Abstract
The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the nonlinear Darcy–Forchheimer–Brinkman system and the linear Stokes system in two complementary Lipschitz domains in \({\mathbb{R}^{3}}\), one of them is a bounded Lipschitz domain \({\Omega}\) with connected boundary, and the other one is the exterior Lipschitz domain \({\mathbb{R}^{3} \setminus \overline{\Omega }}\). We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces.
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References
Adams R.A.: Sobolev Spaces. Academic Press, Amsterdam (2003)
Agranovich M.S.: Elliptic singular integro-differential operators. Russ. Math. Surv. 20, 1–121 (1965)
Alliot F., Amrouche C.: The stokes problem in \({\mathbb{R}^{n}}\): an approach in weighted Sobolev spaces. Math. Models Methods Appl. Sci. 9, 723–754 (1999)
Alliot F., Amrouche C.: Weak solutions for the exterior Stokes problem in weighted Sobolev spaces. Math. Methods Appl. Sci. 23, 575–600 (2000)
Amrouche C., Meslameni M.: Stokes problem with several types of boundary conditions in an exterior domain. Electron. J. Differ. Equ. 2013(196), 1–28 (2013)
Amrouche C., Nguyen H.H.: L p-weighted theory for Navier–Stokes equations in exterior domains. Commun. Math. Anal. 8, 41–69 (2010)
Amrouche C., Razafison U.: On the Oseen problem in three-dimensional exterior domains. Anal. Appl. 4, 133–162 (2006)
Amrouche C., Rodríguez-Bellido M.A.: Stationary Stokes, Oseen and Navier–Stokes equations with singular data. Arch. Ration. Mech. Anal. 199, 597–651 (2011)
Amrouche C., Seloula N.: Lp-theory for vector potentials and Sobolev’s inequalities for vector fields: application to the Stokes equations with pressure boundary conditions. Math. Models Methods Appl. Sci. 23, 37–92 (2013)
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, Department of Hydromechanics and Modelling of Hydrosystems, University Stuttgart, Germany (2014)
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)
Băcuţă C., Hassell M.E., Hsiao G.C., Sayas F.-J.: Boundary integral solvers for an evolutionary exterior Stokes problem. SIAM J. Numer. Anal. 53, 1370–1392 (2015)
Chkadua O., Mikhailov S.E., Natroshvili D.: Localized direct segregated boundary-domain integral equations for variable coefficient transmission problems with interface crack. Mem. Differ. Equ. Math. Phys. 52, 17–64 (2011)
Chkadua O., Mikhailov S.E., Natroshvili D.: Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks. Numer. Meth. PDE 27, 121–140 (2011)
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)
Deuring P.: The resolvent problem for the Stokes system in exterior domains: an elementary approach. Math. Methods 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)
Dautray R., Lions J.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4: Integral Equations and Numerical Methods. Springer, Berlin (1990)
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)
Girault V., Sequeira A.: A well-posed problem for the exterior Stokes equations in two and three dimensions. Arch. Ration. Mech. Anal. 114, 313–333 (1991)
Grisvard P.: Elliptic Problems in Nonsmooth Domains. Pitman Advanced Pub. Program, Boston (1985)
Hanouzet B.: Espaces de Sobolev avec poids—application au problème de Dirichlet dans un demi-espace. Rend. Ser. Math. Univ. Padova 46, 227–272 (1971)
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 Resour. 42, 71–90 (2012)
Jerison D.S., Kenig C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)
Kantorovich L.V., Akilov G.P.: Functional Analysis. Pergamon Press, Oxford (1982)
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
M., Lanza de Cristoforis M., Wendland W.L.: Poisson problems for semilinear Brinkman systems on Lipschitz domains in \({\mathbb{R}^{3}}\). Z. Angew. Math. Phys. 66, 833–864 (2015)
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)
Kohr, M., Mikhailov, S.E.: Dirichlet-transmission problems for the Navier–Stokes and Darcy–Forchheimer–Brinkman systems in Lipschitz domains with interior cuts (in preparation)
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)
Lang J., Méndez O.: Potential techniques and regularity of boundary value problems in exterior non-smooth domains: regularity in exterior domains. Potential Anal. 24, 385–406 (2006)
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)
Maz’ya V.: Sobolev Spaces (with Applications to Elliptic Partial Differential Equations). Springer, Berlin (2011)
McCracken M.: The resolvent problem for the Stokes equations on half-space in \({L_{p}^{*}}\). SIAM J. Math. Anal. 12, 201–228 (1981)
McLean W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Medková D.: Integral equation method for the first and second problems of the Stokes system. Potential Anal. 39, 389–409 (2013)
Medková D.: Integral equations method and the transmission problem for the Stokes system. Kragujevac J. Math. 39, 53–71 (2015)
Mikhailov 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.: Localized direct boundary-domain integro-differential formulations for scalar nonlinear boundary-value problems with variable coefficients. J. Eng. Math. 51, 283–302 (2005)
Mikhailov S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378, 324–342 (2011)
Mikhailov S.E.: Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains. J. Math. Anal. Appl. 400, 48–67 (2013)
Mitrea, D., Mitrea, M., Shi, Q.: Variable coefficient transmission problems and singular integral operators on non-smooth manifolds. J. Integr. 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. N.Y. 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 (2012), viii+241 pp.
Nield D.A., Bejan A.: Convection in Porous Media. 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 Transf. 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 Transf. 38, 2647–2655 (1995)
Razafison U.: The stationary Navier–Stokes equations in 3D exterior domains. An approach in anisotropically weighted L q spaces. J. Differ. Equ. 245, 2785–2801 (2008)
Russo R., Tartaglione A.: On the Robin problem for Stokes and Navier–Stokes systems. Math. Models Methods Appl. Sci. 19, 701–716 (2006)
Sayas F.-J., Selgas V.: Variational views of Stokeslets and stresslets. SeMA 63, 65–90 (2014)
Sohr H.: The Navier–Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser Verlag, Basel (2001)
Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)
Varnhorn W.: The Stokes Equations. Akademie Verlag, Berlin (1994)
Varnhorn W.: An explicit potential theory for the Stokes resolvent boundary value problems in three dimensions. Manuscr. Math. 70, 339–361 (1991)
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Kohr, M., Lanza de Cristoforis, M., Mikhailov, S.E. et al. Integral potential method for a transmission problem with Lipschitz interface in \(\mathbb{R}^{3}\) for the Stokes and Darcy–Forchheimer–Brinkman PDE systems. Z. Angew. Math. Phys. 67, 116 (2016). https://doi.org/10.1007/s00033-016-0696-1
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DOI: https://doi.org/10.1007/s00033-016-0696-1