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Lubrication Theory and Viscous Shallow-Water Equations

  • Didier BreschEmail author
  • Mathieu Colin
  • Xi Lin
  • Pascal Noble
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 17)

Abstract

In this proceeding dedicated to Enrique Fernández-Cara, we indicate how to derive rigorously lubrication equations studied in [A.L. Bertozzi, M.C. Pugh, CPAM (1998)] from the viscous shallow-water equations with drag terms and surface tension effect in the one-dimensional in space case. We consider strong and weak solutions and propose simple adaptation of recent relative entropy tools already developed by the first and third author. Note that the viscous shallow-water equation involves height dependent viscosity which vanishes if height vanishes. We choose such presentation because Enrique Fernández-Cara. worked with Francisco Guillén on non-homogeneous incompressible Navier-Stokes equations with density dependent viscosities and in some sense we want to show that in the compressible setting more general test functions are available so it helps to cover degenerate viscosity when height vanishes contrarily to the incompressible setting where such case is an open question.

Keywords

Shallow-water equations Navier-Stokes equations Global weak solutions Lubrication models 

2000 MR Subject Classification

35B35 

Notes

Acknowledgements

Research of P. Noble was partially supported by French ANR project no. ANR-09-JCJC-0103-01. Research of M. Colin, X. Lin, D. Bresch has been partially supported by the ANR Project Viscap.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Didier Bresch
    • 1
    Email author
  • Mathieu Colin
    • 2
  • Xi Lin
    • 2
  • Pascal Noble
    • 3
  1. 1.Université Grenoble AlpesUniversité Savoie Mont Blanc, CNRS, LAMAChambéryFrance
  2. 2.Equipe INRIA, CARDAMOM, Institut Mathématiques de BordeauxÉquipe EDP 351TalenceFrance
  3. 3.Université de LyonUniversité Lyon 1, Institut Camille Jordan, UMR, CNRSVilleurbanne CedexFrance

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