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Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces
Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres
1. | Steklov Mathematical Institute of the Russian Academy of Sciences, Gubkina str. 8, 119991 Moscow, Russian Federation |
Also we show that similar functions are exact smooth solutions for the recently proposed generalized Enskog equation. They can be referred to as ``particle-like'' or ``soliton-like'' solutions and are analogues of multisoliton solutions of the Korteweg--de Vries equation.
References:
[1] |
L. Arkeryd and C. Cercignani, On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation, Comm. PDE, 14 (1989), 1071-1089.
doi: 10.1080/03605308908820644. |
[2] |
L. Arkeryd and C. Cercignani, Global existence in $L_1$ for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation, J. Stat. Phys., 59 (1990), 845-867.
doi: 10.1007/BF01025854. |
[3] |
N. Bellomo and M. Lachowicz, On the asymptotic theory of the Boltzmann and Enskog equations: A rigorous $H$-theorem for the Enskog equation, Springer Lecture Notes in Mathematics: Mathematical Aspects of Fluid and Plasma Dynamics, 1460 (1991), 15-30.
doi: 10.1007/BFb0091358. |
[4] |
A. V. Bobylev, Tochnye resheniya uravneniya Boltsmana, (Russian) [Exact solutions of the Boltzmann equation], Dokl. Akad. Nauk SSSR, 225 (1975), 1296-1299. |
[5] |
L. Boltzmann, Vorlesungen Über Gastheorie, (German) [Lectures on gas theory], J. A. Barth, Leipzig, 1896/98. |
[6] |
N. V. Brilliantov and T. Pöschel, Kinetic theory of granular gases, Oxford University Press, 2004.
doi: 10.1093/acprof:oso/9780198530381.001.0001. |
[7] |
N. N. Bogolyubov, Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls, Theor. Math. Phys., 24 (1975), 242-247. |
[8] |
N. N. Bogolubov and N. N. Bogolubov, Jr., Introduction to Quantum Statistical Mechanics, Gordon and Breach, New York, 2010.
doi: 10.1142/7623. |
[9] |
M. S. Borovchenkova and V. I. Gerasimenko, On the non-Markovian Enskog equation for granular gases, J. Phys. A: Math. Theor., 47 (2014), 035001, 18pp; arXiv:1307.6676.
doi: 10.1088/1751-8113/47/3/035001. |
[10] |
C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer Academic Publishing, Dordrecht, 1997.
doi: 10.1007/978-94-011-5558-8. |
[11] |
C. Cercignani, On the Boltzmann equation for rigid spheres, Transport Theory and Statistical Physics, 2 (1972), 211-225.
doi: 10.1080/00411457208232538. |
[12] |
C. Cercignani, Theory and Application of the Boltzmann Equation, Elsevier, New York, 1975. |
[13] |
V. I. Gerasimenko and I. V. Gapyak, Hard sphere dynamics and the Enskog equation, Kinet. Relat. Models, 5 (2012), 459-484; arXiv:1107.5572.
doi: 10.3934/krm.2012.5.459. |
[14] |
S.-Y. Ha and S. E. Noh, Global weak solutions and uniform $L^p$-stability of the Boltzmann-Enskog equation, J. Differential Equations, 251 (2011), 1-25.
doi: 10.1016/j.jde.2011.03.021. |
[15] |
R. Illner and M. Pulvirenti, A derivation of the BBGKY hierarchy for hard sphere particle systems, Trans. Theory Stat. Phys., 16 (1987), 997-1012.
doi: 10.1080/00411458708204603. |
[16] |
V. V. Kozlov, Teplovoye Ravnovesie Po Gibbsu i Puankare, (Russian) [Thermal Equilibrium in the sense of Gibbs and Poincaré], Institute of Computer Research, Moscow-Izhevsk, 2002. |
[17] |
V. V. Kozlov, Thermal Equilibrium in the sense of Gibbs and Poincar'e, Dokl. Akad. Nauk, 382 (2002), 602-605. |
[18] |
V. V. Kozlov and D. V. Treshchev, Weak convergence of solutions of the Liouville equation for nonlinear Hamiltonian systems, Theor. Math. Phys., 134 (2003), 339-350.
