Abstract
Criteria (necessary and sufficient conditions) for the Petrovskii parabolicity of the quasi-gasdynamic system of equations with an improved description of heat conduction are derived. A modified quasi-gasdynamic system containing second derivatives with respect to both spatial and time variables is proposed. Necessary and sufficient conditions for its hyperbolicity are deduced. For both systems, the stability of small perturbations against a constant background is analyzed and estimates that are uniform on an infinite time interval are given for relative perturbations in the Cauchy problem and the initial-boundary value problem for the corresponding linearized systems. Similar results are also established in the barotropic case with the general equation of state p = p(ρ).
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References
B. N. Chetveushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations (MAKS, Moscow, 2004) [in Russian].
S. Succi, The Lattice Boltzmann Equation in Fluid Dynamics and Beyond (Clarendon, Oxford, 2001).
M. Tsutakara, N. Takado, and N. Kataoka, Lattice Gas and Lattice Boltzmann Methods — New Methods of Computational Fluid Dynamics (Corona Publishing, Tokyo, 1999).
B. N. Chetverushkin and N. Yu. Romanyukha, “Kinetic and Lattice Boltzmann Schemes,” in Parallel Computational Fluid Dynamics, Multidisciplinary Applications (Elsevier, Amsterdam, 2005), pp. 257–262.
E. T. Elizarova, Mathematical Models and Numerical Methods in Fluid Dynamics (Mosk. Gos. Univ., Moscow, 2005) [in Russian].
Yu. V. Sheretov, Mathematical Modeling of Fluid Flows Based on Quasi-Hydrodynamic and Quasi-Gasdynamic Equations (Tversk. Gos. Univ., Tver, 2000) [in Russian].
A. A. Zlotnik, “Classification of Some Modifications of the Euler System of Equations,” Dokl. Akad. Nauk 407(6), 747–751 (2006) [Dokl. Math. 73, 302–306 (2006)].
B. V. Alekseev, “Physical Principles of the Generalized Boltzmann Kinetic Theory of Gases,” Usp. Fiz. Nauk 170(6), 649–679 (2000) [Phys.-Usp. 43, 601–629 (2000)].
L. V. Dorodnitsyn, L. V. Dorodnitsyn, “On the Stability of Small Oscillations in a Quasi-Gasdynamic System,” Zh. Vychisl. Mat. Mat. Fiz. 44(7), 1299–1305 (2004) [Comput. Math. Math. Phys. 44, 1231–1237 (2004)].
A. A. Zlotnik and I. A. Zlotnik, “Stability Criterion for Small Perturbations for a Quasi-Gasdynamic System of Equations,” Zh. Vychisl. Mat. Mat. Fiz. 46(2), 262–269 (2006) [Comput. Math. Math. Phys. 46, 251–257 (2006)].
A. A. Zlotnik, “On the Parabolicity of the Quasi-Gasdynamic System of Equations and Stability of Small Perturbations for It” (in press).
A. A. Vlasov, Statistical Distribution Functions (Nauka, Moscow, 1966) [in Russian].
I. G. Petrovskii, Selected Works: Systems of Partial Differential Equations and Algebraic Geometry (Nauka, Moscow, 1986) [in Russian].
I. G. Petrovskii, Partial Differential Equations (Fizmatgiz, Moscow, 1961; Saunders, Philadelphia, Pa., 1967).
S. D. Eidel’man, Parabolic Systems (Nauka, Moscow, 1964) [in Russian].
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, R.I., 1968).
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer-Verlag, New York, 1985).
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Original Russian Text © A.A. Zlotnik, B.N. Chetverushkin, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 3, pp. 445–472.
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Zlotnik, A.A., Chetverushkin, B.N. Parabolicity of the quasi-gasdynamic system of equations, its hyperbolic second-order modification, and the stability of small perturbations for them. Comput. Math. and Math. Phys. 48, 420–446 (2008). https://doi.org/10.1134/S0965542508030081
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DOI: https://doi.org/10.1134/S0965542508030081