Communications in Mathematical Sciences

Volume 15 (2017)

Number 5

Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model

Pages: 1265 – 1323

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n5.a5

Authors

John W. Barrett (Department of Mathematics, Imperial College London, South Kensington Campus, London Univted Kingdom)

Yong Lu (Chern Institute of Mathematics, Nankai University, Tianjin, China, and Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic)

Endre Süli (Mathematical Institute, University of Oxford, United Kingdom)

Abstract

A compressible Oldroyd-B type model with stress diffusion is derived from a compressible Navier–Stokes–Fokker–Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a barotropic, compressible, isothermal, viscous Newtonian solvent, are idealized as pairs of massless beads connected with Hookean springs. We develop a priori bounds for the model, including a logarithmic bound, which guarantee the nonnegativity of the elastic extra stress tensor, and we prove the existence of large data global-in-time finite-energy weak solutions in two space dimensions.

Keywords

weak solution, compressible Navier–Stokes equation, Oldroyd-B model

2010 Mathematics Subject Classification

35A01, 35Q35, 76A05

Full Text (PDF format)

The second author acknowledges the support of the project LL1202 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic. The third author is grateful to members of the Nečas Center for Mathematical Modeling and the Faculty of Mathematics and Physics of the Charles University in Prague for their hospitality during his sabbatical leave.

Received 15 October 2016

Published 26 June 2017