doi: 10.1023/A:1022697321418. |
[19] |
V. V. Kozlov and D. V. Treshchev, Evolution of measures in the phase space of nonlinear Hamiltonian systems, Theor. Math. Phys., 136 (2003), 1325-1335.
doi: 10.1023/A:1025607517444. |
[20] |
M.-Y. Lee, T.-P. Liu and S.-H. Yu, Large-time behavior of solutions for the Boltzmann equation with hard potentials, Comm. Math. Phys., 269 (2007), 17-37.
doi: 10.1007/s00220-006-0108-z. |
[21] |
T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608.
doi: 10.1002/cpa.20011. |
[22] |
T.-P. Liu and S.-H. Yu, Green's function of Boltzmann equation, 3-D waves, Bulletin of Institute of Mathematics, Academia Sinica (New Series), 1 (2006), 1-78. |
[23] |
A. I. Mikhailov, Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem, $p$-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), 205-211; arXiv:1305.4057.
doi: 10.1134/S2070046611030046. |
[24] |
D. Ya. Petrina and V. I. Gerasimenko, Mathematical problems of statistical mechanics of a system of elastic balls, Russian Mathematical Surveys, 45 (1990), 153-211.
doi: 10.1070/RM1990v045n03ABEH002360. |
[25] |
D. Ya. Petrina and G. L. Garaffini, Analog of the Liouville equation and BBGKY hierarchy for a system of hard spheres with inelastic collisions, Ukrainian Mathematical Journal, 57 (2005), 967-990.
doi: 10.1007/s11253-005-0242-3. |
[26] |
D. Ya. Petrina and G. L. Garaffini, Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions, Ukrainian Mathematical Journal, 58 (2006), 371-380.
doi: 10.1007/s11253-006-0075-8. |
[27] |
E. V. Piskovskiy, On functional approach to classical mechanics, $p$-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), 243-247.
doi: 10.1134/S2070046611030095. |
[28] |
E. V. Piskovskiy and I. V. Volovich, On the correspondence between Newtonian and functional mechanics, in Quantum Bio-Informatics IV. From Quantum Infarmation to Bio-Informatics, QP-PQ: Quantum Probab. White Noise Anal., vol. 28, World Scientific, Hackensack, NJ, (2011), 363-372.
doi: 10.1142/9789814343763_0028. |
[29] |
P. Resibois, $H$-theorem for the (modified) nonlinear Enskog equation, Phys. Rev. Lett., 40 (1978), 1409-1411.
doi: 10.1103/PhysRevLett.40.1409. |
[30] |
S. Simonella, Evolution of correlation functions in the hard sphere dynamics, J. Stat. Phys., 155 (2014), 1191-1221; arXiv:1205.2789.
doi: 10.1007/s10955-013-0905-7. |
[31] |
H. Spohn, On the integrated form of the BBGKY hierarchy for hard spheres, preprint, arXiv:math-ph/0605068 (written in 1985). |
[32] |
A. S. Trushechkin, Microscopic solutions of the Boltzmann-Enskog equation and the irreversibility problem, Proc. Steklov Inst. Math., 285 (2014), 251-274. |
[33] |
A. S. Trushechkin, Irreversibility and the role of an instrument in the functional formulation of classical mechanics, Theoret. and Math. Phys., 164 (2010), 1198-1201.
doi: 10.1007/s11232-010-0100-9. |
[34] |
A. S. Trushechkin and I. V. Volovich, Functional classical mechanics and rational numbers, p-Adic Numbers Ultrametric Anal. Appl., 1 (2009), 361-367; arXiv:0910.1502.
doi: 10.1134/S2070046609040086. |
[35] |
A. A. Vlasov, Many-Particle Theory and Its Application to Plasma, Gordon and Breach, New York, 1961. |
[36] |
I. V. Volovich, The irreversibilty problem and functional formulation of classical mechanics, preprint,, , ().
|
[37] |
I. V. Volovich, Randomness in classical mechanics and quantum mechanics, Found. Phys., 41 (2011), 516-528.
doi: 10.1007/s10701-010-9450-2. |
[38] |
I. V. Volovich, Bogolyubov equations and functional mechanics, Theoret. and Math. Phys., 164 (2010), 1128-1135.
doi: 10.1007/s11232-010-0090-7. |
show all references
References:
[1] |
L. Arkeryd and C. Cercignani, On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation, Comm. PDE, 14 (1989), 1071-1089.
