Abstract
Classical solutions of the Oseen problem are studied on an exterior domain Ω with Ljapunov boundary in R 3. It is proved a maximum modulus estimate of the following form: If \({{\bf u}\in C^2(\Omega)^3\cap C^0(\overline \Omega)^3}\) and \({p \in C^1(\Omega ), -\Delta {\bf u}+2\lambda \partial_1 {\bf u}+\nabla p=0, \nabla \cdot {\bf u}=0}\) in Ω, and if \({|{\bf u}| \le M}\) on \({\partial \Omega , \limsup |{\bf u}({\bf x})|\le M}\) as \({|{\bf x}|\to \infty }\) , then \({|{\bf u}({\bf x})|\le c M}\) in Ω. Here the constant c depends only on Ω and λ.
Article PDF
Similar content being viewed by others
References
Amrouche C., Bouzit H.: The scalar Oseen operator, \({-\Delta +\partial /\partial x_1}\) in R 2. Appl. Math. 53, 41–80 (2008)
Amrouche Ch., Bouzit H.: L p-inequalities for the scalar Oseen potential. J. Math. Anal. Appl. 337, 753–770 (2008)
Amrouche Ch., Razafison U.: Weighted Sobolev spaces for a scalar model of the stationary Oseen equation in R 3. J. Math. Fluid Mech. 9, 181–210 (2007)
Babenko, K.I.: On stationary solutions of the problem of flow past a body of a viscous incompressible fluid. Mat. Sb. 91(133), 3–27; Engl. Transl. Math. SSSR Sb. 20, 1–25 (1973)
Brown R., Mitrea I., Mitrea M., Wright M.: Mixed boundary value problems for the Stokes system. Trans. Am. Math. Soc. 362, 1211–1230 (2010)
Deuring P.: On volume potentials related to the time-dependent Oseen system. WSEAS Trans. Math. 5, 252–259 (2006)
Deuring, P.: On boundary-driven time-dependent Oseen flows. Banach Center Publ., 81, Part 1, Polish Acad. Sci. Inst. Math., Warsaw, 119–132 (2008)
Deuring P.: Spatial decay of time-dependent Oseen flows. SIAM J. Math. Anal. 41, 886–922 (2009)
Deuring P., Kračmar S.: Artificial boundary conditions for the Oseen system in 3D exterior domains. Analysis 20, 65–90 (2000)
Deuring P., Kračmar S.: Exterior stationary Navier-Stoke flows in 3d with non-zero velocity at infinity: approximation by flows in bounded domains. Math. Nachr. 269(270), 86–115 (2004)
Duistermaat J.J., Kolk J.A.C.: Distributions. Theory and Applications. Birkhäuser, New York (2010)
Enomoto Y., Shibata Y.: On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier–Stokes equation. J. Math. Fluid Mech. 7, 339–367 (2005)
Fabes E.B., Kenig C.E., Verchota G.C.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1988)
Farwig R.: The stationary exterior 3D-problem of Oseen and Navier–Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211, 409–447 (1992)
Farwig, R.: Das stationäre Aussenraumproblem der Navier–Stokes–Gleichungen bei nichtverschwindender anströmgeschwidigkeit in anisotrop gewichteten Sobolev-räumen. SFB 256 preprint n. 110, University of Bonn, Habilitationsschrift (1990)
Farwig, R., Sohr, H.: Weighted estimates for the Oseen equations and the Navier–Stokes equations in exterior domains. Ser. Adv. Math. Appl. Sci., vol. 47, pp. 11–30. World Sci. Publ., River Edge, NJ (1998)
Finn R.: On steady state solutions of the Navier–Stokes partial differential equations. Arch. Ration. Mech. Anal. 3, 381–396 (1959)
Finn R.: Estimates at infinity for stationary solutions of the Navier–Stokes equations. Bull. Math. Soc. Sci. Math. Phys. R. P. Roum. (N.S.) 51, 387–418 (1959)
Finn R.: On the exterior stationary problem for the Navier–Stokes equations and associated perturbation problems. Arch. Ration. Mech. Anal. 19, 363–406 (1965)
Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations I, Linearised Steady Problems. Springer Tracts in Natural Philosophy, vol. 38. Springer, Berlin (1998)
Goldberg S.: Unbounded Linear Operators. Theory and Applications. McGraw-Hill Book Company, USA (1966)
