Skip to main content

The Rot-Div System in Exterior Domains


The goal of this paper is to reconsider the classical elliptic system rot vf, div vg in simply connected domains with bounded connected boundaries (bounded and exterior sets). The main result shows solvability of the problem in the maximal regularity regime in the L p -framework taking into account the optimal/minimal requirements on the smoothness of the boundary. A generalization for the Besov spaces is studied, too, for \({{\bf f} \in \dot B^s_{p,q}(\Omega)}\) for \({-1+\frac 1p < s < \frac 1p}\). As a limit case we prove the result for \({{\bf f} \in \dot B^0_{3,1}(\Omega)}\), provided the boundary is merely in \({B^{2-1/3}_{3,1}}\). The dimension three is distinguished due to the physical interpretation of the system. In other words we revised and extended the classical results of Friedrichs (Commun Pure Appl Math 8;551–590, 1955) and Solonnikov (Zap Nauch Sem LOMI 21:112–158, 1971).


  1. 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(1), 37–92 (2013)

  2. Auchmuty G., Alexander J.C.: L2-well-posedness of 3D div-curl boundary value problems. Q. Appl. Math. 63(3), 479–508 (2005)

    MathSciNet  Google Scholar 

  3. Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, vol. 129. Academic Press, Inc., Boston (1988)

  4. Besov, O.V., I’lin, V.P., Nikolskij, S.M.: Integral Function Representation and Imbedding Theorem, Moscow (1975)

  5. Bahouri H., Chemin J.Y., Danchin R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  6. Bolik, J., von Wahl, W.: Estimating ∇ u in terms of div u, curl u, either (v, u) or v × u and the topology. Math. Methods Appl. Sci. 20, 737–744 (1997)

  7. Danchin R., Mucha P.B.: A critical functional framework for the inhomogeneous Navier–Stokes equations. J. Funct. Anal. 256(3), 881–927 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  8. Danchin R., Mucha P.B.: The divergence equation in rough spaces. J. Math. Anal. Appl. 386(1), 9–31 (2012)

    MathSciNet  Article  Google Scholar 

  9. Delcourte S., Domelevo K., Omnes P.: A discrete duality finite volume approach to Hodge decomposition and div-curl problems on almost arbitrary two-dimensional meshes. SIAM J. Numer. Anal. 45(3), 1142–1174 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  10. Escher J., Giga Y., Ito K.: On a limiting motion and self-intersections for the intermediate surface diffusion flow. J. Evol. Equ. 2(3), 349–364 (2002)

    MathSciNet  Article  Google Scholar 

  11. Escher J., Mucha P.B.: The surface diffusion flow on rough phase spaces. Discrete Contin. Dyn. Syst. 26(2), 431–453 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  12. Escher J., Simonett G.: A center manifold analysis for the Mullins–Sekerka model. J. Differ. Equ. 143, 267–292 (1998)

    MathSciNet  Article  MATH  ADS  Google Scholar 

  13. Friedrichs K.O.: Differential forms on Riemannian manifolds. Commun. Pure Appl. Math. 8, 551–590 (1955)

    MathSciNet  Article  MATH  Google Scholar 

  14. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn. Springer Monographs in Mathematics. Springer, New York (2011)

  15. Griffiths D.J.: Introduction to Electrodynamics. Prentice Hall, Upper Saddle River (1999)

    Google Scholar 

  16. Jackson J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1998)

    Google Scholar 

  17. Kay J.M.: An Introduction to Fluid Mechanics and Heat Transfer: With Applications in Chemical and Mechanical Process Engineering. Cambridge University Press, Cambridge (1963)

    MATH  Google Scholar 

  18. Landau L.D., Lifshitz E.M.: Fluid Mechanics. Butterworth-Heinemann, UK (1987)

    MATH  Google Scholar 

  19. Marcinkiewicz J.: Sur les multiplicateurs des series de Fourier. Studia Math. 8, 78–91 (1939)

    Google Scholar 

  20. Milnor, J.W., Stasheff, J.D.: Characteristic classes. Annals of Mathematics Studies, vol. 76. Princeton University Press, Princeton (1974)

  21. Mucha P.B.: On weak solutions to the Stefan problem with Gibbs–Thomson correction. Differ. Integr. Equ. 20, 769–792 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Mucha P.B.: On the Stefan problem with surface tension in the Lp framework. Adv. Differ. Equ. 10, 861–900 (2005)

    MathSciNet  MATH  Google Scholar 

  23. Mucha P.B., Zajaczkowski W.: On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion. Appl. Math. (Warsaw) 27, 319–333 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Nicolaides, R.A., Wu, X.: Covolume, solutions of three-dimensional div-curl equations. SIAM J. Numer. Anal. 34(6), 2195–2203 (1997)

  25. Runst, T., Sickel, W.: Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. de Gruyter Ser. Nonlinear Anal., Berlin (1996)

  26. Solonnikov V.A.: Overdetermined elliptic boundary value problems. Zap. Nauch. Sem. LOMI 21, 112–158 (1971)

    MathSciNet  MATH  Google Scholar 

  27. Solonnikov V.A.: On the nonstationary motion of an isolated volume of a viscous incompressible fluid. Izv. Akad. Nauk SSSR 51, 1065–1087 (1987)

    Google Scholar 

  28. Tomoro A.: On smoothing effect for higher order curvature flow equations. Adv. Math. Sci. Appl. 20(2), 483–509 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Math. Library, vol. 18. North-Holland Publishing Co., Amsterdam (1978)

  30. von Wahl, W.: Estimating ∇ u by div u and curl u. Math. Methods Appl. Sci. 15, 123–143 (1992)

  31. Zajaczkowski, W.: Existence and regularity of some elliptic systems in domains with edges. Dissertationes Math. (Rozprawy Mat.) 274 (1989)

  32. Zajaczkowski, W.: Global Special Regular Solutions to the Navier-Stokes Equations in a Cylindrical Domain Under Boundary Slip Conditions. Gakuto International Series, Mathematical Sciences and Applications, vol. 21. Gakkotosho, Tokyo (2004)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Piotr B. Mucha.

Additional information

Communicated by K. Pileckas

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mucha, P.B., Pokorný, M. The Rot-Div System in Exterior Domains. J. Math. Fluid Mech. 16, 701–720 (2014).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification

  • Primary 35F35
  • Secondary 35J56
  • 35B45


  • Rot-div system
  • optimal regularity
  • exterior domain
  • Besov spaces
  • a-priori estimates