doi: 10.1080/03605308908820644. |
[2] |
L. Arkeryd and C. Cercignani, Global existence in $L_1$ for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation, J. Stat. Phys., 59 (1990), 845-867.
doi: 10.1007/BF01025854. |
[3] |
N. Bellomo and M. Lachowicz, On the asymptotic theory of the Boltzmann and Enskog equations: A rigorous $H$-theorem for the Enskog equation, Springer Lecture Notes in Mathematics: Mathematical Aspects of Fluid and Plasma Dynamics, 1460 (1991), 15-30.
doi: 10.1007/BFb0091358. |
[4] |
A. V. Bobylev, Tochnye resheniya uravneniya Boltsmana, (Russian) [Exact solutions of the Boltzmann equation], Dokl. Akad. Nauk SSSR, 225 (1975), 1296-1299. |
[5] |
L. Boltzmann, Vorlesungen Über Gastheorie, (German) [Lectures on gas theory], J. A. Barth, Leipzig, 1896/98. |
[6] |
N. V. Brilliantov and T. Pöschel, Kinetic theory of granular gases, Oxford University Press, 2004.
doi: 10.1093/acprof:oso/9780198530381.001.0001. |
[7] |
N. N. Bogolyubov, Microscopic solutions of the Boltzmann-Enskog equation in kinetic theory for elastic balls, Theor. Math. Phys., 24 (1975), 242-247. |
[8] |
N. N. Bogolubov and N. N. Bogolubov, Jr., Introduction to Quantum Statistical Mechanics, Gordon and Breach, New York, 2010.
doi: 10.1142/7623. |
[9] |
M. S. Borovchenkova and V. I. Gerasimenko, On the non-Markovian Enskog equation for granular gases, J. Phys. A: Math. Theor., 47 (2014), 035001, 18pp; arXiv:1307.6676.
doi: 10.1088/1751-8113/47/3/035001. |
[10] |
C. Cercignani, V. I. Gerasimenko and D. Ya. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer Academic Publishing, Dordrecht, 1997.
doi: 10.1007/978-94-011-5558-8. |
[11] |
C. Cercignani, On the Boltzmann equation for rigid spheres, Transport Theory and Statistical Physics, 2 (1972), 211-225.
doi: 10.1080/00411457208232538. |
[12] |
C. Cercignani, Theory and Application of the Boltzmann Equation, Elsevier, New York, 1975. |
[13] |
V. I. Gerasimenko and I. V. Gapyak, Hard sphere dynamics and the Enskog equation, Kinet. Relat. Models, 5 (2012), 459-484; arXiv:1107.5572.
doi: 10.3934/krm.2012.5.459. |
[14] |
S.-Y. Ha and S. E. Noh, Global weak solutions and uniform $L^p$-stability of the Boltzmann-Enskog equation, J. Differential Equations, 251 (2011), 1-25.
doi: 10.1016/j.jde.2011.03.021. |
[15] |
R. Illner and M. Pulvirenti, A derivation of the BBGKY hierarchy for hard sphere particle systems, Trans. Theory Stat. Phys., 16 (1987), 997-1012.
doi: 10.1080/00411458708204603. |
[16] |
V. V. Kozlov, Teplovoye Ravnovesie Po Gibbsu i Puankare, (Russian) [Thermal Equilibrium in the sense of Gibbs and Poincaré], Institute of Computer Research, Moscow-Izhevsk, 2002. |
[17] |
V. V. Kozlov, Thermal Equilibrium in the sense of Gibbs and Poincar'e, Dokl. Akad. Nauk, 382 (2002), 602-605. |
[18] |
V. V. Kozlov and D. V. Treshchev, Weak convergence of solutions of the Liouville equation for nonlinear Hamiltonian systems, Theor. Math. Phys., 134 (2003), 339-350.
doi: 10.1023/A:1022697321418. |
[19] |
V. V. Kozlov and D. V. Treshchev, Evolution of measures in the phase space of nonlinear Hamiltonian systems, Theor. Math. Phys., 136 (2003), 1325-1335.
