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## HILBERT’S 6TH PROBLEM: EXACT AND APPROXIMATE HYDRODYNAMIC MANIFOLDS FOR KINETIC EQUATIONS

Citations: | 5 - 2 self |

### Citations

2121 | An Introduction to Kolmogorov Complexity and its Applications
- Li, Vit'anyi
- 1997
(Show Context)
Citation Context ...ogorov’s axiomatics (1933) [97] is now accepted as standard. Thirty years later, the complexity approach to randomness was invented by Solomonoff and Kolmogorov (see the review [149] and the textbook =-=[108]-=-). The rigorous foundation of equilibrium statistical physics of many particles based on the central limit theorems was proposed [96, 30]. The modern development of the limit theorems in high dimensio... |

535 |
B.Molecular Theory of Gases and Liquids
- Hirschfelder, Curtis, et al.
- 1954
(Show Context)
Citation Context ...0) f (1) M eq = Q−1 M (1 − (DMf M )m)(A(feq M )). A detailed analysis of explicit versions of this formula for the Boltzmann equation and other kinetic equations is presented in many books and papers =-=[24, 79]-=-. Most of the physical applications of kinetic theory, from the transport processes in gases to modern numerical methods (lattice Boltzmann models [137]) give examples of the practical applications an... |

502 |
The Boltzmann Equation and its Applications
- Cercignani
- 1988
(Show Context)
Citation Context ...sary to use the kinetic representation,HILBERT’S 6TH PROBLEM: EXACT HYDRODYNAMICS 45 for rarefied gases the Boltzmann kinetic equation gives the framework for studying the structure of strong shocks =-=[26]-=-. One of the common heuristic ways to use the Boltzmann equation far from local equilibrium consists of three steps: (1) Construction of a specific ansatz for the distribution function for a given phy... |

380 |
Free energy of a nonuniform system
- Cahn, Hilliard
(Show Context)
Citation Context ... (for a review of some further results see [32]). Such terms appear naturally in theories of the phase transitions such as van der Waals liquids [132], Ginzburg–Landau [1] and Cahn–Hilliard equations =-=[17, 16]-=-, and phase fields models [25]. Surprisingly, such terms are also found in the ideal gas dynamics as a consequence of the Chapman–Enskog expansion [134, 133]. In higher-order approximations, the visco... |

330 |
Geometric singular perturbation theory for ordinary differential equations
- Fenichel
- 1979
(Show Context)
Citation Context ...s onto the hydrodynamic fields and the parametrization of this manifold by the hydrodynamic fields are represented. [69, 70]) to ODEs. The recently developed geometric theory of singular perturbation =-=[36, 37, 84]-=- can be considered as a formalization of the Chapman–Enskog approach for the area where complete rigorous theory is achievable. A program of the derivation of (weak) solutions of the Navier–Stokes equ... |

327 |
A model for collision processes in gases I: Small amplitude processes in charged and neutral one-component systems, Phys
- Bhatnagar, Gross, et al.
- 1954
(Show Context)
Citation Context ...tion of dissipation seems to be a universal phenomenon [124, 52, 53, 102, 133, 91, 60]. It appears in all exactly solved reduction problems for kinetic equations [91] and in the Bhatnagar–Gross–Krook =-=[7]-=- (BGK) kinetics [88] and is also proven for various regularizations of the Chapman–Enskog expansion [124, 52, 133, 60]. The answer to Hilbert’s 6th Problem concerning transition from the Boltzmann equ... |

226 |
The Character of Physical Law
- Feynman
- 1998
(Show Context)
Citation Context ...for the creation, validation and destruction of theories for real-world phenomena, where “We are trying to prove ourselves wrong as quickly as possible, because only in that way can we find progress” =-=[38]-=-. There exist no mathematical tools to formalize relations between Theory and Reality in live Physics. Therefore the 6th Problem may be viewed as a tremendous challenge in deep study of ideas of physi... |

204 |
Geometric Singular Perturbation Theory
- Jones
- 1995
(Show Context)
Citation Context ...s onto the hydrodynamic fields and the parametrization of this manifold by the hydrodynamic fields are represented. [69, 70]) to ODEs. The recently developed geometric theory of singular perturbation =-=[36, 37, 84]-=- can be considered as a formalization of the Chapman–Enskog approach for the area where complete rigorous theory is achievable. A program of the derivation of (weak) solutions of the Navier–Stokes equ... |

171 |
On the kinetic theory of rarefied gases
- Grad
- 1949
(Show Context)
Citation Context ...the collision operator. For the Boltzmann equation, Q is a quadratic operator and, therefore, the notation Q(f, f) is often used. The second group of examples are the systems of Grad moment equations =-=[69, 9, 85, 60]-=-. The system of 13-moment Grad equations linearized near equilibrium is (1.2) (1.3) ∂tρ = −∇ · u, ∂tu = −∇ρ − ∇T − ∇ · σ, ∂tT = − 2 (∇ · u + ∇ · q), 3 ∂tσ = −2∇u − 4 1 ∇q − 5 ǫ σ, ∂tq = − 5 2 ∇T − ∇ ·... |

162 |
Extended Irreversible Thermodynamics
- Jou, Casas-Vázques, et al.
- 1993
(Show Context)
Citation Context ...the collision operator. For the Boltzmann equation, Q is a quadratic operator and, therefore, the notation Q(f, f) is often used. The second group of examples are the systems of Grad moment equations =-=[69, 9, 85, 60]-=-. The system of 13-moment Grad equations linearized near equilibrium is (1.2) (1.3) ∂tρ = −∇ · u, ∂tu = −∇ρ − ∇T − ∇ · σ, ∂tT = − 2 (∇ · u + ∇ · q), 3 ∂tσ = −2∇u − 4 1 ∇q − 5 ǫ σ, ∂tq = − 5 2 ∇T − ∇ ·... |

