Hilbert’s 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations

Gorban, Alexander; Karlin, Ilya

doi:10.1090/S0273-0979-2013-01439-3

Hilbert's 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations

Authors: Alexander N. Gorban and Ilya Karlin
Journal: Bull. Amer. Math. Soc. 51 (2014), 187-246
MSC (2010): Primary 76P05, 82B40, 35Q35
DOI: https://doi.org/10.1090/S0273-0979-2013-01439-3
Published electronically: November 20, 2013
MathSciNet review: 3166040
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of a slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman-Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad's moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short-wave limit and the viscosity modification at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilbert's 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newton's iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the Boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts. Finally, a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the Boltzmann equation in the future.

[1] Igor S. Aranson and Lorenz Kramer, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys. 74 (2002), no. 1, 99–143. MR 1895097, https://doi.org/10.1103/RevModPhys.74.99
[2] V. I. Arnold, Proof of a theorem of A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russian Math Surveys 18 (1963), no. 5, 9-36.
[3] V. I. Arnol′d, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 91–192 (Russian). MR 0170705
[4] V. I. Arnol′d, Bifurcations of invariant manifolds of differential equations, and normal forms of neighborhoods of elliptic curves, Funkcional. Anal. i Priložen. 10 (1976), no. 4, 1–12 (Russian). MR 0431285
[5] V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 250, Springer-Verlag, New York-Berlin, 1983. Translated from the Russian by Joseph Szücs; Translation edited by Mark Levi. MR 695786
[6] Claude Bardos, François Golse, and David Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys. 63 (1991), no. 1-2, 323–344. MR 1115587, https://doi.org/10.1007/BF01026608
[7] P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954), no. 3, 511-525.
[8] Wolf-Jürgen Beyn and Winfried Kleß, Numerical Taylor expansions of invariant manifolds in large dynamical systems, Numer. Math. 80 (1998), no. 1, 1–38. MR 1642582, https://doi.org/10.1007/s002110050357
[9] A. V. Bobylëv, On the Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR 262 (1982), no. 1, 71–75 (Russian). MR 647994
[10] A. V. Bobylëv, Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwell gas, Teoret. Mat. Fiz. 60 (1984), no. 2, 280–310 (Russian, with English summary). MR 762269
[11] A. V. Bobylëv, Quasistationary hydrodynamics for the Boltzmann equation, J. Statist. Phys. 80 (1995), no. 5-6, 1063–1083. MR 1349775, https://doi.org/10.1007/BF02179864
[12] T. Bodineau, I. Gallagher and L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Preprint arXiv:1305.3397, 2013.
[13] N. N. Bogoliubov, Problems of a dynamical theory in statistical physics, Studies in Statistical Mechanics, Vol. I, North-Holland, Amsterdam; Interscience, New York, 1962, pp. 1–118. MR 0136381
[14] Ludwig Boltzmann, Lectures on gas theory, Translated by Stephen G. Brush, University of California Press, Berkeley-Los Angeles, Calif., 1964. MR 0158708
[15] S. G. Brush, The kind of motion we call heat: A history of the kinetic theory of gases in the 19th century, Book 2: Statistical Physics and Irreversible Processes, North Holland, Amsterdam The Netherlands, 1976.
[16] J. W. Cahn, Free energy of a nonuniform system. II. Thermodynamic basis, J. Chem. Phys. 30 (1959), 1121-1124.
[17] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys., 28 (1958), 258-266.
