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On the Robin-Transmission Boundary Value Problems for the Nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes Systems

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The purpose of this paper is to study a boundary value problem of Robin-transmission type for the nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes systems in two adjacent bounded Lipschitz domains in \({{\mathbb{R}}^{n} (n\in \{2,3\})}\), with linear transmission conditions on the internal Lipschitz interface and a linear Robin condition on the remaining part of the Lipschitz boundary. We also consider a Robin-transmission problem for the same nonlinear systems subject to nonlinear transmission conditions on the internal Lipschitz interface and a nonlinear Robin condition on the remaining part of the boundary. For each of these problems we exploit layer potential theoretic methods combined with fixed point theorems in order to show existence results in Sobolev spaces, when the given data are suitably small in \({L^2}\)-based Sobolev spaces or in some Besov spaces. For the first mentioned problem, which corresponds to linear Robin and transmission conditions, we also show a uniqueness result. Note that the Brinkman–Forchheimer-extended Darcy equation is a nonlinear equation that describes saturated porous media fluid flows.

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References

  1. Amrouche C., Rodríguez-Bellido M.A.: Stationary Stokes, Oseen and Navier–Stokes equations with singular data. Arch. Ration. Mech. Anal. 199, 597–651 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baber, K.I.: Coupling Free Flow and Flow in Porous Media in Biological and Technical Applications: From a Simple to a Complex Interface Descrition. PhD Thesis (2014) Department of Hydromechanics and Modelling of Hydrosystems, University Stuttgart, Germany

  3. Baber K., Mosthaf K., Flemisch B., Helmig R., Müthing S., Wohlmuth B.: Numerical scheme for coupling two-phase compositional porous-media flow and one-phase compositional free flow. IMA J. Appl. Math. 77, 887–909 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chkadua O., Mikhailov S.E., Natroshvili D.: Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients. Integr. Equ. Oper. Theory. 76, 509–547 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chkadua O., Mikhailov S.E., Natroshvili D.: Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains. Anal. Appl. 11(4), 1350006 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Choe H.J., Kim H.: Dirichlet problem for the stationary Navier–Stokes system on Lipschitz domains. Commun. Partial Differ. Equ. 36, 1919–1944 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Costabel M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cwikel M.: Real and complex interpolation and extrapolation of compact operators. Duke Math. J. 65, 333–343 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dahlberg B.E.J., Kenig C.: Hardy spaces and the Neumann problem in \({L^p}\) for Laplace’s equation in Lipschitz domains. Ann. Math. 125, 437–465 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dalla Riva M., Lanzade Cristoforis M.: Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach. Complex Var. Elliptic Equ. 55, 771–794 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, 1341, Springer, Berlin (1988)

  12. Deuring P.: The Resolvent problem for the Stokes system in exterior domains: an elementary approach. Math. Meth. Appl. Sci. 13, 335–349 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dindos̆ M., Mitrea M.: Semilinear Poisson problems in Sobolev–Besov spaces on Lipschitz domains. Publ. Math. 46, 353–403 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dindos̆ M., Mitrea M.: The stationary Navier–Stokes system in nonsmooth manifolds: the Poisson problem in Lipschitz and \({C^1}\) domains. Arch. Ration. Mech. Anal. 174, 1–47 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Escauriaza L., Mitrea M.: Transmission problems and spectral theory for singular integral operators on Lipschitz domains. J. Funct. Anal. 216, 141–171 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fabes E., Kenig C., Verchota G.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fabes E., Mendez O., Mitrea M.: Boundary layers on Sobolev–Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159, 323–368 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Vol. I, II. Springer, Berlin (1998)

    Google Scholar 

  19. Hsiao G.C., Wendland W.L.: Boundary Integral Equations: Variational Methods. Springer, Heidelberg (2008)

    Book  MATH  Google Scholar 

  20. Jackson A.S., Rybak I., Helmig R., Gray W.G., Miller C.T.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 9. Transition region models. Adv. Water Resourc. 42, 71–90 (2012)

    Article  ADS  Google Scholar 

  21. Jerison D.S., Kenig C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Klingelhöfer, K.: Nonlinear harmonic boundary value problems. I. Arch. Ration. Mech. Anal. 31, 364–371 (1968/1969)

  23. 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)

  24. 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)

  25. Kohr, M., Lanza de Cristoforis, M., Wendland, W.L.: Nonlinear Darcy–Forchheimer–Brinkman system with linear Robin boundary conditions in Lipschitz domains. In: Aliev Azeroglu, T., Golberg, A., Rogosin, S. (eds.) Complex Analysis and Potential Theory, pp. 111–124. Cambridge Scientific Publishers, Cambridge (2014). ISBN:978-1-908106-40-7

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

  27. Kohr M., Pintea C., Wendland W.L.: Layer potential analysis for pseudodifferential matrix operators in Lipschitz domains on compact Riemannian manifolds: applications to pseudodifferential Brinkman operators. Int. Math. Res. Notices 19, 4499–4588 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Kohr M., Pop I.: Viscous Incompressible Flow for Low Reynolds Numbers. WIT Press, Southampton (2004)