160 |
Moment closure hierarchies for kinetic theories
- Levermore
- 1996
(Show Context)
Citation Context ...t equations of various complexity, which was introduced by Grad [69] and elaborated further by many authors. In particular, Levermore proved hyperbolicity of the properly constructed moment equations =-=[107]-=-. This staircase forms the basis of the Extended Irreversible Thermodynamics (EIT [85]). 1.3. Singular perturbation and separation of times in kinetics. The kinetic equations are singularly perturbed ... |

154 | On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation
- Desvillettes, Villani
- 2005
(Show Context)
Citation Context ...m almost to its equilibrium is supported by monotonic entropy growth (Boltzmann’s H-theorem). This qualitative picture is illustrated in Fig. 1. Such a “nonrigorous picture of the Boltzmann dynamics” =-=[29]-=- which operates by the manifolds in the space of probability distributions is a seminal tool for production of qualitative hypotheses. The points (‘states’) in Fig. 1. correspond to the distributions ... |

144 |
Small denominators and problems of stability of motion in classical and celestial mechanics, Usp
- Arnold
- 1963
(Show Context)
Citation Context ...enominators affect the convergence of the Poincaré series in the classical many body problem and the theory of nearly integrable systems. They may even make the perturbation series approach senseless =-=[3]-=-. Many ideas have been proposed and implemented to deal with these singularities: use of the direct iteration method instead of power series in KAM [98, 2, 3], renormalization [39, 21, 117], summation... |

136 |
The world of the complex Ginzburg-Landau equation
- Aranson, Kramer
(Show Context)
Citation Context ...okes equations by Korteweg [100] (for a review of some further results see [32]). Such terms appear naturally in theories of the phase transitions such as van der Waals liquids [132], Ginzburg–Landau =-=[1]-=- and Cahn–Hilliard equations [17, 16], and phase fields models [25]. Surprisingly, such terms are also found in the ideal gas dynamics as a consequence of the Chapman–Enskog expansion [134, 133]. In h... |

130 |
Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems
- Kubo
- 1957
(Show Context)
Citation Context ...description may be omitted. The transition from the microscopic to the macroscopic description without an intermediate kinetic equation is used in many physical theories like the Green–Kubo formalism =-=[101]-=-, the Zubarev method of a nonequilibrium statistical4 ALEXANDER N. GORBAN AND ILYA KARLIN operator [148], and the projection operator techniques [68]. This possibility is demonstrated rigorously for ... |

117 | Lectures on Gas Theory - Boltzmann - 1995 |

111 |
Time evolution of large classical systems
- Lanford
- 1975
(Show Context)
Citation Context ...y. The highest achievement here is the proof that rarefied gas of hard spheres will follow the Boltzmann equation during a fraction of the collision time, starting from a non-correlated initial state =-=[105, 43]-=-. The BBGKY hierarchy [13] provides the general framework for this problem. For the systems close to global thermodynamic equilibrium the global in time estimates are available and the validity of the... |

109 |
Proof of a theorem of A.N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the hamiltonian
- Arnol’d
- 1963
(Show Context)
Citation Context ... make the perturbation series approach senseless [3]. Many ideas have been proposed and implemented to deal with these singularities: use of the direct iteration method instead of power series in KAM =-=[98, 2, 3]-=-, renormalization [39, 21, 117], summation and partial summation and rational approximation of the perturbation series [114, 34] and string theories [144, 28] in quantum field theory [146]. Various ad... |

104 | Renormalization. An introduction to renormalization, the renormalization group, and the operator product expansion - Collins - 1984 |

99 |
Mathematical problems,
- Hilbert
- 1902
(Show Context)
Citation Context ... equation 44 7. Conclusion 48 Acknowledgements 53 About the authors 53 References 53 1. Introduction 1.1. Hilbert’s 6th Problem. The 6th Problem differs significantly from other 22 Hilbert’s problems =-=[77]-=-. The title of the problem itself is mysterious: “Mathematical treatment of the axioms of physics”. Physics, in its essence, is a special activity for the creation, validation and destruction of theor... |

96 |
Matching of asymptotic expansions of solutions of boundary value problems” M.: Nauka, 1989.; translated by
- Il’in
- 1992
(Show Context)
Citation Context ... Some hints may be found in recent works about the universal asymptotics of solutions of PDEs with small dissipation [31] (which develop the ideas of Il’in proposed in the analysis of boundary layers =-=[83]-=-). We hope that problem #1 about the existence of hydrodynamic invariant manifolds for the linearized Boltzmann equation will be solved soon, problem #2 about the full nonlinear Boltzmann equation may... |

90 |
Principles of the kinetic theory of gases
- Grad
- 1958
(Show Context)
Citation Context ...old where the solutions stay after initial layer. The projection of the distributions onto the hydrodynamic fields and the parametrization of this manifold by the hydrodynamic fields are represented. =-=[69, 70]-=-) to ODEs. The recently developed geometric theory of singular perturbation [36, 37, 84] can be considered as a formalization of the Chapman–Enskog approach for the area where complete rigorous theory... |

86 | The CSP Method for Simplifying Kinetics
- Lam, Goussis
- 1994
(Show Context)
Citation Context ... [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors [80, 41, 74, 99]. The Newton-type direct iteration methods in various forms =-=[125, 51, 52, 53, 103]-=- proved their efficiency for model reduction and calculation of slow manifolds in kinetics. There is also a series of numerical methods based on the analysis of motion of an embedded manifold along th... |