[18] Eliodoro Chiavazzo, Alexander N. Gorban, and Iliya V. Karlin, Comparison of invariant manifolds for model reduction in chemical kinetics, Commun. Comput. Phys. 2 (2007), no. 5, 964–992. MR 2355633
[19] Matteo Colangeli, Iliya V. Karlin, and Martin Kröger, From hyperbolic regularization to exact hydrodynamics for linearized Grad’s equations, Phys. Rev. E (3) 75 (2007), no. 5, 051204, 10. MR 2361818, https://doi.org/10.1103/PhysRevE.75.051204
[20] Matteo Colangeli, Iliya V. Karlin, and Martin Kröger, Hyperbolicity of exact hydrodynamics for three-dimensional linearized Grad’s equations, Phys. Rev. E (3) 76 (2007), no. 2, 022201, 4. MR 2365539, https://doi.org/10.1103/PhysRevE.76.022201
[21] John C. Collins, Renormalization, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1984. An introduction to renormalization, the renormalization group, and the operator-product expansion. MR 778558
[22] P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, Applied Mathematical Sciences, vol. 70, Springer-Verlag, New York, 1989. MR 966192
[23] Leo Corry, David Hilbert and the axiomatization of physics (1894–1905), Arch. Hist. Exact Sci. 51 (1997), no. 2, 83–198. MR 1465092, https://doi.org/10.1007/BF00375141
[24] Sydney Chapman and T. G. Cowling, The mathematical theory of non-uniform gases. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, Third edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970. MR 0258399
[25] L. Q. Chen, Phase-field models for microstructure evolution, Annual Review of Materials Research, 32 (2002), no. 1, 113-140.
[26] Carlo Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. MR 1313028
[27] A. Debussche and R. Temam, Inertial manifolds and slow manifolds, Appl. Math. Lett. 4 (1991), no. 4, 73–76. MR 1117775, https://doi.org/10.1016/0893-9659(91)90059-5
[28] P. Deligne, ed. Quantum fields and strings: a course for mathematicians, American Mathematical Society, Providence, RI, 1999.
[29] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math. 159 (2005), no. 2, 245–316. MR 2116276, https://doi.org/10.1007/s00222-004-0389-9
[30] R. L. Dobrushin and Brunello Tirozzi, The central limit theorem and the problem of equivalence of ensembles, Comm. Math. Phys. 54 (1977), no. 2, 173–192. MR 445641
[31] B. Dubrovin and M. Elaeva, On the critical behavior in nonlinear evolutionary PDEs with small viscosity, Russ. J. Math. Phys. 19 (2012), no. 4, 449–460. MR 3001079, https://doi.org/10.1134/S106192081204005X
[32] J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal. 88 (1985), no. 2, 95–133. MR 775366, https://doi.org/10.1007/BF00250907
[33] F. J. Dyson, Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. (2) 85 (1952), 631–632. MR 46927
[34] John Ellis, Einan Gardi, Marek Karliner, and Mark A. Samuel, Padé approximants, Borel transforms and renormalons: the Bjorken sum rule as a case study, Phys. Lett. B 366 (1996), no. 1-4, 268–275. MR 1371554, https://doi.org/10.1016/0370-2693(95)01326-1
[35] D. Enskog, Kinetische theorie der Vorange in massig verdunnten Gasen. I Allgemeiner Teil, Almqvist and Wiksell, Uppsala, 1917.
[36] Neil Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/72), 193–226. MR 287106, https://doi.org/10.1512/iumj.1971.21.21017
[37] Neil Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), no. 1, 53–98. MR 524817, https://doi.org/10.1016/0022-0396(79)90152-9
[38] R. P. Feynman, The character of physical law, MIT Press, Cambridge, MA, 1965.
[39] M. E. Fisher, The renormalization group in the theory of critical behavior, Reviews of Modern Physics 46 (1974), no. 4, 597-616.
[40] C. Foias, M. S. Jolly, I. G. Kevrekidis, G. R. Sell, and E. S. Titi, On the computation of inertial manifolds, Phys. Lett. A 131 (1988), no. 7-8, 433–436. MR 972615, https://doi.org/10.1016/0375-9601(88)90295-2
[41] E. Fontich, Transversal homoclinic points of a class of conservative diffeomorphisms, J. Differential Equations 87 (1990), no. 1, 1–27. MR 1070024, https://doi.org/10.1016/0022-0396(90)90012-E
[42] N. Fröman and P. O. Fröman, JWKB Approximation, North-Holland, Amsterdam, 1965.
[43] I. Gallagher, L. Saint-Raymond, and B. Texier, From Newton to Boltzmann: hard spheres and short-range potentials, Preprint arXiv:1208.5753, 2012.
