Carleman estimate for the Navier-Stokes equations and an application to a lateral Cauchy problem

TL;DR

This paper establishes a Carleman estimate for the linearized Navier-Stokes equations and applies it to prove H"older stability for an inverse problem involving the velocity and pressure fields.

Contribution

It introduces a new Carleman estimate with a regular weight function and applies it to achieve stability results for a lateral Cauchy problem in fluid dynamics.

Findings

Proved a Carleman estimate for linearized Navier-Stokes equations.

Established H"older stability for the inverse problem.

Demonstrated the effectiveness of the estimate in an interior domain.

Abstract

We consider the nonstationary linearized Navier-Stokes equations in a bounded domain and first we prove a Carleman estimate with a regular weight function. Second we apply the Carleman estimate to a lateral Cauchy problem for the Navier-Stokes equations and prove the H\"older stability in determining the velocity and pressure field in an interior domain.