84 |
The small dispersion limit of the Korteweg-de Vries equation
- Lax
- 1983
(Show Context)
Citation Context ...times; (2) not always; (3) possibly. Some hints about the problems with hydrodynamic asymptotics can be found in the series of works about the small dispersion limit of the Korteweg–de Vries equation =-=[106]-=-. Recently, analysis of the exact solution of the model reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local h... |

80 | Supersolutions.’ in “Quantum fields and strings: a course for mathematicians
- Deligne, Freed
- 1999
(Show Context)
Citation Context ...on method instead of power series in KAM [98, 2, 3], renormalization [39, 21, 117], summation and partial summation and rational approximation of the perturbation series [114, 34] and string theories =-=[144, 28]-=- in quantum field theory [146]. Various ad hoc analytical and numerical regularization tricks have been proposed too. Exactly solvable models give the possibility of exhaustive analysis of the solutio... |

76 | Phase-field models of microstructure evolution - Chen |

74 | Integral manifolds and inertial manifolds for dissipative partial differential equations - Constantin, Foias, et al. - 1988 |

71 |
On the Conservation of Conditionally Periodic Motions under Small Perturbation of the Hamiltonian
- Kolmogorov
- 1954
(Show Context)
Citation Context ... make the perturbation series approach senseless [3]. Many ideas have been proposed and implemented to deal with these singularities: use of the direct iteration method instead of power series in KAM =-=[98, 2, 3]-=-, renormalization [39, 21, 117], summation and partial summation and rational approximation of the perturbation series [114, 34] and string theories [144, 28] in quantum field theory [146]. Various ad... |

69 |
Fluid dynamic limits of kinetic equations
- Bardos, Golse, et al.
- 1991
(Show Context)
Citation Context ...then follow this manifold on the path to equilibrium? The answer to all three questions is essentially positive in the asymptotic regime when the Mach number Ma and the Knudsen number Kn tend to zero =-=[6, 46]-=- (see [127, 128]). This is a limit of very slow flows with very small gradients of all fields, i.e. almost no flow at all. Such a flow changes in time very slowly and a rescaling of time told = tnew/ε... |

61 |
Sur la forme que prennent les équations dumouvements des fluides si l’on tient compte des forces capillaires causées par des variations de densité,
- Korteweg
- 1901
(Show Context)
Citation Context ...erms (∂xp) 2 in the energy density are standard in the thermodynamics of phase transitions. The bulk capillarity terms in fluid mechanics were introduced into the Navier– Stokes equations by Korteweg =-=[100]-=- (for a review of some further results see [32]). Such terms appear naturally in theories of the phase transitions such as van der Waals liquids [132], Ginzburg–Landau [1] and Cahn–Hilliard equations ... |

58 |
Projection Operator Techniques in Nonequilibrium Statistical Mechanics,
- Grabert
- 1982
(Show Context)
Citation Context ... physical theories like the Green–Kubo formalism [101], the Zubarev method of a nonequilibrium statistical4 ALEXANDER N. GORBAN AND ILYA KARLIN operator [148], and the projection operator techniques =-=[68]-=-. This possibility is demonstrated rigorously for a rarefied gas near global equilibrium [12]. The reduction from the Boltzmann kinetics to hydrodynamics may be split into three problems: existence of... |

54 |
Divergence of perturbation theory in quantum electrodynamics,” Phys
- Dyson
- 1952
(Show Context)
Citation Context ...ery novel. There are at least three famous examples:18 ALEXANDER N. GORBAN AND ILYA KARLIN • The “ultraviolet catastrophe” in higher order terms because of physical phenomena at very short distances =-=[33]-=- and the perturbation series divergencies [147] are well known in quantum field theory, and many approaches have been developed to deal with these singularities [146]; • Singularities and divergence i... |

53 | Method of invariant manifold for chemical kinetics
- Gorban, Karlin
(Show Context)
Citation Context ...ounded operator K with property (6.4): If −〈PTϕ, PTKPTϕ〉f ∗ ≥ 0 for all ϕ, ψ and all bounded K with property (6.4), then PT is an orthogonal projector with respect to the entropic inner product (6.3) =-=[58, 59]-=-. This projector acts on functions of v. For a local equilibrium f ∗ (x, v) the projector is constructed for each x and acts on functions ϕ(x, v) point-wise at each point x. Liu and Yu [110] also used... |

47 |
The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels
- Golse, Saint-Raymond
(Show Context)
Citation Context ...then follow this manifold on the path to equilibrium? The answer to all three questions is essentially positive in the asymptotic regime when the Mach number Ma and the Knudsen number Kn tend to zero =-=[6, 46]-=- (see [127, 128]). This is a limit of very slow flows with very small gradients of all fields, i.e. almost no flow at all. Such a flow changes in time very slowly and a rescaling of time told = tnew/ε... |

44 |
The Mathematical Theory of Non-uniform Gases, third ed.,
- Chapman, Cowling
- 1990
(Show Context)
Citation Context ...imits of the Boltzmann equation at the formal level”. The theory of singular perturbations was developed starting from complex systems, from the Boltzmann equation (Hilbert [78], Enskog [35], Chapman =-=[24]-=-, Grad8 ALEXANDER N. GORBAN AND ILYA KARLIN Initial conditions Initial layer Solution Projection to Macroscopic variables Slow manifold Parameterization by Macroscopic variables Macroscopic variables... |