[44] Vicente Garzó and Andrés Santos, Kinetic theory of gases in shear flows, Fundamental Theories of Physics, vol. 131, Kluwer Academic Publishers Group, Dordrecht, 2003. Nonlinear transport; With a foreword by James W. Dufty. MR 2012795
[45] J. Gnedenko, Zum sechsten Hilbertschen Problem, In Die Hilbertsche Probleme (ed. by P. Alexandrov), Ostwalds Klassiker der exakten Wissenschaften 252, Leipzig 1979, 144-147.
[46] François Golse and Laure Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels, Invent. Math. 155 (2004), no. 1, 81–161. MR 2025302, https://doi.org/10.1007/s00222-003-0316-5
[47] A. N. Gorban, Order-disorder separation: geometric revision, Physica A 374 (2007), no. 1, 85-102.
[48] A. N. Gorban, P. A. Gorban, and I. V. Karlin, Legendre integrators, post-processing and quasiequilibrium, J. Non-Newtonian Fluid Mech. 120 (2004), 149-167
[49] A. N. Gorban and I. V. Karlin, Structure and approximations of the Chapman-Enskog expansion, Sov. Phys. JETP 73 (1991), 637-641.
[50] Alexander N. Gorban and Iliya V. Karlin, Structure and approximations of the Chapman-Enskog expansion for the linearized Grad equations, Transport Theory Statist. Phys. 21 (1992), no. 1-2, 101–117. MR 1149364, https://doi.org/10.1080/00411459208203524
[51] Alexander N. Gorban and Iliya V. Karlin, Thermodynamic parameterization, Phys. A 190 (1992), no. 3-4, 393–404. MR 1196414, https://doi.org/10.1016/0378-4371(92)90044-Q
[52] Alexander N. Gorban and Iliya V. Karlin, Letter to the editor: nonarbitrary regularization of acoustic spectra, Transport Theory Statist. Phys. 22 (1993), no. 1, 121–124. MR 1198882, https://doi.org/10.1080/00411459308203534
[53] Alexander N. Gorban and Iliya V. Karlin, Method of invariant manifolds and regularization of acoustic spectra, Transport Theory Statist. Phys. 23 (1994), no. 5, 559–632. MR 1272676, https://doi.org/10.1080/00411459408204345
[54] A. N. Gorban and I. V. Karlin, General approach to constructing models of the Boltzmann equation, Physica A 206 (1994), 401-420.
[55] A. N. Gorban and I. V. Karlin, On ``solid liquid'' limit of hydrodynamic equations, Transport Theory and Stat. Phys. 24 (1995), no. 9, 1419-1421.
[56] A. N. Gorban and I. V. Karlin, Scattering rates versus moments: Alternative Grad equations, Phys. Rev. E 54 (1996), R3109-R3112.
[57] A. N. Gorban and I. V. Karlin, Short-wave limit of hydrodynamics: a soluble example, Phys. Rev. Lett. 77 (1996), 282-285.
[58] A. N. Gorban and I. V. Karlin, Method of invariant manifold for chemical kinetics, Chem. Eng. Sci., 58 (2003), 4751-4768.
[59] A. N. Gorban and I. V. Karlin, Uniqueness of thermodynamic projector and kinetic basis of molecular individualism, Physica A 336 (2004), no. 3-4, 391-432.
[60] A. N. Gorban and I. V. Karlin, Invariant manifolds for physical and chemical kinetics, Lecture Notes in Physics, vol. 660, Springer-Verlag, Berlin, 2005. MR 2163015
[61] Alexander N. Gorban and Iliya V. Karlin, Quasi-equilibrium closure hierarchies for the Boltzmann equation, Phys. A 360 (2006), no. 2, 325–364. MR 2186263, https://doi.org/10.1016/j.physa.2005.07.016
[62] Alexander N. Gorban, Iliya V. Karlin, and Andrei Yu. Zinovyev, Constructive methods of invariant manifolds for kinetic problems, Phys. Rep. 396 (2004), no. 4-6, 197–403. MR 2061613, https://doi.org/10.1016/j.physrep.2004.03.006
[63] Alexander N. Gorban, Iliya V. Karlin, and Vladimir B. Zmievskii, Two-step approximation of space-independent relaxation, Transport Theory Statist. Phys. 28 (1999), no. 3, 271–296. MR 1686730, https://doi.org/10.1080/00411459908206037
[64] A. N. Gorban, I. V. Karlin, V. B. Zmievskii, and T. F. Nonnenmacher, Relaxational trajectories: global approximations, Physica A 231 (1996), 648-672.