    MATH  Google Scholar 

  29. Lanza de Cristoforis M., Musolino P.: Asymptotic behaviour of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: a functional analytic approach. Complex Var. Elliptic Equ. 52, 945–977 (2007)

  30. Lanzani L., Méndez O.: The Poisson’s problem for the Laplacian with Robin boundary condition in non-smooth domains. Rev. Mat. Iberoam. 22, 181–204 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lanzani L., Shen Z.: On the Robin boundary condition for Laplace’s equation in Lipschitz domains. Commun. Partial Differ. Equ. 29, 91–109 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Leray J.: Étude de diverses équations intégrales non linéaires et de quelqes problémes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)

    MathSciNet  MATH  Google Scholar 

  33. Medková D.: Transmission problem for the Brinkman system. Complex Var. Elliptic Equ. 59, 1664–1678 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mikhailov S.E.: Direct localized boundary-domain integro-differential formulations for physically nonlinear elasticity of inhomogeneous body. Eng. Anal. Bound. Elem. 29, 1008–1015 (2005)

    Article  MATH  Google Scholar 

  35. Mikhailov S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378, 324–342 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mitrea D., Mitrea M., Qiang S.: Variable coefficient transmission problems and singular integral operators on non-smooth manifolds. J. Integral Equ. Appl. 18, 361–397 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mitrea M., Monniaux S., Wright M.: The Stokes operator with Neumann boundary conditions in Lipschitz domains. J. Math. Sci. (New York) 176(3), 409–457 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mitrea M., Taylor M.: Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev–Besov space results and the Poisson problem. J. Funct. Anal. 176, 1–79 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mitrea M., Taylor M.: Navier–Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann. 321, 955–987 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mitrea, M., Wright, M.: Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque. 344, viii+241 (2012)

  41. Mosthaf, K., Baber, K., Flemisch, B., Helmig, R., Leijnse, A., Rybak, I., Wohlmuth, B.: A coupling concept for two-phase compositional porous-medium and single-phase compositional free flow. Water Resour. Res. 47, W10522 (2011)

  42. Nield D.A., Bejan A.: Convection in Porous Media, 3rd edn. Springer, New York (2013)

    Book  MATH  Google Scholar 

  43. Ochoa-Tapia J.A., Whitacker S.: Momentum transfer at the boundary between porous medium and homogeneous fluid. I. Theoretical development. Int. J. Heat Mass Transfer 38, 2635–2646 (1995)

    Article  MATH  Google Scholar 

  44. Ochoa-Tapia J.A., Whitacker S.: Momentum transfer at the boundary between porous medium and homogeneous fluid. II. Comparison with experiment. Int. J. Heat Mass Transfer 38, 2647–2655 (1995)

    Article  Google Scholar 

  45. Russo, R.: On Stokes’ Problem. In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics, pp. 473–511. Springer, Berlin (2010)

  46. Russo A., Starita G.: On the existence of steady-state solutions to the Navier–Stokes system for large fluxes. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 7, 171–180 (2008)

    MathSciNet  MATH  Google Scholar 

  47. Russo R., Tartaglione A.: On the Robin problem for Stokes and Navier–Stokes systems. Math. Models Methods Appl. Sci. 19, 701–716 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. Scharf B., Schmeiβer H.J., Sickel W.: Traces of vector-valued Sobolev spaces. Math. Nachr. 285, 1082–1106 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. Shen Z.: Resolvent estimates in \({L^p}\) for the Stokes operator in Lipschitz domains. Arch. Ration. Mech. Anal. 205, 395–424 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  51. Steinbach O.: A note on the ellipticity of the single layer potential in two-dimensional elastostatics. J. Math. Anal. Appl. 294, 1–6 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  52. Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publ. Co., Amsterdam (1978)

    MATH  Google Scholar 

  53. Vafai K., Kim S.J.: On the limitations of the Brinkman–Forchheimer-extended Darcy equation. Int. J. Heat Fluid Flow 16, 11–15 (1995)

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

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Correspondence to Wolfgang L. Wendland.

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Communicated by M. Feistauer

M. Kohr acknowledges the support of the Grant PN-II-ID-PCE-2011-3-0994 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI.

M. Lanza de Cristoforis acknowledges the support of “INdAM GNAMPA Project 2015—Un approccio funzionale analitico per problemi di perturbazione singolare e di omogeneizzazione” and of “Progetto di Ateneo: Singular perturbation problems for differential operators—CPDA120171/12”—University of Padova.

The research of this work was also partially supported by the Grant EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK. The authors are indebted to Professor Sergey E. Mikhailov for valuable suggestions and discussions concerning extension and conormal derivative operators. The authors also wish to thank Professor Dr.-Ing. Rainer Helmig for a number of references on potential applications.

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Kohr, M., de Cristoforis, M.L. & 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). https://doi.org/10.1007/s00021-015-0236-3

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