41 | Accurate computations on inertial manifolds
- Jolly, Rosa, et al.
(Show Context)
Citation Context ...ics. There is also a series of numerical methods based on the analysis of motion of an embedded manifold along the trajectories with subtraction of the motion of the manifold ‘parallel to themselves’ =-=[40, 62, 120, 60]-=-. The Chapman–Enskog method [35, 24] was proposed in 1916. This method aims to construct the invariant manifold for the Boltzmann equation in the form of a series in powers of a small parameter, the K... |

38 |
The Chapman-Enskog and Grad methods for solving the Boltzmann equation
- Bobylev
- 1982
(Show Context)
Citation Context ...the collision operator. For the Boltzmann equation, Q is a quadratic operator and, therefore, the notation Q(f, f) is often used. The second group of examples are the systems of Grad moment equations =-=[69, 9, 85, 60]-=-. The system of 13-moment Grad equations linearized near equilibrium is (1.2) (1.3) ∂tρ = −∇ · u, ∂tu = −∇ρ − ∇T − ∇ · σ, ∂tT = − 2 (∇ · u + ∇ · q), 3 ∂tσ = −2∇u − 4 1 ∇q − 5 ǫ σ, ∂tq = − 5 2 ∇T − ∇ ·... |

33 |
Sur Literation et les Solution Asymptotiques des Equations Differentielles
- Hadamard
- 1901
(Show Context)
Citation Context ...apunov series [112], methods of geometric singular perturbation theory [36, 37, 84] and various power series expansions [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 =-=[75]-=- and developed further by many authors [80, 41, 74, 99]. The Newton-type direct iteration methods in various forms [125, 51, 52, 53, 103] proved their efficiency for model reduction and calculation of... |

31 |
The renormalization group in the theory of critical behavior
- Fisher
(Show Context)
Citation Context ...s approach senseless [3]. Many ideas have been proposed and implemented to deal with these singularities: use of the direct iteration method instead of power series in KAM [98, 2, 3], renormalization =-=[39, 21, 117]-=-, summation and partial summation and rational approximation of the perturbation series [114, 34] and string theories [144, 28] in quantum field theory [146]. Various ad hoc analytical and numerical r... |

31 |
Begründung der kinetischen Gastheorie
- Hilbert
- 1912
(Show Context)
Citation Context ...siest of all hydrodynamic limits of the Boltzmann equation at the formal level”. The theory of singular perturbations was developed starting from complex systems, from the Boltzmann equation (Hilbert =-=[78]-=-, Enskog [35], Chapman [24], Grad8 ALEXANDER N. GORBAN AND ILYA KARLIN Initial conditions Initial layer Solution Projection to Macroscopic variables Slow manifold Parameterization by Macroscopic vari... |

29 | Zinovyev, “Constructive methods of invariant manifolds for kinetic problems
- Gorban, Karlin, et al.
- 2004
(Show Context)
Citation Context ...ics. There is also a series of numerical methods based on the analysis of motion of an embedded manifold along the trajectories with subtraction of the motion of the manifold ‘parallel to themselves’ =-=[40, 62, 120, 60]-=-. The Chapman–Enskog method [35, 24] was proposed in 1916. This method aims to construct the invariant manifold for the Boltzmann equation in the form of a series in powers of a small parameter, the K... |

24 | The method of invariant manifold and regularization of acoustic spectra,
- Gorban, Karlin
- 1994
(Show Context)
Citation Context ...mic ones if the linearized collision operator has a spectral gap between the five times degenerated zero and the rest of the spectrum. The saturation of dissipation seems to be a universal phenomenon =-=[124, 52, 53, 102, 133, 91, 60]-=-. It appears in all exactly solved reduction problems for kinetic equations [91] and in the Bhatnagar–Gross–Krook [7] (BGK) kinetics [88] and is also proven for various regularizations of the Chapman–... |

23 |
Kinetic Theory of Gases in Shear Flows. Nonlinear Transport
- Garzó, Santos
- 2003
(Show Context)
Citation Context ... obtained are qualitatively the same as in the case of Maxwell molecules. Precisely the same result for the nonlinear elongational viscosity obtained first from the Grad equations [89] was derived in =-=[129, 44]-=- from the solution to the BGK kinetic equation in the regime of so-called homoenergetic extension flow. This remarkable fact gives more credit to the derivation of hydrodynamic manifolds from nonlinea... |

22 | Hydrodynamics from Grad’s equations: what can we learn from exact solutions?
- Karlin, Gorban
- 2002
(Show Context)
Citation Context ...del reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics =-=[91, 87, 88, 19, 20]-=- supports and extends this observations: the hydrodynamic invariant manifold may exist but sometimes does not exist, and the hydrodynamic equations when Ma ↛ 0 may differ essentially from the Euler an... |

20 | Exact solutions of the nonlinear Boltzmann equation and the theory of relaxation of a Maxwellian gas - Bobylev - 1984 |

18 | General approach to constructing models of the Boltzmann equation - Gorban, Karlin - 1994 |

16 | Numerical Taylor expansions of invariant manifolds in large dynamical systems
- Beyn, Kleß
- 1998
(Show Context)
Citation Context ...Several approaches to computation of invariant manifolds have been developed: Lyapunov series [112], methods of geometric singular perturbation theory [36, 37, 84] and various power series expansions =-=[35, 24, 8]-=-. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors [80, 41, 74, 99]. The Newton-type direct iteration methods in various forms [125, 51, 52... |

