BD entropy and Bernis–Friedman entropy

Didier Bresch; Mathieu Colin; Khawla Msheik; Pascal Noble; Xi Song

doi:10.1016/j.crma.2018.11.009

Logic

BD entropy and Bernis–Friedman entropy
[BD entropie et entropie de Bernis–Friedman]

Présenté par : Pierre-Louis Lions

Didier Bresch ¹ ; Mathieu Colin ² ; Khawla Msheik ¹ ; Pascal Noble ³ ; Xi Song ²

¹ LAMA UMR5127 CNRS, Université Grenoble Alpes, Université Savoie Mont-Blanc, 73376 Le Bourget-du-Lac, France
² Équipe INRIA CARDAMOM, IMB Équipes EDP, 351, cours de la Libération, 33405 Talence, France
³ IMT, INSA Toulouse, 135, avenue de Rangueil, 31077 Toulouse cedex 9, France

Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 1-6.

Résumé
Abstract

Dans cette note, on propose un lien général entre la BD entropie introduite par D. Bresch et B. Desjardins pour les équations de Saint-Venant visqueuses et l'entropie dissipative de Bernis–Friedman (notée BF) introduite pour étudier les équations de lubrification. Différentes entropies dissipatives sont obtenues suivant le choix des termes de traînée sur Saint-Venant visqueux. Ce lien entre ces deux outils mathématiques aide, par exemple, à prouver l'existence de solutions faibles positives pour les équations de lubrification en partant de l'existence de solutions faibles positives pour des équations de Saint-Venant visqueuses bien choisies.

In this note, we propose in the full generality a link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis–Friedman (called BF) dissipative entropy introduced to study the lubrication equations. Different dissipative entropies are obtained playing with the drag terms on the viscous shallow-water equations. It helps for instance to prove the global existence of nonnegative weak solutions to the lubrication equations starting from the global existence of nonnegative weak solutions to appropriate viscous shallow-water equations.

Reçu le : 2018-11-07
Accepté le : 2018-11-16
Publié le : 2018-11-27

DOI : 10.1016/j.crma.2018.11.009

Affiliations des auteurs :

Didier Bresch ¹ ; Mathieu Colin ² ; Khawla Msheik ¹ ; Pascal Noble ³ ; Xi Song ²

@article{CRMATH_2019__357_1_1_0,
     author = {Didier Bresch and Mathieu Colin and Khawla Msheik and Pascal Noble and Xi Song},
     title = {BD entropy and {Bernis{\textendash}Friedman} entropy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1--6},
     publisher = {Elsevier},
     volume = {357},
     number = {1},
     year = {2019},
     doi = {10.1016/j.crma.2018.11.009},
     language = {en},
}

TY  - JOUR
AU  - Didier Bresch
AU  - Mathieu Colin
AU  - Khawla Msheik
AU  - Pascal Noble
AU  - Xi Song
TI  - BD entropy and Bernis–Friedman entropy
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 1
EP  - 6
VL  - 357
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crma.2018.11.009
LA  - en
ID  - CRMATH_2019__357_1_1_0
ER  -

%0 Journal Article
%A Didier Bresch
%A Mathieu Colin
%A Khawla Msheik
%A Pascal Noble
%A Xi Song
%T BD entropy and Bernis–Friedman entropy
%J Comptes Rendus. Mathématique
%D 2019
%P 1-6
%V 357
%N 1
%I Elsevier
%R 10.1016/j.crma.2018.11.009
%G en
%F CRMATH_2019__357_1_1_0

Didier Bresch; Mathieu Colin; Khawla Msheik; Pascal Noble; Xi Song. BD entropy and Bernis–Friedman entropy. Comptes Rendus. Mathématique, Volume 357 (2019) no. 1, pp. 1-6. doi : 10.1016/j.crma.2018.11.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.11.009/

Bibliographie
Cité par

[1] F. Bernis; A. Friedman Higher-order nonlinear degenerate parabolic equations, J. Differ. Equ., Volume 83 (1990) no. 1, pp. 179-206

[2] A.L. Bertozzi; M. Pugh The lubrication approximation for thin viscous films: the moving contact line with a “porous media” cut-off of van der Waals interactions, Nonlinearity, Volume 7 (1994) no. 6, pp. 1535-1564

[3] A.L. Bertozzi; M.C. Pugh Long-wave instabilities and saturation in thin film equations, Commun. Pure Appl. Math., Volume 51 (1998) no. 6, pp. 625-661

[4] Y. Brenier; X. Duan From conservative to dissipative systems through quadratic change of time, with application to the curve-shrotening flow, Arch. Ration. Mech. Anal., Volume 227 (2018), pp. 545-565

[5] D. Bresch, M. Colin, K. Msheik, P. Noble, X. Song, Lubrication/shallow-water system: BD and Bernis–Friedman entropies, 2018, forthcoming.

[6] D. Bresch; B. Desjardins Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Commun. Math. Phys., Volume 238 (2003) no. 1–2, pp. 211-223

[7] D. Bresch; B. Desjardins Quelques modèles diffusifs capillaires de type Korteweg, C. R. Mecanique, Volume 332 (2004), pp. 881-886

[8] J.A. Carillo; A. Wróblewska-Kamińska; Z. Zatorska On long-time asymptotic for viscous hydrodynamic models of collective behaviour with damping and nonlocal interactions, Math. Models Methods Appl. Sci. (2018) (in press)

[9] M.A. Fontelos; G. Kitavtsev; R.M. Taranets Asymptotic decay and non-rupture of viscous sheets, Z. Angew. Math. Phys. (2018), pp. 69-79

[10] G. Grün On Bernis' interpolation inequalities in multiple space dimensions, Z. Anal. Anwend., Volume 20 (2001), pp. 987-998

[11] C. Imbert; A. Mellet Electrified thin films: global existence of non-negative solutions, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 29 (2012) no. 11, pp. 413-433

[12] G. Kitavtsev; P. Laurençot; B. Niethammer Weak solutions to lubrication equations in the presence of strong slippage, Methods Appl. Anal., Volume 18 (2011) no. 2, pp. 183-202

[13] H.L. Li; J. Li; Z.P. Xin Vanishing of vacuum states and blow-up phenomena of the compressible Navier–Stokes equations, Commun. Math. Phys., Volume 281 (2008), pp. 401-444

[14] A. Münch; B.A. Wagner; T.P. Witelski Lubrication models with small to large slip lengths, J. Eng. Math., Volume 53 (2005) no. 3–4, pp. 359-383

Cité par Sources :

Commentaires - Politique