Hilbert's 6th problem: Exact and approximate hydrodynamic manifolds for kinetic equations

@article{Gorban2013Hilberts6P,
  title={Hilbert's 6th problem: Exact and approximate hydrodynamic manifolds for kinetic equations},
  author={Alexander N. Gorban and Iliya V. Karlin},
  journal={Bulletin of the American Mathematical Society},
  year={2013},
  volume={51},
  pages={187-246}
}
  • A. Gorban, I. Karlin
  • Published 1 October 2013
  • Mathematics
  • Bulletin of the American Mathematical Society
The problem of the derivation of hydrodynamics from the Boltz- mann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few in- stances 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… 

Figures from this paper

Linear hydrodynamics and stability of the discrete velocity Boltzmann equations
The discrete velocity Boltzmann equations (DVBE) underlie the attainable properties of all numerical lattice Boltzmann methods (LBM). To that regard, a thorough understanding of their intrinsic
The Chapman Enskog expansion
A mathematical PDE perspective on the Chapman–Enskog expansion
This paper presents in a synthetic way some recent advances on hydrodynamic limits of the Boltzmann equation. It aims at bringing a new light to these results by placing them in the more general
Beyond Navier–Stokes equations: capillarity of ideal gas
Abstract The system of Navier–Stokes–Fourier equations is one of the most celebrated systems of equations in modern science. It describes dynamics of fluids in the limit when gradients of density,
The Problem with Hilbert's 6th Problem
This paper reviews earlier results of the author regarding the hydrodynamic limit problem for the Boltzmann equation. In particular the key points are that the work of Gorban&Karlin suggests that
Conservative regularization of neutral fluids and plasmas
Ideal systems of equations such as Euler and MHD may develop singular structures like shocks, vortex/current sheets. Among these, vortical singularities arise due to vortex stretching which can lead
Quantum Corrections to Classical Kinetics: the Weight of Rotation
Hydrodynamics of gases in the classical domain are examined from the perspective that the gas has a well-defined wavefunction description at all times. Specifically, the internal energy and volume
Boltzmann equation and hydrodynamics beyond Navier–Stokes
  • A. Bobylev
  • Mathematics
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2018
TLDR
The author’s approach based on successive changes of hydrodynamic variables is presented in more detail for the Burnett level and it is shown that the best results in this case can be obtained by using the ‘diagonal’ equations ofhydrodynamics.
Derivation of regularized Grad's moment system from kinetic equations: modes, ghosts and non-Markov fluxes
  • I. Karlin
  • Physics
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2018
TLDR
The results of this study clarify the intrinsic structure of the R13 system and imply partition of R13 fluxes into two types of contributions: dissipative fluxes (both linear and non linear) and nonlinear streamline convective fluxes.
Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics
  • J. Krommes
  • Mathematics
    Journal of Plasma Physics
  • 2018
An introduction to the use of projection-operator methods for the derivation of classical fluid transport equations for weakly coupled, magnetised, multispecies plasmas is given. In the present work,
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 208 REFERENCES
Short-Wave Limit of Hydrodynamics: A Soluble Example.
TLDR
This Letter considers the CE procedure for a simple model of nonhydrodynamic description (one-dimensional linearized 10-moment Grad equations) and the CE series, which is due to a nonlinear procedure even here and which also suffers the Bobylev instability in low-order approximations.
A mathematical PDE perspective on the Chapman–Enskog expansion
This paper presents in a synthetic way some recent advances on hydrodynamic limits of the Boltzmann equation. It aims at bringing a new light to these results by placing them in the more general
Quasistationary hydrodynamics for the Boltzmann equation
The Boltzmann equation solutions are considered for small Knudsen number. The main attention is devoted to certain deviations from the classical Navier-Stokes description. The equations for the
Fluid dynamic limits of kinetic equations. I. Formal derivations
The connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equations are based on a
On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation
As part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations, started in [21], we derive estimates on the rate of convergence to equilibrium for solutions of the
The return of the quartic oscillator. The complex WKB method
The semi-classical treatment of the one-dimensional Schrodinger equation is made free from all approximation. For an analytic potential indeed, the WKB method in complex parameters can be formalized
Hydrodynamics from Grad's equations: What can we learn from exact solutions?
A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models
From Boltzmann to Euler: Hilbert's 6th problem revisited
From hyperbolic regularization to exact hydrodynamics for linearized Grad's equations.
TLDR
The method is described in detail for a simple kinetic model -- a 13 moment Grad system and is based on a dynamic invariance principle which derives exact constitutive relations for the stress tensor and heat flux, and a transformation which renders the exact equations of hydrodynamics hyperbolic and stable.
From the BGK model to the Navier–Stokes equations
...
1
2
3
4
5
...