Hostname: page-component-76c49bb84f-7l7zb Total loading time: 0 Render date: 2025-07-12T18:57:27.263Z Has data issue: false hasContentIssue false
  • English
  • Français

On the identification of a single body immersed in a Navier-Stokes fluid

Published online by Cambridge University Press:  01 February 2007

A. DOUBOVA ,
E. FERNÁNDEZ-CARA  and
J. H. ORTEGA
A. DOUBOVA
Affiliation:
Universidad de Sevilla, Dpto. E.D.A.N., Aptdo. 1160, 41080 Sevilla, SPAIN emails: doubova@us.es, cara@us.es
E. FERNÁNDEZ-CARA
Affiliation:
Universidad de Sevilla, Dpto. E.D.A.N., Aptdo. 1160, 41080 Sevilla, SPAIN emails: doubova@us.es, cara@us.es
J. H. ORTEGA
Affiliation:
Universidad del Bío-Bío, Facultad de Ciencias, Dpto. de Ciencias Básicas, Casilla 447, Campus Fernando May, Chillán, Chile and Universidad de Chile, Centro de Modelamiento Matemático UMI 2807 CNRS-UChile, Casilla 170/3, Correo 3, Santiago, Chile email: jortega@dim.uchile.cl
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work we consider the inverse problem of the identification of a single rigid body immersed in a fluid governed by the stationary Navier-Stokes equations. It is assumed that friction forces are known on a part of the outer boundary. We first prove a uniqueness result. Then, we establish a formula for the observed friction forces, at first order, in terms of the deformation of the rigid body. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs we use unique continuation and regularity results for the Navier-Stokes equations and domain variation techniques.

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 1 , February 2007 , pp. 57 - 80
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Alessandrini, G., Beretta, E., Rosset, E. & Vesella, S. (2000) Optimal stability for inverse elliptic boundary value problems with unknown boundaries. Ann. Scuola. Norm. Sup. Pisa CI Sci. 29 (4), 755–806.Google Scholar
[2]Alessandrini, G. & Isakov, V. (1997) Analyticity and uniqueness for the inverse conductivity problem. Rend. Istit. Mat. Univ. Trieste 28 (1–2), 351–369.Google Scholar
[3]Alessandrini, G., Morassi, A. & Rosset, E. (2002) Detecting cavities by electrostatic boundary measurements. Inverse Problems, 18, 1333–53.Google Scholar
[4]Alessandrini, G., Morassi, A. & Rosset, E. (2004) Detecting inclusion in an elastic body by boundary measurements. SIAM Rev. 46, 477–498.Google Scholar
[5]Aparicio, N. D. & Pidcock, M. K. (1996) The boundary inverse problem for the Laplace equation in two dimensions. Inverse Problems, 12, 565–577.Google Scholar
[6]Alvarez, C., Conca, C., Friz, L., Kavian, O. & Ortega, J. H. (2005) Identification of inmersed obstacle via boundary measurements. Inverse Problems, 21, 1531–1552.Google Scholar
[7]Andrieux, S., Abda, A. B. & Jaoua, M. (1993) Identifiabilité de frontière inaccessible par des mesures de surface. C. R. Acad. Sci. Paris Sér. I Math. 316 (5), 429–434.Google Scholar
[8]Bello, J. A. (1996) L r regularity for the Stokes and Navier-Stokes problems. Ann. Mat. Pura Appl. 170 (4), 187–206.CrossRefGoogle Scholar
[9]Bello, J. A., Fernández-Cara, E., Lemoine, J. & Simon, J. (1997) The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow. SIAM J. Control Optim. 35 (2), 626–640.Google Scholar
[10]Beretta, E. & Vesella, S. (1998) Stable determination of boundaries from Cauchy data. SIAM J. Math. Anal. 30, 220–232.CrossRefGoogle Scholar
[11]Bukhgeim, A. L., Cheng, J. & Yamamoto, M. (2000) Conditional stability in an inverse problem of determining a non-smooth boundary. J. Math. Anal. Appl. 242 (1), 57–74.Google Scholar
[12]Canuto, B. & Kavian, O. (2001) Determining coefficients in a class of heat equations via boundary measurements. SIAM J. Math. Anal. 32 (5), 963–986.Google Scholar
[13]Cheng, J., Hon, Y. C. & Yamamoto, M. (2001) Conditional stability estimation for an inverse boundary problem with non-smooth boundary in R3. Trans. Amer. Math. Soc. 353 (10), 4123–4138.CrossRefGoogle Scholar
[14]Constantin, P. & Foias, C. (1988) Navier-Stokes Equations. University of Chicago Press.Google Scholar
[15]Fabre, C. (1996/95) Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems. ESAIM Control Optim. Calc. 1, 267–302.Google Scholar
[16]Fabre, C. & Lebeau, G. (1996) Prolongement unique des solutions de l'équation de Stokes. Comm. Part. Diff. Eq. 21, 573–596.Google Scholar
[17]Galdi, G. P. (1994) An introduction to the mathematical theory of the Navier-Stokes equations. Springer-Verlag.Google Scholar
[18]Girault, V. & Raviart, P. A. (1986) Finite element methods for Navier-Stokes equations. Theory and algorithms. Springer-Verlag.Google Scholar
[19]Kavian, O. (2002) Four lectures on parameter identification in elliptic partial differential operators. Lectures at the University of Sevilla (Spain).Google Scholar
[20]Kaup, P. G., Santosa, F. & Vogelius, M. (1996) Method for imaging corrosion damage in thin plates from electrostatic data. Inverse Problems, 12, 279–293.Google Scholar
[21]Kwon, O. & Seo, J. K. (2001) Total size estimation and identification of multiple anomalies in the inverse conductivity problem. Inverse Problems, 17 (1), 59–75.Google Scholar
[22]Ladyzhenskaya, O. A. (1969) Theory of Viscous Incompressible Flow. Gordon and Breach.Google Scholar
[23]Lions, P.-L. (1996) Mathematical Topics in Fluid Mechanics. Clarendon Press.Google Scholar
[24]Murat, F. & Simon, J. (1974) Quelques résultats sur le contrôle par un domaine géométrique. Rapport du L.A. 189 No. 74003. Université Paris VI.Google Scholar
[25]Murat, F. & Simon, J. (1976) Sur le contrôle par un domaine géométrique. Rapport du L.A. 189 (76015). Université Paris VI.Google Scholar
[26]Serre, D. (1983) Equation de Navier-Stokes Stationnaires avec Données pue Régulieère Ann. Sc. Norm. Pisa, 10 (4), 543–559.Google Scholar
[27]Simon, J. (1980) Differentiation with respect to the domain in boundary value problems. Numer. Func. Anal. Optim. 2, 649–687.Google Scholar
[28]Temam, R. (1985) Navier-Stokes Equations: Theory And Numerical Analysis. North-Holland.Google Scholar