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On Dissipative Solutions to a System Arising in Viscoelasticity

  • Martin KalousekEmail author
Article
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Abstract

We consider a model for an incompressible visoelastic fluid. It consists of the Navier–Stokes equations involving an elastic term in the stress tensor and a transport equation for the evolution of the deformation gradient. The novel feature of the paper is the introduction of the notion of a dissipative solution and its analysis. We show that dissipative solutions exist globally in time for arbitrary finite energy initial data and that a dissipative solution and a strong solution emanating from the same initial data coincide as long as the latter exists.

Keywords

Incompressible vicoelastic system Dissipative solution Existence Weak-strong uniqueness 

Mathematics Subject Classification

Primary 35Q35 35Q74 Secondary 76A10 

Notes

Acknowledgements

The research leading to these results was supported by DFG grant SCHL 1706/4-1.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany

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