15 |
Padé Approximants, Borel Transforms and Renormalons: the Bjorken Sum Rule as a Case Study,” Phys
- Ellis, Gardi, et al.
- 1996
(Show Context)
Citation Context ...ies: use of the direct iteration method instead of power series in KAM [98, 2, 3], renormalization [39, 21, 117], summation and partial summation and rational approximation of the perturbation series =-=[114, 34]-=- and string theories [144, 28] in quantum field theory [146]. Various ad hoc analytical and numerical regularization tricks have been proposed too. Exactly solvable models give the possibility of exha... |

15 | Uniqueness of thermodynamic projector and kinetic basis of molecular individualism
- Gorban, Karlin
- 2003
(Show Context)
Citation Context ...the preservation of entropy production for hydrodynamics produced by projection of kinetics onto an approximate invariant manifold even for rough approximations10 ALEXANDER N. GORBAN AND ILYA KARLIN =-=[59]-=-. This construction is presented in Sec. 6. In Conclusion, we discuss solved and unsolved problems and formulate several hypotheses. 2. Invariance equation and Chapman–Enskog expansion 2.1. The idea o... |

15 | A fast method for approximating invariant manifolds
- Guckenheimer, Vladimirsky
(Show Context)
Citation Context ...ingular perturbation theory [36, 37, 84] and various power series expansions [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors =-=[80, 41, 74, 99]-=-. The Newton-type direct iteration methods in various forms [125, 51, 52, 53, 103] proved their efficiency for model reduction and calculation of slow manifolds in kinetics. There is also a series of ... |

14 |
The Kind of Motion we Call Heat - A History of the Kinetic Theory
- Brush
- 1976
(Show Context)
Citation Context ...tion of the inverse operator Q −1 M . (Nevertheless these calculations may be very bulky and one of the creators of the method, S. Chapman, compared reading his book [24] to “chewing glass”, cited by =-=[15]-=-). Differentiability is needed also because the transport operator A should be bounded to provide strong sense to the manipulations (see the discussion in [128]). The second order in ǫ hydrodynamic eq... |

14 | Extended detailed balance for systems with irreversible reactions
- Gorban, Yablonskii
(Show Context)
Citation Context ...ures in Wonderland.) The small parameter disappears (we take ǫ = 1) but the effect of its presence persists in the analytic invariant manifold. There are some other effects of such a grin in kinetics =-=[67]-=-. The use of the term “slow manifold” for the case ǫ = 1 seems to be an abuse of language. Nevertheless, this manifold has some offprints of slowness, at least for smooth solutions bounded by small nu... |

13 |
Scattering rates versus moments: Alternative Grad equations
- Gorban, Karlin
(Show Context)
Citation Context ...rs are more suitable from thermodynamic point of view. In addition, the flexible choice of projectors allows us to treat various nonlinear functionals (like scattering rates) as macroscopic variables =-=[56, 65]-=-.12 ALEXANDER N. GORBAN AND ILYA KARLIN If f M is a solution to the invariance equation (2.2) then the reduced model (2.3) has two important properties: • Preservation of conservation laws. If a diff... |

12 |
From Newton to Boltzmann: hard spheres and short-range potentials
- Gallagher, Saint-Raymond, et al.
(Show Context)
Citation Context ...y. The highest achievement here is the proof that rarefied gas of hard spheres will follow the Boltzmann equation during a fraction of the collision time, starting from a non-correlated initial state =-=[105, 43]-=-. The BBGKY hierarchy [13] provides the general framework for this problem. For the systems close to global thermodynamic equilibrium the global in time estimates are available and the validity of the... |

12 |
Thermodynamic parameterization, Physica A
- Gorban, Karlin
- 1992
(Show Context)
Citation Context ...dissipation inequalities may occur when one uses the approximate solutions of the invariance equation. For these situations, two techniques are invented: modification of the projection operation (see =-=[51, 59]-=- and Sec. 6 below) and modification of the entropy functional [72, 71]. They allow to retain the dissipation inequality for the approximate equations. It is obvious that the invariance equation (2.2) ... |

12 |
Spaces and questions, GAFA 2000 (Tel Aviv
- Gromov
- 1999
(Show Context)
Citation Context ...icles based on the central limit theorems was proposed [96, 30]. The modern development of the limit theorems in high dimensions is based on the geometrical ideas of the measure concentration effects =-=[73, 138]-=- and gives new insights into the foundation of statistical physics (see, for example, [47, 139]). Despite many open questions, this part of the Hilbert programme is essentially fulfilled – the probabi... |

11 | The Brownian motion as the limit of a deterministic system of hard-spheres. arXiv:1305.3397 [math.AP
- Bodineau, Gallagher, et al.
- 2013
(Show Context)
Citation Context ... to global thermodynamic equilibrium the global in time estimates are available and the validity of the linearized Boltzmann equation is proven recently in this limit for rarefied gas of hard spheres =-=[12]-=-. The second part, model reduction in dissipative systems, from kinetics to macroscopic dynamics, is ready for a mathematical treatment. Some limit theorems about this model reduction are already prov... |

11 |
The central limit theorem and the problem of equivalence of ensembles
- Dobrushin, Tirozzi
- 1977
(Show Context)
Citation Context ... Solomonoff and Kolmogorov (see the review [149] and the textbook [108]). The rigorous foundation of equilibrium statistical physics of many particles based on the central limit theorems was proposed =-=[96, 30]-=-. The modern development of the limit theorems in high dimensions is based on the geometrical ideas of the measure concentration effects [73, 138] and gives new insights into the foundation of statist... |

11 |
On the thermomechanics of interstitial working. Archive for Rational Mechanics and Analysis,
- Dunn, Serrin
- 1985
(Show Context)
Citation Context ...in the thermodynamics of phase transitions. The bulk capillarity terms in fluid mechanics were introduced into the Navier– Stokes equations by Korteweg [100] (for a review of some further results see =-=[32]-=-). Such terms appear naturally in theories of the phase transitions such as van der Waals liquids [132], Ginzburg–Landau [1] and Cahn–Hilliard equations [17, 16], and phase fields models [25]. Surpris... |

