Study of some inverse problems for the Stokes system. Application to the lungs.

Anne-Claire Egloffe 1
1 REO - Numerical simulation of biological flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, UPMC - Université Pierre et Marie Curie - Paris 6
Abstract : In this work, we are interested in the resolution of some inverse problems arising from a multi-scale modeling of the airflow in the lungs. As a first step, we focus on a simplified model of the airflow in the lungs: we consider the incompressible Stokes equations with Robin boundary conditions on a part of the boundary. We want to identify the Robin coefficient defined on this non accessible part of the boundary from measurements of the velocity and the pressure available on another part of the boundary. We first prove quantification results for the unique continuation property for the Stokes system, then we establish two logarithmic stability inequalities, one valid in dimension 2 and the other one valid in any dimension. Both are based on Carleman estimates, global in the first case and local in the second one. Our stability estimates are first established for the stationary problem and the semigroup theory allows to deduce from the stationary case stability estimates for the non-stationary problem. Moreover, under the a priori assumption that the Robin coefficient is piecewise constant, we provide a Lipschitz stability estimate for the stationary problem. We conclude by coming back to the initial model which involves non-standard boundary conditions with the flux. In particular, we obtain encouraging first numerical results concerning the identification of some parameters of the model.
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https://tel.archives-ouvertes.fr/tel-00756334
Contributor : Anne-Claire Egloffe <>
Submitted on : Friday, November 23, 2012 - 4:05:34 PM
Last modification on : Tuesday, May 14, 2019 - 10:12:28 AM
Long-term archiving on : Saturday, December 17, 2016 - 2:14:47 PM

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  • HAL Id : tel-00756334, version 1

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Anne-Claire Egloffe. Study of some inverse problems for the Stokes system. Application to the lungs.. Analysis of PDEs [math.AP]. Université Pierre et Marie Curie - Paris VI, 2012. English. ⟨tel-00756334⟩

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