Abstract
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.
Similar content being viewed by others
References
Amrouche C., Rodríguez-Bellido M.A.: Stationary Stokes, Oseen and Navier–Stokes equations with singular data. Arch. Ration. Mech. Anal. 199, 597–651 (2011)
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
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)
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)
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)
Choe H.J., Kim H.: Dirichlet problem for the stationary Navier–Stokes system on Lipschitz domains. Commun. Partial Differ. Equ. 36, 1919–1944 (2011)
Costabel M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)
Cwikel M.: Real and complex interpolation and extrapolation of compact operators. Duke Math. J. 65, 333–343 (1992)
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)
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)
Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, 1341, Springer, Berlin (1988)
Deuring P.: The Resolvent problem for the Stokes system in exterior domains: an elementary approach. Math. Meth. Appl. Sci. 13, 335–349 (1990)
Dindos̆ M., Mitrea M.: Semilinear Poisson problems in Sobolev–Besov spaces on Lipschitz domains. Publ. Math. 46, 353–403 (2002)
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)
Escauriaza L., Mitrea M.: Transmission problems and spectral theory for singular integral operators on Lipschitz domains. J. Funct. Anal. 216, 141–171 (2004)
Fabes E., Kenig C., Verchota G.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1988)
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)
Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Vol. I, II. Springer, Berlin (1998)
Hsiao G.C., Wendland W.L.: Boundary Integral Equations: Variational Methods. Springer, Heidelberg (2008)
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)
Jerison D.S., Kenig C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)
Klingelhöfer, K.: Nonlinear harmonic boundary value problems. I. Arch. Ration. Mech. Anal. 31, 364–371 (1968/1969)
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)
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)
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
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)
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)
Kohr M., Pop I.: Viscous Incompressible Flow for Low Reynolds Numbers. WIT Press, Southampton (2004)
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)
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)
Lanzani L., Shen Z.: On the Robin boundary condition for Laplace’s equation in Lipschitz domains. Commun. Partial Differ. Equ. 29, 91–109 (2004)
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)
Medková D.: Transmission problem for the Brinkman system. Complex Var. Elliptic Equ. 59, 1664–1678 (2014)
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)
Mikhailov S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378, 324–342 (2011)
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)
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)
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)
Mitrea M., Taylor M.: Navier–Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann. 321, 955–987 (2001)
Mitrea, M., Wright, M.: Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque. 344, viii+241 (2012)
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)
Nield D.A., Bejan A.: Convection in Porous Media, 3rd edn. Springer, New York (2013)
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)
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)
Russo, R.: On Stokes’ Problem. In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics, pp. 473–511. Springer, Berlin (2010)
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)
Russo R., Tartaglione A.: On the Robin problem for Stokes and Navier–Stokes systems. Math. Models Methods Appl. Sci. 19, 701–716 (2006)
Scharf B., Schmeiβer H.J., Sickel W.: Traces of vector-valued Sobolev spaces. Math. Nachr. 285, 1082–1106 (2012)
Shen Z.: Resolvent estimates in \({L^p}\) for the Stokes operator in Lipschitz domains. Arch. Ration. Mech. Anal. 205, 395–424 (2012)
Sohr H.: The Navier–Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser Verlag, Basel (2001)
Steinbach O.: A note on the ellipticity of the single layer potential in two-dimensional elastostatics. J. Math. Anal. Appl. 294, 1–6 (2004)
Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publ. Co., Amsterdam (1978)
Vafai K., Kim S.J.: On the limitations of the Brinkman–Forchheimer-extended Darcy equation. Int. J. Heat Fluid Flow 16, 11–15 (1995)
Varnhorn W.: The Stokes Equations. Akademie Verlag, Berlin (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-015-0236-3