[65] A.N. Gorban and O. Radulescu, Dynamic and static limitation in multiscale reaction networks, revisited, Advances in Chemical Engineering 34 (2008), 103-173.
[66] A. N. Gorban and G. S. Yablonsky, Extended detailed balance for systems with irreversible reactions, Chemical Engineering Science 66 (2011), 5388-5399.
[67] Hermann Grabert, Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics, vol. 95, Springer-Verlag, Berlin-New York, 1982. MR 670593
[68] Harold Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), 331–407. MR 33674, https://doi.org/10.1002/cpa.3160020403
[69] Harold Grad, Principles of the kinetic theory of gases, Handbuch der Physik (herausgegeben von S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958, pp. 205–294. MR 0135535
[70] M. Grmela, Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering, Advances in Chemical Engineering 39 (2010), 75-129.
[71] Miroslav Grmela, Role of thermodynamics in multiscale physics, Comput. Math. Appl. 65 (2013), no. 10, 1457–1470. MR 3061716, https://doi.org/10.1016/j.camwa.2012.11.019
[72] Misha Gromov, Spaces and questions, Geom. Funct. Anal. Special Volume (2000), 118–161. GAFA 2000 (Tel Aviv, 1999). MR 1826251
[73] John Guckenheimer and Alexander Vladimirsky, A fast method for approximating invariant manifolds, SIAM J. Appl. Dyn. Syst. 3 (2004), no. 3, 232–260. MR 2114735, https://doi.org/10.1137/030600179
[74] J. Hadamard, Sur l'iteration et les solutions asymptotiques des equations diffŕentielles, Bull. Soc. Math. France 29 (1901), 224-228.
[75] E. H. Hauge, Exact and Chapman-Enskog solutions of the Boltzmann equation for the Lorentz model, Phys. Fluids 13 (1970), 1201–1208. MR 278672, https://doi.org/10.1063/1.1693050
[76] David Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), no. 10, 437–479. MR 1557926, https://doi.org/10.1090/S0002-9904-1902-00923-3
[77] David Hilbert, Begründung der kinetischen Gastheorie, Math. Ann. 72 (1912), no. 4, 562–577 (German). MR 1511713, https://doi.org/10.1007/BF01456676
[78] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular theory of gases and liquids, J. Wiley, New York, 1954.
[79] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
[80] I. Hosokawa and S. Inage, Local entropy balance through the shock wave, J. Phys. Soc. Japan 55 (1986), no. 10, 3402-3409.
[81] Tadanori Hyouguchi, Ryohei Seto, Masahito Ueda, and Satoshi Adachi, Divergence-free WKB theory, Ann. Physics 312 (2004), no. 1, 177–267. MR 2067810, https://doi.org/10.1016/j.aop.2004.01.005
[82] A. M. Il′in, Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs, vol. 102, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by V. Minachin [V. V. Minakhin]. MR 1182791
[83] Christopher K. R. T. Jones, Geometric singular perturbation theory, Dynamical systems (Montecatini Terme, 1994) Lecture Notes in Math., vol. 1609, Springer, Berlin, 1995, pp. 44–118. MR 1374108, https://doi.org/10.1007/BFb0095239
[84] D. Jou, J. Casas-Vázquez, and G. Lebon, Extended irreversible thermodynamics, Springer-Verlag, Berlin, 1993. MR 1271780
[85] Iliya V. Karlin, Simplest nonlinear regularization, Transport Theory Statist. Phys. 21 (1992), no. 3, 291–293. MR 1165529, https://doi.org/10.1080/00411459208203925
[86] Iliya V. Karlin, Exact summation of the Chapman-Enskog expansion from moment equations, J. Phys. A 33 (2000), no. 45, 8037–8046. MR 1804291, https://doi.org/10.1088/0305-4470/33/45/303
[87] I. V. Karlin, M. Colangeli, and M. Kröger, Exact linear hydrodynamics from the Boltzmann Equation, Phys. Rev. Lett. 100 (2008), 214503.
