Stability estimates for Navier-Stokes equations and application to inverse problems

TL;DR

This paper develops new Carleman inequalities for Stokes and Navier-Stokes equations, leading to stability estimates for inverse problems and reconstruction methods based on boundary and interior measurements.

Contribution

It introduces novel Carleman inequalities for non-homogeneous boundary conditions and applies them to inverse boundary coefficient problems and solution reconstruction methods.

Findings

Log-type stability inequalities for Navier-Stokes inverse problems.

Stability estimates for recovering boundary coefficients from measurements.

Convergence rates for reconstruction algorithms using Cauchy data.

Abstract

In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method.