Detecting a moving obstacle in an ideal fluid by a boundary measurement

Carlos Conca; Patricio Cumsille; Jaime Ortega; Lionel Rosier

doi:10.1016/j.crma.2008.06.007

Partial Differential Equations/Optimal Control

Detecting a moving obstacle in an ideal fluid by a boundary measurement
[Détection d'un obstacle en mouvement dans un fluide parfait à partir d'une mesure sur le bord du domaine]

Présenté par : Gilles Lebeau

Carlos Conca ¹ ; Patricio Cumsille ^1,² ; Jaime Ortega ^1,² ; Lionel Rosier ^1,³

¹ Departamento de Ingeniería Matemática (DIM) and Centro de Modelamiento Matemático (CMM), Universidad de Chile (UMI CNRS 2807), Avda Blanco Encalada 2120, Casilla 170-3, Correo 3, Santiago, Chile
² Depto. Ciencias Básicas, Fac. Ciencias, U. del Bío-Bío, Avda Andrés Bello, Chillán, Chile
³ Institut Élie-Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, 54506 Vandœuvre-lès-Nancy cedex, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 839-844.

Résumé
Abstract

Dans cette Note, on s'intéresse au problème de la détection d'un obstacle en mouvement dans un fluide parfait incompressible à partir de la mesure de la vitesse du fluide sur une partie du bord du domaine. Lorsque l'obstacle est une boule, on montre que la position et la vitesse de son centre de gravité peuvent être identifiées à l'aide d'une seule mesure. La stabilité linéaire par rapport à la mesure est prouvée par des techniques de différentiation par rapport au domaine.

In this Note we investigate the problem of the detection of a moving obstacle in a perfect fluid occupying a bounded domain in $R^{2}$ from the measurement of the velocity of the fluid on one part of the boundary. We show that when the obstacle is a ball, we may identify the position and the velocity of its center of mass from a single boundary measurement. Linear stability estimates are also established by using shape differentiation techniques.

Reçu le : 2008-01-07
Accepté le : 2008-06-18
Publié le : 2008-07-25

DOI : 10.1016/j.crma.2008.06.007

Affiliations des auteurs :

Carlos Conca ¹ ; Patricio Cumsille ^{1, 2} ; Jaime Ortega ^{1, 2} ; Lionel Rosier ^{1, 3}

@article{CRMATH_2008__346_15-16_839_0,
     author = {Carlos Conca and Patricio Cumsille and Jaime Ortega and Lionel Rosier},
     title = {Detecting a moving obstacle in an ideal fluid by a boundary measurement},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {839--844},
     publisher = {Elsevier},
     volume = {346},
     number = {15-16},
     year = {2008},
     doi = {10.1016/j.crma.2008.06.007},
     language = {en},
}

TY  - JOUR
AU  - Carlos Conca
AU  - Patricio Cumsille
AU  - Jaime Ortega
AU  - Lionel Rosier
TI  - Detecting a moving obstacle in an ideal fluid by a boundary measurement
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 839
EP  - 844
VL  - 346
IS  - 15-16
PB  - Elsevier
DO  - 10.1016/j.crma.2008.06.007
LA  - en
ID  - CRMATH_2008__346_15-16_839_0
ER  -

%0 Journal Article
%A Carlos Conca
%A Patricio Cumsille
%A Jaime Ortega
%A Lionel Rosier
%T Detecting a moving obstacle in an ideal fluid by a boundary measurement
%J Comptes Rendus. Mathématique
%D 2008
%P 839-844
%V 346
%N 15-16
%I Elsevier
%R 10.1016/j.crma.2008.06.007
%G en
%F CRMATH_2008__346_15-16_839_0

Carlos Conca; Patricio Cumsille; Jaime Ortega; Lionel Rosier. Detecting a moving obstacle in an ideal fluid by a boundary measurement. Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 839-844. doi : 10.1016/j.crma.2008.06.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.06.007/

Bibliographie
Cité par

[1] G. Alessandrini; A. Diaz Valenzuela Unique determination of multiple cracks by two measurements, SIAM J. Control Optim., Volume 34 (1996) no. 3, pp. 913-921

[2] C. Alvarez; C. Conca; L. Friz; O. Kavian; J.H. Ortega Identification of immersed obstacles via boundary measurements, Inverse Problems, Volume 21 (2005) no. 5, pp. 1531-1552

[3] C. Conca; P. Cumsille; J. Ortega; L. Rosier On the detection of a moving obstacle in an ideal fluid by a boundary measurement, Inverse Problems, Volume 24 (2008), p. 045001 (18 p.)

[4] A. Doubova; E. Fernández-Cara; J.H. Ortega On the identification of a single body immersed in a Navier–Stokes fluid, Eur. J. Appl. Math., Volume 18 (2007) no. 1, pp. 57-80

[5] H. Heck; G. Uhlmann; J.-N. Wang Reconstruction of obstacles immersed in an incompressible fluid, Inverse Probl. Imaging, Volume 1 (2007) no. 1, pp. 63-76

[6] P. Henrici Applied and Computational Complex Analysis, Volume 1: Power Series—Integration—Conformal Mapping—Location of Zeros, Pure and Applied Mathematics, John Wiley & Sons, New York, 1974

[7] A. Majda Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math., Volume 39 (1986) no. Suppl., p. S187-S220

[8] J.H. Ortega; L. Rosier; T. Takahashi On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 24 (2007) no. 1, pp. 139-165

[9] J. Simon Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., Volume 2 (1980) no. 7–8, pp. 649-687

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Bloch wave homogenization in a medium perforated by critical holes

Jaime Ortega; Jorge San Martín; Loredana Smaranda

C. R. Méca (2007)

Uniqueness and partial identification in a geometric inverse problem for the Boussinesq system

Anna Doubova; Enrique Fernández-Cara; Manuel González-Burgos; ...

C. R. Math (2006)