11 | Quasi-equilibrium closure hierarchies for the Boltzmann equation, Physica A 360
- Gorban, Karlin
- 2006
(Show Context)
Citation Context ...rs are more suitable from thermodynamic point of view. In addition, the flexible choice of projectors allows us to treat various nonlinear functionals (like scattering rates) as macroscopic variables =-=[56, 65]-=-.12 ALEXANDER N. GORBAN AND ILYA KARLIN If f M is a solution to the invariance equation (2.2) then the reduced model (2.3) has two important properties: • Preservation of conservation laws. If a diff... |

11 | Effects of a Saturating Dissipation in Burgers-Type Equations
- Kurganov, Rosenau
- 1997
(Show Context)
Citation Context ...mic ones if the linearized collision operator has a spectral gap between the five times degenerated zero and the rest of the spectrum. The saturation of dissipation seems to be a universal phenomenon =-=[124, 52, 53, 102, 133, 91, 60]-=-. It appears in all exactly solved reduction problems for kinetic equations [91] and in the Bhatnagar–Gross–Krook [7] (BGK) kinetics [88] and is also proven for various regularizations of the Chapman–... |

10 |
Transversal homoclinic points of a class of conservative diffeomorphisms,”
- Fontich
- 1990
(Show Context)
Citation Context ...ingular perturbation theory [36, 37, 84] and various power series expansions [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors =-=[80, 41, 74, 99]-=-. The Newton-type direct iteration methods in various forms [125, 51, 52, 53, 103] proved their efficiency for model reduction and calculation of slow manifolds in kinetics. There is also a series of ... |

10 |
Structure and approximation of the Chapman–Enskog expansion for linearized Grad equations,
- Gorban, Karlin
- 1991
(Show Context)
Citation Context ... 1500) , 153k 2 + 35 867k 2 k 2 + 2105k 2 + 300 . The corresponding hydrodynamics are non-local but stable and were first obtained by a partial summation (regularization) of the Chapman–Enskog series =-=[49]-=-. A numerical solution of the invariance equation (4.13) is also straightforward and does not produce any serious problem. Select of the proper (Chapman–Enskog) branch of the solution, is set by the a... |

10 |
Short-wave limit of hydrodynamics: A soluble example
- Gorban, Karlin
- 1996
(Show Context)
Citation Context ...und in the series of works about the small dispersion limit of the Korteweg–de Vries equation [106]. Recently, analysis of the exact solution of the model reduction problem for a simple kinetic model =-=[57, 136]-=- has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics [91, 87, 88, 19, 20] supports and extends this observation... |

10 | Two–step approximation of space– independent relaxation, Transport Theory Stat. Phys - Gorban, Karlin, et al. - 1999 |

10 |
Invariance principle for extension of hydrodynamics: Nonlinear viscosity
- Karlin, Dukek, et al.
- 1997
(Show Context)
Citation Context ...of the approximation obtained are qualitatively the same as in the case of Maxwell molecules. Precisely the same result for the nonlinear elongational viscosity obtained first from the Grad equations =-=[89]-=- was derived in [129, 44] from the solution to the BGK kinetic equation in the regime of so-called homoenergetic extension flow. This remarkable fact gives more credit to the derivation of hydrodynami... |

9 |
Problems of a Dynamical Theory
- Bogoliubov
- 1962
(Show Context)
Citation Context ... is the proof that rarefied gas of hard spheres will follow the Boltzmann equation during a fraction of the collision time, starting from a non-correlated initial state [105, 43]. The BBGKY hierarchy =-=[13]-=- provides the general framework for this problem. For the systems close to global thermodynamic equilibrium the global in time estimates are available and the validity of the linearized Boltzmann equa... |

8 | David Hilbert and the Axiomatization of Physics (1894-1905)." Archive for History of Exact Sciences
- Corry
- 1997
(Show Context)
Citation Context ...r, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics”. This is definitely “a programmatic call” =-=[23]-=- for the axiomatization of the formal parts of existent physical theories and no new universal logical framework for the representation ofHILBERT’S 6TH PROBLEM: EXACT HYDRODYNAMICS 3 reality is neces... |

8 | Legendre integrators, post-processing and quasiequilibrium
- Gorban, Gorban, et al.
(Show Context)
Citation Context ...details of the thermodynamic projector construction we refer to [59, 60]. Some examples with construction of the thermodynamic projector with preservation of linear conservation laws are presented in =-=[48]-=-. Another possible modification is a modification of the entropy functional. Recently, Grmela [71, 72] proposed to modify the entropy functional after each step of the Chapman–Enskog expansion in orde... |

8 |
Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering
- Grmela
- 2010
(Show Context)
Citation Context ...ions of the invariance equation. For these situations, two techniques are invented: modification of the projection operation (see [51, 59] and Sec. 6 below) and modification of the entropy functional =-=[72, 71]-=-. They allow to retain the dissipation inequality for the approximate equations. It is obvious that the invariance equation (2.2) for dynamical systems usually has too many solutions, at least locally... |

8 |
Singular PDEs and the problem of finding invariant manifolds for nonlinear dynamical systems
- Kazantzis
(Show Context)
Citation Context ...kerm) ′ at point f 0 . Recently, the approach to invariant manifolds based on the invariance equation in combination with the Lyapunov auxiliary theorem were used for the reduction of kinetic systems =-=[93, 94, 95]-=-. 2.2. The Chapman–Enskog expansion. The Chapman–Enskog and geometric singular perturbation approach assume the special singularly perturbed structure of the equations and look for the invariant manif... |

