On the uniqueness of bounded very weak solutions to the Navier–Stokes Cauchy problem

Abstract

In this article, we study the uniqueness of very weak solutions to the Navier–Stokes Cauchy problem assuming that the solutions belong to L ^∞((0, T) × ℝ ⁿ ). We find sufficient conditions which allow to deduce uniqueness. By virtue of a counterexample, our conditions can be considered sharp.

Keywords:

• uniqueness
• weak solutions
• Navier–Stokes equations

AMS Subject Classifications::

• 35Q30
• 76D05
• 76N10

Acknowledgements

The author would like to thank Prof G.P. Galdi, who introduced him to the uniqueness question presented in this article, and recently had an interesting talk on the problem which has been very valuable. The author is supported partially by GNFM (INdAM) and partially by MIUR-COFIN 2008–2010 ‘Ottimizzazione non lineare, disequazioni variazionali, e problemi di equilibrio’.

Notes

Notes

1. for the definition and the properties of Riesz transform we refer the reader to 14,15.

2. The claim is a consequence of the singular transformation properties established in 14,15. An extensive proof of the claim can be found in 16.