[88] I. V. Karlin, G. Dukek, and T. F. Nonnenmacher, Invariance principle for extension of hydrodynamics: Nonlinear viscosity, Phys. Rev. E 55 (1997), no. 2, 1573-1576.
[89] I. V. Karlin, G. Dukek, and T. F. Nonnenmacher, Gradient expansions in kinetic theory of phonons, Phys. Rev. B 55 (1997), 6324-6329.
[90] Iliya V. Karlin and Alexander N. Gorban, Hydrodynamics from Grad’s equations: what can we learn from exact solutions?, Ann. Phys. 11 (2002), no. 10-11, 783–833. MR 1957348, https://doi.org/10.1002/1521-3889(200211)11:10/11<783::AID-ANDP783>3.0.CO;2-V
[91] I. V. Karlin, A. N. Gorban, G. Dukek, and T. F. Nonnenmacher, Dynamic correction to moment approximations, Phys. Rev. E 57 (1998), 1668-1672.
[92] Nikolaos Kazantzis, Singular PDEs and the problem of finding invariant manifolds for nonlinear dynamical systems, Phys. Lett. A 272 (2000), no. 4, 257–263. MR 1774786, https://doi.org/10.1016/S0375-9601(00)00451-5
[93] N. Kazantzis and T. Good, Invariant manifolds and the calculation of the long-term asymptotic response of nonlinear processes using singular PDEs, Computers and Chemical Engineering 26 (2002), no. 7, 999-1012.
[94] Nikolaos Kazantzis, Costas Kravaris, and Lemonia Syrou, A new model reduction method for nonlinear dynamical systems, Nonlinear Dynam. 59 (2010), no. 1-2, 183–194. MR 2585285, https://doi.org/10.1007/s11071-009-9531-y
[95] A. I. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover Publications, Inc., New York, N. Y., 1949. Translated by G. Gamow. MR 0029808
[96] A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin-New York, 1977 (German). Reprint of the 1933 original. MR 0494348
[97] A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), 527–530 (Russian). MR 0068687
[98] A. A. Kornev, On a method of “graph transformation” type for the numerical construction of invariant manifolds, Tr. Mat. Inst. Steklova 256 (2007), no. Din. Sist. i Optim., 237–251 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 256 (2007), no. 1, 223–237. MR 2336902, https://doi.org/10.1134/S0081543807010129
[99] D. J. Korteweg, Sur la forme que prennent les équations du mouvements des fluides si l'on tient compte des forces capillaires causées par des variations de densité, Archives Néerl. Sci. Exactes Nat. Ser II 6 (1901), 1-24.
[100] Ryogo Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan 12 (1957), 570–586. MR 98482, https://doi.org/10.1143/JPSJ.12.570
[101] A. Kurganov and P. Rosenau, Effects of a saturating dissipation in Burgers-type equations, Comm. Pure Appl. Math. 50 (1997), no. 8, 753–771. MR 1454172, https://doi.org/10.1002/(SICI)1097-0312(199708)50:8<753::AID-CPA2>3.0.CO;2-5
[102] S. H. Lam, D. A. Goussis, The CSP method for simplifying kinetics, International Journal of Chemical Kinetics 26 (1994), 461-486.
[103] M. Lampis, New approach to the Mott-Smith method for shock waves, Meccanica 12 (1977), 171-173.
