Skip to main content

The Robin problem for the Brinkman system and for the Darcy–Forchheimer–Brinkman system

Abstract

In this paper, we study the Neumann problem and the Robin problem for the Darcy–Forchheimer–Brinkman system in \(W^{1,q}(\Omega ,{\mathbb {R}}^m)\times L^q(\Omega )\) for a bounded domain \(\Omega \subset {\mathbb {R}}^m\) with Lipschitz boundary. First, we study the Neumann problem and the Robin problem for the Brinkman system by the integral equation method. If \(\Omega \subset {\mathbb {R}}^m\) is a bounded domain with Lipschitz boundary and \(2\le m\le 3\), then we prove the unique solvability of the Neumann problem and the Robin problem for the Brinkman system in \(W^{1,q}(\Omega ,{\mathbb {R}}^m)\times L^q(\Omega )\), where \(3/2<q<3\). Then we get results for the Darcy–Forchheimer–Brinkman system from the results for the Brinkman system using the fixed point theorem. If \(\Omega \subset {\mathbb {R}}^m\) is a bounded domain with Lipschitz boundary, \(2\le m\le 3\), \(3/2<q<3\), then we prove the existence of a solution of the Neumann problem and the Robin problem for the Darcy–Forchheimer–Brinkman system in \(W^{1,q}(\Omega ,{\mathbb {R}}^m)\times L^q(\Omega )\) for small given data.

This is a preview of subscription content, access via your institution.

References

  1. Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin Heidelberg (1996)

    Book  Google Scholar 

  2. Behzadan, A., Holst, M.: Multiplication in Sobolev Spaces. revisited, arXiv:1512.07379v1

  3. Berg, J., Löström, J.: Interpolation Spaces. An Introduction. Springer, Berlin-Heidelberg-New York (1976)

    Book  Google Scholar 

  4. Dobrowolski, M.: Angewandte Functionanalysis. Functionanalysis, Sobolev-Räume und elliptische Differentialgleichungen. Springer, Berlin Heidelberg (2006)

    Google Scholar 

  5. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady State Problems. Springer, New York–Dordrecht–Heidelberg–London (2011)

    MATH  Google Scholar 

  6. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. SIAM, Philadelphia (2011)

    Book  Google Scholar 

  7. Grosan, T., Kohr, M., Wendland, W.L.: Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in Lipschitz domains. Math. Methods Appl. Sci. 38, 3615–3628 (2015)

    MathSciNet  Article  Google Scholar 

  8. Gutt, R., Grosan, T.: On the lid-driven problem in a porous cavity: a theoretical and numerical approach. Appl. Math. Comput. 266, 1070–1082 (2015)

    MathSciNet  Google Scholar 

  9. Gutt, R., Kohr, M., Mikhailov, S.E., Wendland, W.L.: On the mixed problem for the semilinear Darcy–Forchheimer–Brinkman PDE system in Besov spaces on creased Lipschitz domains. Math. Methods Appl. Sci. 40(18), 7780–7829 (2017)

    MathSciNet  Article  Google Scholar 

  10. Jonsson, A., Wallin, H.: Function Spaces on Subsets of \(R^n\). Harwood Academic Publishers, London (1984)

    MATH  Google Scholar 

  11. Kohr, M., Lanza de Cristoforis, M., Mikhailov, S.E., Wendland, W.L.: Integral potential method for a transmission problem with Lipschitz interface in \(R^3\) for the Stokes and Darcy–Forchheimer–Brinkman PDE systems. Z. Angew. Math. Phys. 67, 116 (2016)

    Article  Google Scholar 

  12. Kohr, M., Lanza de Cristoforis, M., Wendland, W.L.: Nonlinear Neumann-transmission problems for Stokes and Brinkman equations on Euclidean Lipschitz domains. Potential Anal. 38, 1123–1171 (2013)

    MathSciNet  Article  Google Scholar 

  13. Kohr, M., Lanza de Cristoforis, M., Wendland, W.L.: Boundary value problems of Robin type for the Brinkman and Darcy–Forchheimer–Brinkman systems in Lipschitz domains. J. Math. Fluid Mech. 16, 595–630 (2014)

