Instability of an inverse problem for the stationary Navier-Stokes equations

Fabien Caubet (1)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

This paper provides a theoretical study of the detection of an object immersed in a fluid when the fluid motion is governed by the stationary Navier-Stokes equations with non homogeneous Dirichlet boundary conditions. To solve this inverse problem, we make a boundary measurement on a part of the exterior boundary. First, we present an identifiability result. We then use a shape optimization method: in order to identify the obstacle, we minimize a nonlinear least squares criterion. Thus, we prove the existence of the first order shape derivative of the state, we characterize it and deduce the gradient of the least squares functional. Finally, we study the stability of this setting doing a shape sensitivity analysis of order two. Hence, we prove the existence of the second order shape derivatives and we give the expression of the shape Hessian at possible solutions of the original inverse problem. Then, the compactness of the Riesz operator corresponding to this shape Hessian is shown and the ill-posedness of the identification problem follows. This explains the need of regularization to solve numerically this problem.

Mots clés: stationary Navier-Stokes equations; order two shape sensitivity; geometric inverse problem; sensitivity with respect to the domain; shape calculus

Domaines: Optimisation et contrôle [math.OC]

Fabien Caubet. Instability of an inverse problem for the stationary Navier-Stokes equations. SIAM Journal on Control and Optimization, 2013, 51 (4), pp.2949--2975. ⟨hal-00696174⟩