EDP Sciences Journals List
Advanced Search
Subscriber Authentication Point
Home All issues Volume 18 Article
Free access article

Issue ESAIM: Proc.
Volume 18, 2007
Paris-sud working group on modelling and scientific computing 2006-2007
Page(s) 99 - 119
DOI http://dx.doi.org/10.1051/proc:071809
Published online 12 September 2007

ESAIM: Proc., 2007, Vol. 18, pp. 99-119
DOI: 10.1051/proc:071809

Asymptotics of Higher Order Entropies

Vincent Giovangigli

CMAP-CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France, vincent.giovangigli@polytechnique.edu


(28 mai 2006 / Published online: 12 September 2007)

Abstract
Higher order entropies are kinetic entropy estimators for fluid models. These quantities are quadratic in the velocity v and temperature T derivatives and have temperature dependent coefficients. We investigate asymptotic expansions of higher order entropies for incompressible flows in terms of the Knudsen ${\epsilon_\textsc{k}}$ and Mach ${\epsilon_\textsc{m}}$ numbers. The correspoding entropic inequalities are obtained when $\parallel\ \log T\parallel\ _{BMO}+ {\epsilon_\textsc{m}}\parallel\ v/\sqrt{T}\parallel\ _{L^{\mskip -2mu\infty}_{}}$ is small enough, provided that the temperature dependence of the thermal conductivity $\lambda$ and the viscosity $\eta$ is that given by the kinetic theory. As an example of application of higher order entropic estimates we establish an existence theorem for small Mach number flows.


Résumé
Les entropies d'ordre supérieur sont des estimateurs d'entropie cinétique pour les modèles fluides. Ces quantités sont quadratiques en les dérivées de la vitesse v et la température T avec des coefficients dépendants de T. On s'intéresse aux développements asymptotiques des entropies d'ordre supérieur lorsque les nombres de Knudsen ${\epsilon_\textsc{k}}$ et de Mach ${\epsilon_\textsc{m}}$ sont petits. Les inégalités entropiques correspondantes sont établies lorsque $\parallel\ \log T\parallel\ _{BMO}+ {\epsilon_\textsc{m}}\parallel\ v/\sqrt{T}\parallel\ _{L^{\mskip -2mu\infty}_{}}$ est assez petite, pourvu que la dépendance de la conductivité thermique $\lambda$ et de la viscosité $\eta$ en la température soit celle de la théorie cinétique. Comme exemple d'application, on établit un résultat d'existence de solutions lorsque le nombre de Mach est faible.


Mathematics Subject Classification. 35K, 76N, 82B40

Key words: Entropy, Kinetic, Fluid, Higher order, Estimates


© EDP Sciences, ESAIM 2007


What is OpenURL?

The OpenURL standard is a protocol for transmission of metadata describing the resource that you wish to access. An OpenURL link contains article metadata and directs it to the OpenURL server of your choice. The OpenURL server can provide access to the resource and also offer complementary services (specific search engine, export of references...). The OpenURL link can be generated by different means.
  • If your librarian has set up your subscription with an OpenURL resolver, OpenURL links appear automatically on the abstract pages.
  • You can define your own OpenURL resolver with your EDPS Account. In this case your choice will be given priority over that of your library.
  • You can use an add-on for your browser (Firefox or I.E.) to display OpenURL links on a page (see http://www.openly.com/openurlref/). You should disable this module if you wish to use the OpenURL server that you or your library have defined.