7 |
Bifurcations of Invariant Manifolds of Differential Equations and Normal Forms in Neighborhoods of Elliptic Curves
- Arnold
- 1977
(Show Context)
Citation Context ... recognized in mechanics by Lyapunov in his thesis (1892) [112]. The problem of persistence and bifurcations of invariant manifolds under perturbations is one of the most seminal problems in dynamics =-=[98, 2, 3, 4, 80, 140]-=-. Several approaches to computation of invariant manifolds have been developed: Lyapunov series [112], methods of geometric singular perturbation theory [36, 37, 84] and various power series expansion... |

7 | Order–disorder separation: Geometric revision
- Gorban
(Show Context)
Citation Context ...limit theorems in high dimensions is based on the geometrical ideas of the measure concentration effects [73, 138] and gives new insights into the foundation of statistical physics (see, for example, =-=[47, 139]-=-). Despite many open questions, this part of the Hilbert programme is essentially fulfilled – the probability theory and the foundations of equilibrium statistical physics are now well-established cha... |

7 |
Kravaris C: A New Model Reduction Method for Nonlinear Dynamical Systems using Singular PDE Theory. Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena SpringerGorban
- Kazantzis
(Show Context)
Citation Context ...kerm) ′ at point f 0 . Recently, the approach to invariant manifolds based on the invariance equation in combination with the Lyapunov auxiliary theorem were used for the reduction of kinetic systems =-=[93, 94, 95]-=-. 2.2. The Chapman–Enskog expansion. The Chapman–Enskog and geometric singular perturbation approach assume the special singularly perturbed structure of the equations and look for the invariant manif... |

6 | Quasistationary Hydrodynamics for the Boltzmann Equation - Bobylev - 1995 |

6 |
Inertial manifolds and slow manifolds
- Debussche, Temam
- 1991
(Show Context)
Citation Context ...uage. Nevertheless, this manifold has some offprints of slowness, at least for smooth solutions bounded by small number. The definition of slow manifolds for a single system may be a non-trivial task =-=[27, 60]-=-. There is a problem with a local definition because for a given vector field the “slowness” of a submanifold cannot be invariant with respect to diffeomorphisms in a vicinity of a regular point. Ther... |

6 | Relaxational trajectories: global approximations, Physica A 231 - Gorban, Karlin, et al. - 1996 |

6 |
Role of thermodynamics in multiscale physics
- Grmela
(Show Context)
Citation Context ...ions of the invariance equation. For these situations, two techniques are invented: modification of the projection operation (see [51, 59] and Sec. 6 below) and modification of the entropy functional =-=[72, 71]-=-. They allow to retain the dissipation inequality for the approximate equations. It is obvious that the invariance equation (2.2) for dynamical systems usually has too many solutions, at least locally... |

6 |
Exact summation of the Chapman-Enskog expansion from moment equations
- Karlin
(Show Context)
Citation Context ...del reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics =-=[91, 87, 88, 19, 20]-=- supports and extends this observations: the hydrodynamic invariant manifold may exist but sometimes does not exist, and the hydrodynamic equations when Ma ↛ 0 may differ essentially from the Euler an... |

6 |
Kinchin, Mathematical foundation of statistical mechanics
- I
- 1949
(Show Context)
Citation Context ... Solomonoff and Kolmogorov (see the review [149] and the textbook [108]). The rigorous foundation of equilibrium statistical physics of many particles based on the central limit theorems was proposed =-=[96, 30]-=-. The modern development of the limit theorems in high dimensions is based on the geometrical ideas of the measure concentration effects [73, 138] and gives new insights into the foundation of statist... |

5 |
From hyperbolic regularization to exact hydrodynamics for linearized Grad equations, Phys
- Colangeli, Karlin, et al.
- 2007
(Show Context)
Citation Context ...del reduction problem for a simple kinetic model [57, 136] has demonstrated that a hydrodynamic invariant manifold may exist and produce non-local hydrodynamics. Analysis of more complicated kinetics =-=[91, 87, 88, 19, 20]-=- supports and extends this observations: the hydrodynamic invariant manifold may exist but sometimes does not exist, and the hydrodynamic equations when Ma ↛ 0 may differ essentially from the Euler an... |

5 |
Kinetische theorie der Vorange in massig verdunnten Gasen
- Enskog
- 1917
(Show Context)
Citation Context ...hydrodynamic limits of the Boltzmann equation at the formal level”. The theory of singular perturbations was developed starting from complex systems, from the Boltzmann equation (Hilbert [78], Enskog =-=[35]-=-, Chapman [24], Grad8 ALEXANDER N. GORBAN AND ILYA KARLIN Initial conditions Initial layer Solution Projection to Macroscopic variables Slow manifold Parameterization by Macroscopic variables Macrosc... |

5 | Dynamic and Static Limitation - Gorban, Radulescu - 2008 |

5 | Gradient expansions in kinetic theory of phonons, Phys - Karlin, Dukek, et al. - 1997 |

4 |
Hyperbolicity of exact hydrodynamics for threedimensional linearized Grads equations, Phys
- Colangeli, Karlin, et al.
(Show Context)
Citation Context |