    MathSciNet  Article  Google Scholar 

  14. Kohr, M., Lanza de Cristoforis, M., Wendland, W.L.: Poisson problems for semilinear Brinkman systems on Lipschitz domains in \(R^n\). Z. Angew. Math. Phys. 66, 833–864 (2015)

    MathSciNet  Article  Google Scholar 

  15. Kohr, M., Lanza de Cristoforis, M., Wendland, W.L.: On the Robin-transmission boundary value problems for the nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes systems. J. Math. Fluid Mech. 18, 293–329 (2016)

    MathSciNet  Article  Google Scholar 

  16. Kohr, M., Medková, D., Wendland, W.L.: On the Oseen–Brinkman flow around an \((m-1)\)-dimensional solid obstacle. Monatsh. Math. 183, 269–302 (2017)

    MathSciNet  Article  Google Scholar 

  17. Kohr, M., Mikhailov, S.E., Wendland, W.L.: Transmission problems for the Navier–Stokes and Darcy–Forchheimer–Brinkman systems in Lipschitz domains on compact Riemannian manifolds. J. Math. Fluid Mech. 19, 203–238 (2017)

    MathSciNet  Article  Google Scholar 

  18. Kufner, A., John, O., Fučík, S.: Function Spaces. Academia, Prague (1977)

    MATH  Google Scholar 

  19. Maz’ya, V., Mitrea, M., Shaposhnikova, T.: The inhomogenous Dirichlet problem for the Stokes system in Lipschitz domains with unit normal close to \(VMO^*\). Funct. Anal. Appl. 43, 217–235 (2009)

    MathSciNet  Article  Google Scholar 

  20. Medková, D.: Regularity of solutions of the Neumann problem for the Laplace equation. Le Matematiche LXI, 287–300 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Medková, D.: Bounded solutions of the Dirichlet problem for the Stokes resolvent system. Complex Var. Elliptic Equ. 61, 1689–1715 (2016)

    MathSciNet  Article  Google Scholar 

  22. Mitrea, I., Mitrea, M.: Multi-Layer Potentials and Boundary Problems for Higher-Order Elliptic Systems in Lipschitz Domains. Springer, Berlin Heidelberg (2013)

    Book  Google Scholar 

  23. Mitrea, M., Wright, M.: Boundary Value Problems for the Stokes System in Arbitrary Lipschitz Domains, vol. 344. Astérisque, Paris (2012)

    MATH  Google Scholar 

  24. Nield, D.A., Bejan, A.: Convection in Porous Media. Springer, New York (2013)

    Book  Google Scholar 

  25. Sohr, H.: The Navier–Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser, Basel–Boston–Berlin (2001)

    MATH  Google Scholar 

  26. Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin Heidelberg (2007)

    MATH  Google Scholar 

  27. Temam, R.: Navier–Stokes Equations. North Holland, Amsterdam (1979)

    MATH  Google Scholar 

  28. Triebel, H.: Höhere Analysis. VEB Deutscher Verlag der Wissenschaften, Berlin (1972)

    MATH  Google Scholar 

  29. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin (1978)

    MATH  Google Scholar 

  30. Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel–Boston–Stuttgart (1983)

    Book  Google Scholar 

  31. Triebel, H.: Theory of Function Spaces III. Birkhäuser, Basel (2006)

    MATH  Google Scholar 

  32. Varnhorn, W.: The Stokes Equations. Akademie Verlag, Berlin (1994)

    MATH  Google Scholar 

  33. Wolf, J.: On the local pressure of the Navier–Stokes equations and related systems. Adv. Differ. Equ. 22, 305–338 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Yosida, K.: Functional Analysis. Springer, Berlin (1965)

    MATH  Google Scholar 

  35. Ziemer, W.P.: Weakly Differentiable Functions. Springer, New York (1989)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dagmar Medková.

Additional information

The work was supported by RVO: 67985840 and GAČR Grant No. 17-01747S.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Medková, D. The Robin problem for the Brinkman system and for the Darcy–Forchheimer–Brinkman system. Z. Angew. Math. Phys. 69, 132 (2018). https://doi.org/10.1007/s00033-018-1020-z

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-018-1020-z

Mathematics Subject Classification

  • 35Q35

Keywords

  • Brinkman system
  • Neumann problem
  • Robin problem
  • Darcy–Forchheimer–Brinkman system
  • Boundary layer potentials