[104] Oscar E. Lanford III, Time evolution of large classical systems, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) Springer, Berlin, 1975, pp. 1–111. Lecture Notes in Phys., Vol. 38. MR 0479206
[105] Peter D. Lax and C. David Levermore, The small dispersion limit of the Korteweg-de Vries equation. I, Comm. Pure Appl. Math. 36 (1983), no. 3, 253–290. MR 697466, https://doi.org/10.1002/cpa.3160360302
[106] C. David Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys. 83 (1996), no. 5-6, 1021–1065. MR 1392419, https://doi.org/10.1007/BF02179552
[107] Ming Li and Paul Vitányi, An introduction to Kolmogorov complexity and its applications, 2nd ed., Graduate Texts in Computer Science, Springer-Verlag, New York, 1997. MR 1438307
[108] Pierre-Louis Lions and Nader Masmoudi, Une approche locale de la limite incompressible, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 5, 387–392 (French, with English and French summaries). MR 1710123, https://doi.org/10.1016/S0764-4442(00)88611-5
[109] Tai-Ping Liu and Shih-Hsien Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys. 246 (2004), no. 1, 133–179. MR 2044894, https://doi.org/10.1007/s00220-003-1030-2
[110] Tai-Ping Liu, Tong Yang, and Shih-Hsien Yu, Energy method for Boltzmann equation, Phys. D 188 (2004), no. 3-4, 178–192. MR 2043729, https://doi.org/10.1016/j.physd.2003.07.011
[111] A. M. Lyapunov, The general problem of the stability of motion, Taylor & Francis, Ltd., London, 1992. Translated from Edouard Davaux’s French translation (1907) of the 1892 Russian original and edited by A. T. Fuller; With an introduction and preface by Fuller, a biography of Lyapunov by V. I. Smirnov, and a bibliography of Lyapunov’s works compiled by J. F. Barrett; Lyapunov centenary issue; Reprint of Internat. J. Control 55 (1992), no. 3 [ MR1154209 (93e:01035)]; With a foreword by Ian Stewart. MR 1229075
[112] U. Maas and S. B. Pope, Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, Combustion and Flame, 88 (1992), 239-264.
[113] Richard D. Mattuck, A guide to Feynman diagrams in the many-body problem, Dover Publications, Inc., New York, 1992. Reprint of the second (1976) edition. MR 1191398
[114] H. P. McKean Jr., A simple model of the derivation of fluid mechanics from the Boltzmann equation, Bull. Amer. Math. Soc. 75 (1969), 1–10. MR 235792, https://doi.org/10.1090/S0002-9904-1969-12128-2
[115] J. D. Mengers and J. M. Powers, One-dimensional slow invariant manifolds for fully coupled reaction and micro-scale diffusion, SIAM J. Appl. Dyn. Syst. 12 (2013), no. 2, 560–595. MR 3044099, https://doi.org/10.1137/120877118
[116] Tim R. Morris, The exact renormalization group and approximate solutions, Internat. J. Modern Phys. A 9 (1994), no. 14, 2411–2449. MR 1277033, https://doi.org/10.1142/S0217751X94000972
[117] Jürgen Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136–176. MR 208078, https://doi.org/10.1007/BF01399536
[118] Clément Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations 31 (2006), no. 7-9, 1321–1348. MR 2254617, https://doi.org/10.1080/03605300600635004
[119] J. Nafe and U. Maas, A general algorithm for improving ILDMs, Combust. Theory Modelling 6 (2002), 697-709.
[120] V. A. Palin and E. V. Radkevich, The Navier-Stokes approximation and Chapman-Enskog projection problems for kinetic equations, Tr. Semin. im. I. G. Petrovskogo 25 (2006), 184–225, 326 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 135 (2006), no. 1, 2721–2748. MR 2271911, https://doi.org/10.1007/s10958-006-0140-8
[121] H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Vols. 1-3. Gauthier-Villars, Paris, 1892/1893/1899.
[122] Baldwin Robertson, Equations of motion in nonequilibrium statistical mechanics, Phys. Rev. (2) 144 (1966), 151–161. MR 198898
[123] Philip Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A (3) 40 (1989), no. 12, 7193–7196. MR 1031939, https://doi.org/10.1103/PhysRevA.40.7193
[124] M. R. Roussel and S. J. Fraser, Geometry of the steady-state approximation: Perturbation and accelerated convergence methods, J. Chem. Phys. 93 (1990), 1072-1081.