4 |
On critical behavior in nonlinear evolutionary pdes with small viscosity, preprint
- Dubrovin, Elaeva
- 2013
(Show Context)
Citation Context ...ehaviour of the hydrodynamic solutions becomes dependent only on the extreme cases. Some hints may be found in recent works about the universal asymptotics of solutions of PDEs with small dissipation =-=[31]-=- (which develop the ideas of Il’in proposed in the analysis of boundary layers [83]). We hope that problem #1 about the existence of hydrodynamic invariant manifolds for the linearized Boltzmann equat... |

4 |
Zum sechsten Hilbertschen Problem
- Gnedenko
- 1979
(Show Context)
Citation Context ...iteria either but some important parts of them have been already claimed as solved. Several axiomatic approaches to probability have been developed and the equivalence of some of them has been proven =-=[45]-=-. Kolmogorov’s axiomatics (1933) [97] is now accepted as standard. Thirty years later, the complexity approach to randomness was invented by Solomonoff and Kolmogorov (see the review [149] and the tex... |

4 |
Local entropy balance through the shock wave
- Hosokawa, Inage
- 1986
(Show Context)
Citation Context ...d the Tamm–Mott-Smith lifting operation provides the approximate equations on the Tamm–Mott-Smith manifold with positive entropy production. Several other projections have been tested computationally =-=[81]-=-. All of them violate second law of thermodynamics because for some initial conditions the entropy production for them becomes negative at some points. Indeed, introduction of the entropy density as a... |

4 |
New approach to the Mott-Smith method for shock waves, Meccanica 12
- Lampis
- 1977
(Show Context)
Citation Context ... how to project the Boltzmann equation onto the linear manifold (6.1) and create the macroscopic equation. To respect second law of thermodynamics and provide positivity of entropy production, Lampis =-=[104]-=- used the entropy density s as a new variable. The entropy density is defined as a functional of f(v), s(x) = − ∫ f(x, v)ln f(x, v)d3v. For each distribution f the time derivative of s is defined by t... |

3 | Simplest nonlinear regularization, Transport Theory and Stat - Karlin - 1992 |

3 |
Exact linear hydrodynamics from the Boltzmann Equation
- Karlin, Colangeli, et al.
- 2008
(Show Context)
Citation Context |

3 |
Invariant manifolds and the calculation of the long-term asymptotic response of nonlinear processes using singular PDEs, Computers and chemical engineering 26
- Kazantzis, Good
(Show Context)
Citation Context ...kerm) ′ at point f 0 . Recently, the approach to invariant manifolds based on the invariance equation in combination with the Lyapunov auxiliary theorem were used for the reduction of kinetic systems =-=[93, 94, 95]-=-. 2.2. The Chapman–Enskog expansion. The Chapman–Enskog and geometric singular perturbation approach assume the special singularly perturbed structure of the equations and look for the invariant manif... |

3 |
An English translation by Nathan Morrison, Foundations of the Theory of Probability
- Kolmogorov, Wahrscheinlichkeitsrechnung, et al.
- 1933
(Show Context)
Citation Context ...s of them have been already claimed as solved. Several axiomatic approaches to probability have been developed and the equivalence of some of them has been proven [45]. Kolmogorov’s axiomatics (1933) =-=[97]-=- is now accepted as standard. Thirty years later, the complexity approach to randomness was invented by Solomonoff and Kolmogorov (see the review [149] and the textbook [108]). The rigorous foundation... |

2 |
Persistence and smooothness of invariant manifolds for flows
- Fenichel
- 1971
(Show Context)
Citation Context ...s onto the hydrodynamic fields and the parametrization of this manifold by the hydrodynamic fields are represented. [69, 70]) to ODEs. The recently developed geometric theory of singular perturbation =-=[36, 37, 84]-=- can be considered as a formalization of the Chapman–Enskog approach for the area where complete rigorous theory is achievable. A program of the derivation of (weak) solutions of the Navier–Stokes equ... |

2 | Nonarbitrary regularization of acoustic spectra, Transport Theory and Stat - Gorban, Karlin - 1993 |

2 |
On “Solid liquid” limit of hydrodynamic equations, Transport Theory and Stat
- Gorban, Karlin
- 1995
(Show Context)
Citation Context ...finite dissipation is quite typical for the application of the Newton method to solution of the invariance equation. The ‘solid jet’ limit for the extremely fast compressions gives us another example =-=[55]-=- (see also Sec. 5.2). 4.4. Invariant manifolds, entanglement of hydrodynamic and non-hydrodynamic modes and saturation of dissipation for the 3D 13 moments Grad system . The thirteen moments linear Gr... |

2 | Exact and Chapman–Enskog solutions of the Boltzmann equation for the Lorentz model, Phys. Fluids 13 - Hauge - 1970 |

2 |
Divergence-free WKB theory
- Hyouguchi, Seto, et al.
(Show Context)
Citation Context ...n in quantum field theory, and many approaches have been developed to deal with these singularities [146]; • Singularities and divergence in the semiclassical Wentzel-Kramers-Brillouin (WKB) approach =-=[42, 145, 82]-=-; • The small denominators affect the convergence of the Poincaré series in the classical many body problem and the theory of nearly integrable systems. They may even make the perturbation series appr... |

2 | Dynamic correction to moment approximations, Phys - Karlin, Gorban, et al. - 1998 |

2 |
A method of graph transformation type for numerical simulation of invariant manifolds
- Kornev
(Show Context)
Citation Context ...ingular perturbation theory [36, 37, 84] and various power series expansions [35, 24, 8]. The graph transformation approach was invented by Hadamard in 1901 [75] and developed further by many authors =-=[80, 41, 74, 99]-=-. The Newton-type direct iteration methods in various forms [125, 51, 52, 53, 103] proved their efficiency for model reduction and calculation of slow manifolds in kinetics. There is also a series of ... |