# On Dissipative Solutions to a System Arising in Viscoelasticity

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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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