Global Lipschitz stability for an inverse source problem for the Navier–Stokes equations

Abstract

For linearized Navier–Stokes equations, we consider an inverse source problem of determining a spatially varying divergence-free factor. We prove the global Lipschitz stability by interior data over a time interval and velocity field at $t_0>0$ over the spatial domain. The key machinery are Carleman estimates for the Navier–Stokes equations and the operator rot.

Keywords:

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

AMS subject classifications:

• 35R30
• 35R25
• 35Q30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported partly by NSF grant DMS 1312900. The second author was supported by Grant-in-Aid for Scientific Research (S) 15H05740 and Grant-in-Aid (A) 20H00117 of Japan Society for the Promotion of Science, The National Natural Science Foundation of China (no. 11771270, 91730303), and the RUDN University Strategic Academic Leadership Program.