Carleman estimate for the Navier–Stokes equations and an application to a lateral Cauchy problem. (5th January 2016) - Bellassoued, Mourad; Imanuvilov, Oleg; Yamamoto, Masahiro
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 Hölder stability in determining the velocity and pressure field in an interior domain. In the final section, we apply the results for the linearized Navier–Stokes equations to the fully nonlinear Navier–Stokes equations and establish a similar Hölder stability estimate within sufficiently smooth solutions, and prove the uniqueness of Leray–Hopf weak solutions by surface stresses on an arbitrarily chosen sub-boundary.