[125] Laure Saint-Raymond, From the BGK model to the Navier-Stokes equations, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 2, 271–317 (English, with English and French summaries). MR 1980313, https://doi.org/10.1016/S0012-9593(03)00010-7
[126] Laure Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Lecture Notes in Mathematics, vol. 1971, Springer-Verlag, Berlin, 2009. MR 2683475
[127] L. Saint-Raymond, A mathematical PDE perspective on the Chapman-Enskog expansion, Bull. Amer. Math. Soc., 51 (2014), no. 2, 247-275.
[128] A. Santos, Nonlinear viscosity and velocity distribution function in a simple longitudinal flow, Phys. Rev. E 62 (2000), 6597-6607.
[129] Lee A. Segel and Marshall Slemrod, The quasi-steady-state assumption: a case study in perturbation, SIAM Rev. 31 (1989), no. 3, 446–477. MR 1012300, https://doi.org/10.1137/1031091
[130] M. A. Shubin, Pseudodifferential operators and spectral theory, 2nd ed., Springer-Verlag, Berlin, 2001. Translated from the 1978 Russian original by Stig I. Andersson. MR 1852334
[131] M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), no. 4, 301–315. MR 683192, https://doi.org/10.1007/BF00250857
[132] M. Slemrod, Renormalization of the Chapman-Enskog expansion: isothermal fluid flow and Rosenau saturation, J. Statist. Phys. 91 (1998), no. 1-2, 285–305. MR 1632510, https://doi.org/10.1023/A:1023048322851
[133] Marshall Slemrod, Constitutive relations for monatomic gases based on a generalized rational approximation to the sum of the Chapman-Enskog expansion, Arch. Ration. Mech. Anal. 150 (1999), no. 1, 1–22. MR 1738169, https://doi.org/10.1007/s002050050178
[134] Marshall Slemrod, Chapman-Enskog ⇒ viscosity-capillarity, Quart. Appl. Math. 70 (2012), no. 3, 613–624. MR 2986137, https://doi.org/10.1090/S0033-569X-2012-01305-1
[135] M. Slemrod, From Boltzmann to Euler: Hilbert’s 6th problem revisited, Comput. Math. Appl. 65 (2013), no. 10, 1497–1501. MR 3061719, https://doi.org/10.1016/j.camwa.2012.08.016
[136] Sauro Succi, The lattice Boltzmann equation for fluid dynamics and beyond, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2001. Oxford Science Publications. MR 1857912
[137] Michel Talagrand, A new look at independence, Ann. Probab. 24 (1996), no. 1, 1–34. MR 1387624, https://doi.org/10.1214/aop/1042644705
[138] Michel Talagrand, Rigorous low-temperature results for the mean field 𝑝-spins interaction model, Probab. Theory Related Fields 117 (2000), no. 3, 303–360. MR 1774067, https://doi.org/10.1007/s004400050009
[139] Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967
[140] A. N. Tihonov, Systems of differential equations containing small parameters in the derivatives, Mat. Sbornik N. S. 31(73) (1952), 575–586 (Russian). MR 0055515
[141] François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1, Plenum Press, New York-London, 1980. Pseudodifferential operators; The University Series in Mathematics. MR 597144
[142] G. E. Uhlenbeck, Some notes on the relation between fluid mechanics and statistical physics, Annual review of fluid mechanics, Vol. 12, Annual Reviews, Palo Alto, Calif., 1980, pp. 1–9. MR 565387
[143] G. Veneziano, Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories, Nuovo Cimento A 57 (1968), 190-197.
[144] A. Voros, The return of the quartic oscillator: the complex WKB method, Ann. Inst. H. Poincaré Sect. A (N.S.) 39 (1983), no. 3, 211–338 (English, with French summary). MR 729194
[145] S. Weinberg, The quantum theory of fields, Cambridge University Press (1995).
[146] J. Zinn-Justin, Quantum field theory and critical phenomena, 2nd ed., International Series of Monographs on Physics, vol. 85, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1227790
[147] D. N. Zubarev, Nonequilibrium statistical thermodynamics, Consultants Bureau, New York-London, 1974. Translated from the Russian by P. J. Shepherd; Studies in Soviet Science. MR 0468659
[148] A. K. Zvonkin and L. A. Levin, The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms, Uspehi Mat. Nauk 25 (1970), no. 6(156), 85–127 (Russian). MR 0307889