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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References
L. R. Volevich and E. V. Radkevich, “Uniform Estimates for Solutions of the Cauchy Problem for Hyperbolic Equations with a Small Parameter Multiplying the Higher Derivatives,” Differ. Uravn. 39(4), 486–499 (2003) [Differ. Equ. 39 (4), 521–535 (2003)].
L. R. Volevich and E. V. Radkevich, “Stable Pencils of Hyperbolic Polynomials, and the Cauchy Problem for Hyperbolic Equations with a Small Parameter Multiplying the Highest Derivatives,” Tr. Mosk. Mat. Obs. 65, 69–113 (2004) [Trans. Moscow Math. Soc. 2004, 63–104].
S. Chapman and T. Cowling, Mathematical Theory on Non-Uniform Gases, 3rd. ed. (Cambridge Univ. Press, 1970).
Gui-Qiang Chen, C. D. Levermore, and Tai-Ping Luui, “Hyperbolic Conservation Laws with Stiff Relaxation Terms and Entropy,” Comm. Pure Appl. Math. XLVII, 787–830 (1994).
H. Struchtrup and W. Weiss, “Temperature Jump and Velocity Slip in the Moment Method,” Contin. Mech. Thermodyn. 12, 1–18 (2000).
E. V. Radkevich, “Irreducible Chapman-Enskog Projections and Navier-Stokes Approximations,” in Instability in Models Connected with Fluid Flows. II, Ed. by C. Bardos and A. Fursikov, Int. Math. Ser. (N.Y.) 6 (Springer, New York, 2007), pp. 85–151.
E. V. Radkevich, “Kinetic Equation and Chapman-Enskog Projection Problems,” in Differ. Uravn. i Din. Sist., Tr. Mat. Inst. Steklova 250, 219–225 (2005) [Proc. Steklov Inst. Math. 250 (3), 204–210 (2005)].
E. V. Radkevich, Mathematical Problems of Nonequilibrium Processes, White Series 4 (Izd-vo Tamara Rozhkovskaya, Novosibirsk, 2007).
V. V. Palin and E. V. Radkevich, “The Navier-Stokes Approximation and Chapman-Enskog Projection Problems for Kinetic Equations,” Tr. Semin. im. I.G. Petrovskogo 25, 184–225 (2005) [J. Math. Sci. (N. Y.) 135 (1), 2721–2748 (2006)].
V. V. Palin, “Solvability of Quadratic Matrix Equations,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. (to appear).
R. Peierls, “Zur kinetischen Theorie der Wärmeleitung in Kristallen,” Ann. Phys. 395, Folge 5 (3), 1055–1101 (1929).
W. Dreyer and H. Struchtrup, “Heat Pulse Experiments Revisted,” Contin. Mech. Thermodyn. 5, 3–50 (1993).
I. Müller and T. Ruggeri, Extended Thermodynamics (Springer, New York, 1993).
C. D. Levermore, “Moment Closure Hierarchies for Kinetic Theories,” J. Statist. Phys. 83, 1021–1065 (1996).
J. Goodman and P. D. Lax, “On Dispersive Difference Schemes. I,” Comm. Pure Appl. Math. XLI, 591–613 (1988).
I. Edelman and S. A. Shapiro, “An Analytical Approach to the Description of Fluid Injection Induced Microseismicity in Porous Rock,” Doklady Earth Sciences 399(8), 1108–1112 (2004).
N. A. Zhura and A. N. Oraevskii, “The Cauchy Problem for a Hyperbolic System with Constant Coefficients,” Dokl. Akad. Nauk 396(5), 590–594 (2004).
N. A. Zhura, “First-Order Hyperbolic Systems and Quantum Mechanics,” Mat. Zametki (to appear).
A. N. Gorban and I. V. Karlin, Invariant Manifolds for Physical and Chemical Kinetics (Springer, Berlin, 2005).
A. V. Bobylev, “On the Chapman-Enskog and Grad Methods for Solving the Boltzmann Equation,” Dokl. Akad. Nauk SSSR 262(1), 71–75 (1982) [Soviet Phys. Dokl. 27 (1), 29–31 (1982)].
J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954).
M. Reissig and J. Wirth, “L p → L q Estimate for Wave Equation with Monotone Time-Dependent Dissipation,” in Proceedings of the RIMS Symposium on Mathematical Models of Phenomena and Evolution Equations, Kyoto, October 2005.
M. Ruzhansky and J. Smith, Global Time Estimates for Solutions to Equations of Dissipative Type, Journées “Équations aux dérivées partielles” 12 (2005) (math/0612758).
S. I. Gelfand, “On the Number of Solutions of a Quadratic Equation,” in Globus General Mathematical Seminar (MTsN II, Moscow, 2004), Vol. 1, pp. 124–133.
V. V. Kozlov, “Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization,” Funktsional. Anal. i Prilozhen. 39(4), 32–47 (2005) [Funct. Anal. Appl. 39 (4), 271–283 (2005)].
L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudodifferential Operators (Springer-Verlag, Berlin-New York, 1985; Mir, Moscow, 1987).
V. V. Palin, “Separation of Dynamics in Systems of Conservation Laws with Relaxation” (to appear).
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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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DOI: https://doi.org/10.1134/S1061920808030051