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

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.