533

## On a regularization of the magnetic gas dynamics system of equations

 1 CEA, DAM, DIF, F-91297 Arpajon 2 Department of Mathematics at Faculty of Economics Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow

Received  August 2012 Revised  January 2013 Published  May 2013

A brief derivation of a specific regularization for the magnetic gas dynamic system of equations is given in the case of general equations of gas state (in presence of a body force and a heat source). The entropy balance equation in two forms is also derived for the system. For a constant magnetic regularization parameter and under a standard condition on the heat source, we show that the entropy production rate is nonnegative.
Citation: Bernard Ducomet, Alexander Zlotnik. On a regularization of the magnetic gas dynamics system of equations. Kinetic & Related Models, 2013, 6 (3) : 533-543. doi: 10.3934/krm.2013.6.533
##### References:
 [1] B. N. Chetverushkin, "Kinetic Schemes and Quasi-Gas Dynamic System of Equations,", CIMNE, (2008).   Google Scholar [2] T. G. Elizarova., "Quasi-Gas Dynamic Equations,", Computational Fluid and Solid Mechanics, (2009).  doi: 10.1007/978-3-642-00292-2.  Google Scholar [3] T. G. Elizarova, Time averaging as an approximate technique for constructing quasi-gas-dynamic and quasi-hydrodynamic equations,, Comp. Math. Math. Phys., 51 (2011), 1973.  doi: 10.1134/S0965542511110078.  Google Scholar [4] T. G. Elizarova and S. D. Ustjugov, Quasi-gas dynamics algorithm to solve equations of magnetohydrodynamics. One-dimensional case,, (in Russian) Keldysh Inst. Appl. Math. Moscow, (2011).   Google Scholar [5] T. G. Elizarova and S. D. Ustjugov, Quasi-gas dynamics algorithm to solve equations of magnetohydrodynamics. Multidimensional case,, (in Russian) Keldysh Inst. Appl. Math. Moscow, (2011).   Google Scholar [6] A. C. Eringen and G. A. Maugin, "Electrodynamics of Continua II: Fluids and Complex Media,", Springer-Verlag, (1990).   Google Scholar [7] E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids,", Advances in Mathematical Fluid Mechanics, (2009).  doi: 10.1007/978-3-7643-8843-0.  Google Scholar [8] I. A. Kvasnikov, "Thermodynamics and Statistical Physics, Vol. 1: Theory of Equilibrium Systems and Thermodynamics,", (in Russian) Editorial URSS, (2002).   Google Scholar [9] L. D. Landau and E. M. Lifshitz, "Electrodynamics of Continuous Media,", Pergamon Press, (1960).   Google Scholar [10] Yu. V. Sheretov, "Continuum Dynamics Under Spatiotemporal Averaging,", (in Russian) RKhD, (2009).   Google Scholar [11] A. A. Zlotnik, Classification of some modifications of the Euler system of equations,, Doklady Math., 73 (2006), 302.  doi: 10.1134/S1064562406020396.  Google Scholar [12] A. A. Zlotnik, Energy equalities and estimates for barotropic quasi-gas-dynamic and quasi-hydrodynamic systems of equations,, Comp. Maths. Math. Phys., 50 (2010), 310.  doi: 10.1134/S0965542510020120.  Google Scholar [13] A. A. Zlotnik, A quasi-gas-dynamic system of equations with general state equations,, Doklady Math., 81 (2010), 312.  doi: 10.1134/S1064562410020419.  Google Scholar [14] A. A. Zlotnik, Linearized stability of equilibrium solutions to the quasi-gas-dynamic system of equations,, Doklady Math., 82 (2010), 811.  doi: 10.1134/S1064562410050352.  Google Scholar [15] A. A. Zlotnik, On the quasi-gas-dynamic system of equations with general equations of state and a heat source,, (in Russian) Math. Modelling, 22 (2010), 53.   Google Scholar [16] A. A. Zlotnik, On construction of quasi-gas-dynamic systems of equations and the barotropic system with the potential body force,, (in Russian) Math. Modelling, 24 (2012), 65.   Google Scholar [17] A. A. Zlotnik and B. N. Chetverushkin, Parabolicity of the quasi-gas-dynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them,, Comp. Maths. Math. Phys., 48 (2008), 420.  doi: 10.1134/S0965542508030081.  Google Scholar [18] A. Zlotnik and V. Gavrilin, On quasi-gas-dynamic system of equations with general equations of state and its application,, Math. Modelling Anal., 16 (2011), 509.  doi: 10.3846/13926292.2011.627382.  Google Scholar

