Acta Applicandae Mathematicae

, Volume 160, Issue 1, pp 101–128 | Cite as

Global Well-Posedness of an Initial-Boundary Value Problem of the 2-D Incompressible Navier-Stokes-Darcy System

  • Pan LiuEmail author
  • Wenjuan Liu


We investigate in the paper the initial boundary value problem of the two dimensional incompressible Navier-Stokes-Darcy system in a strip domain. It is shown that, without any small initial data assumption, the Navier-Stokes-Darcy equations have a unique global strong solution in the strip domain with a flat interface. The key is to establish the global-in-time regularity uniformly by pursuing the properties of Dirichlet-Neumann operator.


Navier-Stokes-Darcy system Global well-posedness Dirichlet-Neumann operator 

Mathematics Subject Classification

35A07 74F10 76D03 



The authors would like to thank Professor Guilong Gui for helpful discussions. This work of the authors is partially supported by the National Natural Science Foundation of China under the grants 11571279 and 11331005. The authors would like to thank the referees for constructive suggestions and comments.


  1. 1.
    Amrouche, C., Girault, V., Giroire, J.: Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator an approach in weighted Sobolev spaces. J. Math. Pures Appl. 76, 55–81 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arbogast, T., Brunson, D.: A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Comput. Geosci. 11, 207–218 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Badea, L., Discacciati, M., Quarteroni, A.: Numerical analysis of the Navier-Stokes/Darcy coupling. Numer. Math. 115, 195–227 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally impermeable wall. J. Fluid Mech. 30, 197–207 (1967) CrossRefGoogle Scholar
  5. 5.
    Behrndt, J., Rohleder, J.: Spectral analysis of self-adjoint elliptic differential operators, Dirichlet-to-Neumann maps, and abstract Weyl functions. Adv. Math. 285, 1301–1338 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Behrndt, J., Terelst, A.F.M.: Dirichlet-to-Neumann maps on bounded Lipschitz domains. J. Differ. Equ. 259, 5903–5926 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brenner, S.C.: Korn’s inequalities for piecewise \(H^{1}\) vector fields. Math. Comput. 73, 1067–1087 (2004) CrossRefzbMATHGoogle Scholar
  8. 8.
    Cesmelioglu, A., Girault, V., Rivivére, B.: Time-dependent coupling of Navier-Stokes and Darcy flows. Math. Model. Numer. Anal. 47, 539–554 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cesmelioglu, A., Riviére, B.: Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. J. Numer. Math. 16(4), 249–280 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chidyagwai, P.: A multilevel decoupling method for the Navier-Stokes/Darcy model. J. Comput. Appl. Math. 325, 74–96 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Du, G., Zuo, L.: Local and parallel finite element method for the mixed Navier-Stokes/Darcy model with Beavers-Joseph-Saffman interface conditions. Acta Math. Sci. Ser. B Engl. Ed. 37, 1331–1347 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Girault, V., Riviére, B.: DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Numer. Anal. 47, 2052–2098 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Knüpfer, H., Masmoudi, N.: Well-posedness and uniform bounds for a nonlocal third order evolution operator on an infinite wedge. Commun. Math. Phys. 320, 395–424 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Knüpfer, H., Masmoudi, N.: Darcy’s flow with prescribed contact angle: well-posedness and lubrication approximation. Arch. Ration. Mech. Anal. 218, 589–646 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2003) zbMATHGoogle Scholar
  17. 17.
    Mardal, K.A., Tai, X., Winther, R.: A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal. 40, 1605–1631 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ren, X., Xiang, Z., Zhang, Z.: Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain. Nonlinearity 29, 1257–1291 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Riviére, B.: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22, 479–500 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Riviére, B., Yotov, I.: Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal. 42, 1959–1977 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Saffman, P.G.: On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50, 93–101 (1971) CrossRefzbMATHGoogle Scholar
  22. 22.
    Stoker, S.F., Müller, P., Cicalese, L., et al.: A diffuse interface method for the Navier-Stokes/Darcy equations: perfusion profile for a patient-specific human liver based on MRI scans. Comput. Methods Appl. Mech. Eng. 321, 70–102 (2017) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of MathematicsNorthwest UniversityXi’anChina

Personalised recommendations