Global Well-posedness for an incompressible flow with intrinsic degrees of freedom in bounded domains

Abstract

We consider a mathematical model for an incompressible Newtonian fluid with intrinsic degrees of freedom in a smooth bounded domain. We first show that there exists a unique local strong solution for large initial data. Then, we prove that the local strong solution is indeed global provided that the initial data is sufficiently small. Furthermore, we prove that when the strong solution exists, all the global weak solutions constructed by Lions must be equal to the unique strong solution with the same initial data.

Keywords:

• Well-posedness
• Navier–Stokes equations
• intrinsic degrees of freedom
• weak–strong uniqueness

AMS Subject Classifications:

• 35Q35
• 76D05

Acknowledgements

The author thanks Prof. Cheng He for helpful discussions.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

The author was supported in part by the National Natural Science Foundation of China [grant number 11226169]; Doctor Fund of Henan Polytechnic University [grant number B2012-110].