Regular flows of weakly compressible viscoelastic fluids and the incompressible limit

Colette Guillopé; Zaynab Salloum; Raafat Talhouk

doi:10.3934/dcdsb.2010.14.1001

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2010, Volume 14, Issue 3: 1001-1028. Doi: 10.3934/dcdsb.2010.14.1001

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Regular flows of weakly compressible viscoelastic fluids and the incompressible limit

1.
Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050, 61 avenue du Général de Gaulle, 94010 Créteil Cedex
2.
Université Libanaise, Département de Mathématiques, Faculté de Sciences, Beyrouth

Received: August 31, 2009

Revised: February 28, 2010

Published: September 2010

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Abstract

We consider compressible viscoelastic fluids satisfying the Oldroyd constitutive law. We prove the local existence (and uniqueness) of flows by a classical fixed point argument. We also prove some global properties of the solutions. In particular, we obtain some a priori estimates which are uniform in the Mach number and prove global existence of weakly compressible fluids flows. We show that weakly compressible flows with well-prepared initial data converge to incompressible ones when the Mach number converges to zero.

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Mathematics Subject Classification: Primary: 76A05, 76A10, 76N10; Secondary: 35B30.

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