American Institute of Mathematical Sciences

Advanced
January  2009, 24(1): 95-114. doi: 10.3934/dcds.2009.24.95

Persistence of Boltzmann entropy in fluid models

Vincent Giovangigli 1,

1. 

CMAP-CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Received  July 2007 Revised  December 2007 Published  January 2009

Higher order entropies are kinetic entropy estimators suggested by Enskog expansion of Boltzmann entropy. These quantities are quadratic in the density $\rho$, velocity $v$ and temperature $T$ renormalized derivatives. We investigate asymptotic expansions of higher order entropies for compressible flows in terms of the Knudsen $\epsilon_k$ and Mach $\epsilon_m$ numbers in the natural situation where the volume viscosity, the shear viscosity, and the thermal conductivity depend on temperature, essentially in the form $T^x$. Entropic inequalities are obtained when ||$\log \rho$||BMO,$\quad$ $\epsilon_m$||$v/\sqrt{T}$|| L ∞ ,$\quad$ ||$\log T$||$BMO$,$\quad$ $\epsilon_k$||$h\partial_{x} \rho$/$\rho$|| L ∞ , $\epsilon_k$$\epsilon_m$||$h\partial_{x} v$/$\sqrt{T}$ || L ∞ , $\epsilon_k$||$h\partial_{x}T$/$T$|| L ∞ , and $\epsilon_k^2$||$h^2\partial^2_x T$/$T$|| L ∞ are small enough, where $h = 1/(\rho T^{(1/2) -x)}$ is a weight associated with the dependence on density and temperature of the mean free path.
Keywords: Boltzmann, Fluid, Entropy, Enskog., Kinetic.
Mathematics Subject Classification: Primary: 35Q30, 76N10, 82B4.
Citation: Vincent Giovangigli. Persistence of Boltzmann entropy in fluid models. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 95-114. doi: 10.3934/dcds.2009.24.95
[1]

Anton Trushechkin. Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres. Kinetic & Related Models, 2014, 7 (4) : 755-778. doi: 10.3934/krm.2014.7.755
[2]

Mario Pulvirenti, Sergio Simonella, Anton Trushechkin. Microscopic solutions of the Boltzmann-Enskog equation in the series representation. Kinetic & Related Models, 2018, 11 (4) : 911-931. doi: 10.3934/krm.2018036
[3]

Seung-Yeal Ha, Doron Levy. Particle, kinetic and fluid models for phototaxis. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 77-108. doi: 10.3934/dcdsb.2009.12.77
[4]

Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009
[5]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[6]

Martial Agueh, Guillaume Carlier, Reinhard Illner. Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds. Kinetic & Related Models, 2015, 8 (2) : 201-214. doi: 10.3934/krm.2015.8.201
[7]

Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic & Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787
[8]

Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic & Related Models, 2012, 5 (3) : 459-484. doi: 10.3934/krm.2012.5.459
[9]

Nicola Zamponi. Some fluid-dynamic models for quantum electron transport in graphene via entropy minimization. Kinetic & Related Models, 2012, 5 (1) : 203-221. doi: 10.3934/krm.2012.5.203
[10]

Florian Schneider, Jochen Kall, Graham Alldredge. A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry. Kinetic & Related Models, 2016, 9 (1) : 193-215. doi: 10.3934/krm.2016.9.193
[11]

Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic & Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79
[12]

Michael Herty, Lorenzo Pareschi, Mohammed Seaïd. Enskog-like discrete velocity models for vehicular traffic flow. Networks & Heterogeneous Media, 2007, 2 (3) : 481-496. doi: 10.3934/nhm.2007.2.481
[13]

Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic & Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033
[14]

Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017
[15]

Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237
[16]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
[17]

Vincent Giovangigli, Wen-An Yong. Erratum: ``Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion''. Kinetic & Related Models, 2016, 9 (4) : 813-813. doi: 10.3934/krm.2016018
[18]

Reiner Henseler, Michael Herrmann, Barbara Niethammer, Juan J. L. Velázquez. A kinetic model for grain growth. Kinetic & Related Models, 2008, 1 (4) : 591-617. doi: 10.3934/krm.2008.1.591
[19]

Qi Wang, Tianyu Zhang. Kinetic theories for biofilms. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1027-1059. doi: 10.3934/dcdsb.2012.17.1027
[20]

Anatoly N. Kochubei, Yuri Kondratiev. Fractional kinetic hierarchies and intermittency. Kinetic & Related Models, 2017, 10 (3) : 725-740. doi: 10.3934/krm.2017029

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences