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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 25, Issue 1


On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

Jorge San Martín
  • Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR 2071, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170/3-Correo 3, Santiago, Chile
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/ Erica L. Schwindt
  • Departamento de Matemática, Facultad de Ciencias Exactas, Físico-Químicas y Naturales, Universidad Nacional de Río Cuarto, 5800 - Río Cuarto, Córdoba, Argentina
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/ Takéo Takahashi
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  • Inria, Villers-lès-Nancy, F-54600, France; and Université de Lorraine, IECL, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; and CNRS, IECL, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France
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Published Online: 2015-12-09 | DOI: https://doi.org/10.1515/jiip-2014-0056


We consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. We deal with the case where the fluid equations are the nonstationary Stokes system and using the enclosure method, we can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, we can obtain the same result in the case of a linear fluid-structure system composed by a rigid body and a viscous incompressible fluid. We also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid-structure system with free boundary. Using complex spherical waves, we obtain some partial information on the distance from a point to the obstacle.

Keywords: Geometrical inverse problems; fluid-structure interaction; Navier–Stokes system; enclosure method; complex geometrical solutions

MSC 2010: 35R30; 35Q35; 76D07; 35R35; 74F10; 76D05


  • [1]

    Alvarez C., Conca C., Friz L., Kavian O. and Ortega J. H., Identification of immersed obstacles via boundary measurements, Inverse Problems 21 (2005), no. 5, 1531–1552. Google Scholar

  • [2]

    Alves C. J. S., Kress R. and Silvestre A. L., Integral equations for an inverse boundary value problem for the two-dimensional Stokes equations, J. Inverse Ill-Posed Probl. 15 (2007), no. 5, 461–481. Google Scholar

  • [3]

    Badra M., Caubet F. and Dambrine M., Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci. 21 (2011), no. 10, 2069–2101. Google Scholar

  • [4]

    Bonnesen T. and Fenchel W., Theory of Convex Bodies, BCS Associates, Moscow, 1987. Google Scholar

  • [5]

    Conca C., Cumsille P., Ortega J. and Rosier L., On the detection of a moving obstacle in an ideal fluid by a boundary measurement, Inverse Problems 24 (2008), no. 4, Article ID 045001. Google Scholar

  • [6]

    Conca C., Malik M. and Munnier A., Detection of a moving rigid solid in a perfect fluid, Inverse Problems 26 (2010), no. 9, Article ID 095010. Google Scholar

  • [7]

    Conca C., Schwindt E. L. and Takahashi T., On the identifiability of a rigid body moving in a stationary viscous fluid, Inverse Problems 28 (2012), no. 1, Article ID 015005. Google Scholar

  • [8]

    Dos Santos Ferreira D., Kenig C. E., Sjöstrand J. and Uhlmann G., Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys. 271 (2007), no. 2, 467–488. Google Scholar

  • [9]

    Doubova A., Fernández-Cara E., González-Burgos M. and Ortega J. H., A geometric inverse problem for the Boussinesq system, Discrete Contin. Dyn. Syst. Ser. B 6 (2006), no. 6, 1213–1238. Google Scholar

  • [10]

    Doubova A., Fernández-Cara E. and Ortega J. H., On the identification of a single body immersed in a Navier–Stokes fluid, European J. Appl. Math. 18 (2007), no. 1, 57–80. Google Scholar

  • [11]

    Duvaut G., Mécanique des Milieux Continus, Dunod, Paris, 1998. Google Scholar

  • [12]

    Eskin G. and Ralston J., On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems 18 (2002), no. 3, 907–921. Google Scholar

  • [13]

    Gaitan P., Isozaki H., Poisson O., Siltanen S. and Tamminen J., Probing for inclusions in heat conductive bodies, Inverse Probl. Imaging 6 (2012), no. 3, 423–446. Google Scholar

  • [14]

    Galdi G. P., An Introduction to the Mathematical Theory of the Navier–Stokes Equations, 2nd ed., Springer Monogr. Math., Springer, New York, 2011. Google Scholar

  • [15]

    Heck H., Li X. and Wang J.-N., Identification of viscosity in an incompressible fluid, Indiana Univ. Math. J. 56 (2007), no. 5, 2489–2510. Google Scholar

  • [16]

    Heck H., Uhlmann G. and Wang J.-N., Reconstruction of obstacles immersed in an incompressible fluid, Inverse Probl. Imaging 1 (2007), no. 1, 63–76. Google Scholar

  • [17]

    Hesla T. I., Collisions of smooth bodies in viscous fluids: A mathematical investigation, Ph.D. thesis, University of Minnesota, 2005. Google Scholar

  • [18]

    Hillairet M., Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1345–1371. Google Scholar

  • [19]

    Hillairet M. and Takahashi T., Collisions in three-dimensional fluid structure interaction problems, SIAM J. Math. Anal. 40 (2009), no. 6, 2451–2477. Google Scholar

  • [20]

    Ikehata M., How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inverse Ill-Posed Probl. 7 (1999), no. 3, 255–271. Google Scholar

  • [21]

    Ikehata M., Reconstruction of a source domain from the Cauchy data, Inverse Problems 15 (1999), no. 2, 637–645. Google Scholar

  • [22]

    Ikehata M., Reconstruction of the support function for inclusion from boundary measurements, J. Inverse Ill-Posed Probl. 8 (2000), no. 4, 367–378. Google Scholar

  • [23]

    Ikehata M., Inverse scattering problems and the enclosure method, Inverse Problems 20 (2004), no. 2, 533–551. Google Scholar

  • [24]

    Ikehata M., An inverse source problem for the heat equation and the enclosure method, Inverse Problems 23 (2007), no. 1, 183–202. Google Scholar

  • [25]

    Ikehata M., The framework of the enclosure method with dynamical data and its applications, Inverse Problems 27 (2011), no. 6, Article ID 065005. Google Scholar

  • [26]

    Ikehata M. and Itou H., On reconstruction of a cavity in a linearized viscoelastic body from infinitely many transient boundary data, Inverse Problems 28 (2012), no. 12, Article ID 125003. Google Scholar

  • [27]

    Ikehata M. and Kawashita M., The enclosure method for the heat equation, Inverse Problems 25 (2009), no. 7, Article ID 075005. Google Scholar

  • [28]

    Ikehata M. and Kawashita M., On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval, Inverse Problems 26 (2010), no. 9, Article ID 095004. Google Scholar

  • [29]

    Kenig C. E., Sjöstrand J. and Uhlmann G., The Calderón problem with partial data, Ann. of Math. (2) 165 (2007), no. 2, 567–591. Google Scholar

  • [30]

    Nakamura G. and Uhlmann G., Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math. 118 (1994), no. 3, 457–474. Google Scholar

  • [31]

    Nakamura G. and Uhlmann G., Erratum: “Global uniqueness for an inverse boundary value problem arising in elasticity” [Invent. Math. 118 (1994), no. 3, 457–474], Invent. Math. 152 (2003), no. 1, 205–207. Google Scholar

  • [32]

    Raviart P.-A. and Thomas J.-M., Introduction à l’analyse numérique des équations aux dérivées partielles, Coll. Math. Appl. Maîtrise, Masson, Paris, 1983. Google Scholar

  • [33]

    Raymond J.-P., Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 6, 921–951. Google Scholar

  • [34]

    San Martín J. A., Starovoitov V. and Tucsnak M., Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal. 161 (2002), no. 2, 113–147. Google Scholar

  • [35]

    Sylvester J. and Uhlmann G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169. Google Scholar

  • [36]

    Takahashi T., Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations 8 (2003), no. 12, 1499–1532. Google Scholar

  • [37]

    Takahashi T. and Tucsnak M., Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid, J. Math. Fluid Mech. 6 (2004), no. 1, 53–77. Google Scholar

About the article

Received: 2014-08-04

Revised: 2015-09-25

Accepted: 2015-10-14

Published Online: 2015-12-09

Published in Print: 2017-02-01

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 1, Pages 1–21, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2014-0056.

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