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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



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
Published electronically: November 20, 2013
MathSciNet review: 3166040
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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.

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Additional Information

Alexander N. Gorban
Affiliation: Department of Mathematics, University of Leicester, Leicester, United Kingdom

Ilya Karlin
Affiliation: Department of Mechanical and Process Engineering, ETH Zürich, Switzerland

Received by editor(s): August 28, 2013
Received by editor(s) in revised form: September 25, 2013
Published electronically: November 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.