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##### References:
 [1] B. N. Chetverushkin, "Kinetic Schemes and Quasi-Gas Dynamic System of Equations,", CIMNE, (2008).   Google Scholar [2] T. G. Elizarova., "Quasi-Gas Dynamic Equations,", Computational Fluid and Solid Mechanics, (2009).  doi: 10.1007/978-3-642-00292-2.  Google Scholar [3] T. G. Elizarova, Time averaging as an approximate technique for constructing quasi-gas-dynamic and quasi-hydrodynamic equations,, Comp. Math. Math. Phys., 51 (2011), 1973.  doi: 10.1134/S0965542511110078.  Google Scholar [4] T. G. Elizarova and S. D. Ustjugov, Quasi-gas dynamics algorithm to solve equations of magnetohydrodynamics. One-dimensional case,, (in Russian) Keldysh Inst. Appl. Math. Moscow, (2011).   Google Scholar [5] T. G. Elizarova and S. D. Ustjugov, Quasi-gas dynamics algorithm to solve equations of magnetohydrodynamics. Multidimensional case,, (in Russian) Keldysh Inst. Appl. Math. Moscow, (2011).   Google Scholar [6] A. C. Eringen and G. A. Maugin, "Electrodynamics of Continua II: Fluids and Complex Media,", Springer-Verlag, (1990).   Google Scholar [7] E. Feireisl and A. Novotný, "Singular Limits in Thermodynamics of Viscous Fluids,", Advances in Mathematical Fluid Mechanics, (2009).  doi: 10.1007/978-3-7643-8843-0.  Google Scholar [8] I. A. Kvasnikov, "Thermodynamics and Statistical Physics, Vol. 1: Theory of Equilibrium Systems and Thermodynamics,", (in Russian) Editorial URSS, (2002).   Google Scholar [9] L. D. Landau and E. M. Lifshitz, "Electrodynamics of Continuous Media,", Pergamon Press, (1960).   Google Scholar [10] Yu. V. Sheretov, "Continuum Dynamics Under Spatiotemporal Averaging,", (in Russian) RKhD, (2009).   Google Scholar [11] A. A. Zlotnik, Classification of some modifications of the Euler system of equations,, Doklady Math., 73 (2006), 302.  doi: 10.1134/S1064562406020396.  Google Scholar [12] A. A. Zlotnik, Energy equalities and estimates for barotropic quasi-gas-dynamic and quasi-hydrodynamic systems of equations,, Comp. Maths. Math. Phys., 50 (2010), 310.  doi: 10.1134/S0965542510020120.  Google Scholar [13] A. A. Zlotnik, A quasi-gas-dynamic system of equations with general state equations,, Doklady Math., 81 (2010), 312.  doi: 10.1134/S1064562410020419.  Google Scholar [14] A. A. Zlotnik, Linearized stability of equilibrium solutions to the quasi-gas-dynamic system of equations,, Doklady Math., 82 (2010), 811.  doi: 10.1134/S1064562410050352.  Google Scholar [15] A. A. Zlotnik, On the quasi-gas-dynamic system of equations with general equations of state and a heat source,, (in Russian) Math. Modelling, 22 (2010), 53.   Google Scholar [16] A. A. Zlotnik, On construction of quasi-gas-dynamic systems of equations and the barotropic system with the potential body force,, (in Russian) Math. Modelling, 24 (2012), 65.   Google Scholar [17] A. A. Zlotnik and B. N. Chetverushkin, Parabolicity of the quasi-gas-dynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them,, Comp. Maths. Math. Phys., 48 (2008), 420.  doi: 10.1134/S0965542508030081.  Google Scholar [18] A. Zlotnik and V. Gavrilin, On quasi-gas-dynamic system of equations with general equations of state and its application,, Math. Modelling Anal., 16 (2011), 509.  doi: 10.3846/13926292.2011.627382.  Google Scholar
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