Lubrication and shallow-water systems Bernis-Friedman and BD entropies

¹ Laboratoire de Mathématiques UMR5127 CNRS, Université Savoie Mont-Blanc ;
e-mail: Didier.Bresch@univ-smb.fr
² Equipe INRIA CARDAMOM, IMB Equipes EDP, 351 cours de la libération, 33405 Talence, France ;
e-mail: Mathieu.Colin@math.u-bordeaux.fr
³ Laboratoire de Mathématiques UMR5127 CNRS, Université Savoie Mont-Blanc ;
e-mail: khawla.msheik@univ-smb.fr
⁴ IMT, INSA Toulouse, 135 avenue de Rangueil, 31077 Toulouse Cedex 9, France ;
e-mail: pascal.noble@math.univ-toulouse.fr
⁵
e-mail: xi.song@hotmail.com

Abstract

This paper concerns the results recently announced by the authors, in C.R. Acad. Sciences Maths volume 357, Issue 1, 1-6 (2019), which make the link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF in our paper) dissipative entropy introduced to study the lubrication equations. More precisely different dissipative BF entropies are obtained from the BD entropies playing with drag terms and capillarity formula for viscous shallow water type equations. This is the main idea in the paper which makes the link between two communities. The limit processes employ the standard compactness arguments taking care of the control in the drag terms. It allows in one dimension for instance to prove global existence of nonnegative weak solutions for lubrication equations starting from the global existence of nonnegative weak solutions for appropriate viscous shallow-water equations (for which we refer to appropriate references). It also allows to prove global existence of nonnegative weak solutions for fourth-order equation including the Derrida-Lebowitz-Speer-Spohn equation starting from compressible Navier-Stokes type equations.