Kenig, C.E.: Recent progress on boundary value problems on Lipschitz domains. Pseudodifferential operators and Applications. In: Proceedings of Symposium Notre Dame/ Indiana 1984. Proc. Symp. Pure Math. vol. 43, pp. 175–205 (1985)
Kobayashi T., Shibata Y.: On the Oseen equation in the three dimensional exterior domains. Math. Ann. 310, 1–45 (1998)
Kračmar S., Novotný A., Pokorný M.: Estimates of three-dimensional Oseen kernels in weighted L p spaces. In: Sequiera, A., da Veiga, H.B., Videman, J.H. (eds) Applied Nonlinear Analysis, Kluwer/Plenum Publishers, New York (1999)
Kračmar S., Novotný A., Pokorný M.: Estimates of Oseen kernels in weighted L p spaces. J. Math. Soc. Jpn. 53, 59–111 (2001)
Kratz W.: On the maximum modulus theorem for Stokes functions. Appl. Anal. 58, 293–302 (1995)
Kratz W.: The maximum modulus theorem for the Stokes system in a ball. Math. Z. 226, 389–403 (1997)
Kratz W.: An extremal problem related to the maximum modulus theorem for Stokes functions. Z. Anal. Anwend. 17, 599–613 (1998)
Ladyzenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)
Maremonti P.: On the Stokes equations: the maximum modulus theorem. Math. Models Meth. Appl. Sci. 10, 1047–1072 (2000)
Maremonti P., Russo R.: On the maximum modulus theorem for the Stokes system. Ann. Sc. Norm. Super. Pisa XXI, 629–643 (1994)
Medková D.: Integral representation of a solution of the Neumann problem for the Stokes system. Numer. Algorithm. 54, 459–484 (2010)
Medková D., Varnhorn W.: Boundary value problems for the Stokes equations with jumps in open sets. Appl. Anal. 87, 829–849 (2008)
Mitrea D.: A generalization of Dahlberg’s theorem concerning the regularity of harmonic Green potentials. Trans. Am. Math. Soc. 360, 3771–3793 (2008)
Odquist F.K.G.: Über die Randwertaufgaben in der Hydrodynamik zäher Flüssigkeiten. Math. Z. 32, 329–375 (1930)
Oseen C.W.: Über die Stokesche Formel und Über eine Verwandte Aufgabe in der Hydrodynamik. Ark. Mat. Astron. Fys. 29, 1–20 (1910)
Pokorný M.: Comportement Asymptotique des Solutions de Quelques Equations aux Derivees Partielles Decrivant l’ecoulement de Fluides dans les Domaines Non-bornes. These de Doctorat. Universite de Toulon et Du Var, Universite Charles de Prague, Prague (1999)
Pólya G.: Liegt die Stelle der gröbsten Beanspruchung an der Oberfläche?. Zeitschr. Ang. Math. Mech. 10, 353–360 (1930)
Shilov, G.E.: Mathematical Analysis. Second special course. Nauka, Moskva (1965) (Russian)
Schulze B.W., Wildenhein G.: Methoden der Potentialtheorie für elliptisch Differentialgleichungen beliebiger Ordnung. Akademie, Berlin (1977)
Smirnov, W. I.: Lehrgang der höheren Mathematik IV, V. Berlin (1961 and 1962)
Verchota G.: Layer potentials and regularity for Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)
Vladimirov V.S.: Uravnenia Matematicheskoj Fiziki. Nauka, Moscow (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of Š. N. and D. M. was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan no. AV0Z10190503 and by the Grant Agency of the Czech Republic, grant No. P201/11/1304. The research of S. K. was supported by the research plans of the Ministry of Education of the Czech Republic No. 6840770010 and by the Grant Agency of the Czech Republic, grant No. P201/11/1304. The research of W.V. was partially supported by Nečas Centrum for Mathematical Modeling LC06052 financed by MSMT.
Rights and permissions
About this article
Cite this article
Kračmar, S., Medková, D., Nečasová, Š. et al. A maximum modulus theorem for the Oseen problem. Annali di Matematica 192, 1059–1076 (2013). https://doi.org/10.1007/s10231-012-0258-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-012-0258-x
Keywords
- Oseen problem
- Maximum modulus theorem
- Oseen potentials
- Uniqueness
- Non-tangential limit
- Theorem of Liouville type