doi: 10.1023/A:1025607517444. |
[20] |
M.-Y. Lee, T.-P. Liu and S.-H. Yu, Large-time behavior of solutions for the Boltzmann equation with hard potentials, Comm. Math. Phys., 269 (2007), 17-37.
doi: 10.1007/s00220-006-0108-z. |
[21] |
T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608.
doi: 10.1002/cpa.20011. |
[22] |
T.-P. Liu and S.-H. Yu, Green's function of Boltzmann equation, 3-D waves, Bulletin of Institute of Mathematics, Academia Sinica (New Series), 1 (2006), 1-78. |
[23] |
A. I. Mikhailov, Functional mechanics: Evolution of the moments of distribution function and the Poincaré recurrence theorem, $p$-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), 205-211; arXiv:1305.4057.
doi: 10.1134/S2070046611030046. |
[24] |
D. Ya. Petrina and V. I. Gerasimenko, Mathematical problems of statistical mechanics of a system of elastic balls, Russian Mathematical Surveys, 45 (1990), 153-211.
doi: 10.1070/RM1990v045n03ABEH002360. |
[25] |
D. Ya. Petrina and G. L. Garaffini, Analog of the Liouville equation and BBGKY hierarchy for a system of hard spheres with inelastic collisions, Ukrainian Mathematical Journal, 57 (2005), 967-990.
doi: 10.1007/s11253-005-0242-3. |
[26] |
D. Ya. Petrina and G. L. Garaffini, Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions, Ukrainian Mathematical Journal, 58 (2006), 371-380.
doi: 10.1007/s11253-006-0075-8. |
[27] |
E. V. Piskovskiy, On functional approach to classical mechanics, $p$-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), 243-247.
doi: 10.1134/S2070046611030095. |
[28] |
E. V. Piskovskiy and I. V. Volovich, On the correspondence between Newtonian and functional mechanics, in Quantum Bio-Informatics IV. From Quantum Infarmation to Bio-Informatics, QP-PQ: Quantum Probab. White Noise Anal., vol. 28, World Scientific, Hackensack, NJ, (2011), 363-372.
doi: 10.1142/9789814343763_0028. |
[29] |
P. Resibois, $H$-theorem for the (modified) nonlinear Enskog equation, Phys. Rev. Lett., 40 (1978), 1409-1411.
doi: 10.1103/PhysRevLett.40.1409. |
[30] |
S. Simonella, Evolution of correlation functions in the hard sphere dynamics, J. Stat. Phys., 155 (2014), 1191-1221; arXiv:1205.2789.
doi: 10.1007/s10955-013-0905-7. |
[31] |
H. Spohn, On the integrated form of the BBGKY hierarchy for hard spheres, preprint, arXiv:math-ph/0605068 (written in 1985). |
[32] |
A. S. Trushechkin, Microscopic solutions of the Boltzmann-Enskog equation and the irreversibility problem, Proc. Steklov Inst. Math., 285 (2014), 251-274. |
[33] |
A. S. Trushechkin, Irreversibility and the role of an instrument in the functional formulation of classical mechanics, Theoret. and Math. Phys., 164 (2010), 1198-1201.
doi: 10.1007/s11232-010-0100-9. |
[34] |
A. S. Trushechkin and I. V. Volovich, Functional classical mechanics and rational numbers, p-Adic Numbers Ultrametric Anal. Appl., 1 (2009), 361-367; arXiv:0910.1502.
doi: 10.1134/S2070046609040086. |
[35] |
A. A. Vlasov, Many-Particle Theory and Its Application to Plasma, Gordon and Breach, New York, 1961. |
[36] |
I. V. Volovich, The irreversibilty problem and functional formulation of classical mechanics, preprint,, , ().
|
[37] |
I. V. Volovich, Randomness in classical mechanics and quantum mechanics, Found. Phys., 41 (2011), 516-528.
doi: 10.1007/s10701-010-9450-2. |
[38] |
I. V. Volovich, Bogolyubov equations and functional mechanics, Theoret. and Math. Phys., 164 (2010), 1128-1135.
doi: 10.1007/s11232-010-0090-7. |
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