Inverse source problem for linearized Navier–Stokes equations with data in arbitrary sub-domain

Abstract

We consider an inverse problem of determining a spatially varying factor in a source term in the non-stationary linearized Navier–Stokes equations by observation data in an arbitrarily fixed sub-domain over some time interval. We prove the Lipschitz stability provided that the t-dependent factor satisfies a non-degeneracy condition. Our proof is based on a new Carleman estimate for the linearized Navier–Stokes equations.

Keywords:

• inverse source problem
• Navier-Stokes equations
• Carleman estimate
• Lipschitz stability

AMS Subject Classifications::

• Primary 35R30
• Secondary 35Q30

Acknowledgements

Most of this article has been written during the stay of the fourth author at the University of Metz and the stay of the second author at the University of Tokyo, and they thank the universities for the hospitality. The second author was partly supported by NSF grant DMS 0808130. The authors thank the anonymous referees for valuable comments.