Skip to main content
SpringerLink
Log in

The Order Completion Method for Systems of Nonlinear PDEs Revisited

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion as presented in Oberguggenberger and Rosinger (Solution of continuous nonlinear PDEs through order completion, North-Holland, Amsterdam, 1994). In particular, spaces of generalized functions are constructed that contain generalized solutions to a large class of systems of continuous, nonlinear PDEs. In terms of the existence and uniqueness results previously obtained for such systems of equations (van der Walt, Acta Appl. Math. 103:1–17, 2008), one may interpret the existence of generalized solutions presented here as a regularity result. Furthermore, it is indicated how the methods developed in this paper may be adapted to solve initial and/or boundary value problems. In particular, we consider the Navier-Stokes equations in three spacial dimensions, subject to an initial condition on the velocity. In this regard, we obtain the existence of a generalized solution to a large class of such initial value problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anguelov, R.: Dedekind order completion of C(X) by Hausdorff continuous functions. Quaest. Math. 27, 153–170 (2004)

    MATH  MathSciNet  Google Scholar 

  2. Anguelov, R., Rosinger, E.E.: Hausdorff continuous solutions of nonlinear PDEs through the order completion method. Quaest. Math. 28(3), 271–285 (2005)

    MATH  MathSciNet  Google Scholar 

  3. Anguelov, R., van der Walt, J.H.: Order convergence on \(\mathcal{C}(X)\) . Quaest. Math. 28(4), 425–457 (2005)

    MATH  MathSciNet  Google Scholar 

  4. Anguelov, R., Markov, S., Sendov, B.: The set of Hausdorff continuous functions—the largest linear space of interval functions. Reliab. Comput. 12, 337–363 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Arnold, V.I.: Lectures on PDEs. Springer Universitext (2004)

  6. Baire, R.: Lecons sur les fonctions discontinues. Collection Borel, Paris (1905)

  7. Bartle, R.G.: The Elements of Real Analysis. Wiley, New York (1976)

    MATH  Google Scholar 

  8. Beattie, R., Butzmann, H.-P.: Convergence Structures and Applications to Functional Analysis. Kluwer Academic, Dordrecht (2002)

    MATH  Google Scholar 

  9. Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dilworth, R.P.: The normal completion of the lattice of continuous functions. Trans. Am. Math. Soc. 68, 427–438 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  11. Forster, O.: Analysis 3, Integralrechnung im ℝn mit Anwendungen. Vieweg, Wiesbaden (1981)

    Google Scholar 

  12. Lin, F.-H.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Commun. Pure Appl. Math. 51, 241–257 (1998)

    Article  MATH  Google Scholar 

  13. Luxemburg, W.A., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)

    MATH  Google Scholar 

  14. Micheal, E.: Continuous selections I. Ann. Math. 63, 361–382 (1956)

    Article  Google Scholar 

  15. Neuberger, J.W.: Sobolev Gradients and Differential Equations. Springer Lecture Notes in Mathematics, vol. 1670. Springer, Berlin (1997)

    MATH  Google Scholar 

  16. Neuberger, J.W.: Continuous Newton’s method for polynomials. Math. Intell. 21, 18–23 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Neuberger, J.W.: A near minimal hypothesis Nash-Moser theorem. Int. J. Pure Appl. Math. 4, 269–280 (2003)

    MATH  MathSciNet  Google Scholar 

  18. Neuberger, J.W.: Prospects of a central theory of partial differential equations. Math. Intell. 27(3), 47–55 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Oberguggenberger, M.B., Rosinger, E.E.: Solution of Continuous Nonlinear PDEs through Order Completion. North-Holland, Amsterdam (1994)

    MATH  Google Scholar 

  20. Oxtoby, J.C.: Measure and Category, 2nd edn. Springer, New York (1980)

    MATH  Google Scholar 

  21. Sobolev, S.L.: Le probleme de Cauchy dans l’espace des functionelles. Dokl. Acad. Sci. URSS 7(3), 291–294 (1935)

    Google Scholar 

  22. Sobolev, S.L.: Methode nouvelle a resondre le probleme de Cauchy pour les equations lineaires hyperbokiques normales. Mat. Sb. 1(43), 39–72 (1936)

    MATH  Google Scholar 

  23. van der Walt, J.H.: Order convergence in sets of Hausdorff continuous functions. Honors Essay, University of Pretoria (2004)

  24. van der Walt, J.H.: Order convergence on Archimedean vector lattices. M.Sc. Thesis, University of Pretoria (2006)

  25. van der Walt, J.H.: The uniform order convergence structure on \(\mathcal{ML}(X)\) . Quaest. Math. 31, 55–77 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. van der Walt, J.H.: The order completion method for systems of nonlinear PDEs: Pseudo-topological perspectives. Acta Appl. Math. 103, 1–17 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. van der Walt, J.H.: On the completion of uniform convergence spaces and an application to nonlinear PDEs. Technical Report UPWT 2007/14

  28. Wyler, O.: Ein komplettieringsfunktor für uniforme limesräume. Math. Nachr. 40, 1–12 (1970)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

    Jan Harm van der Walt

Authors
  1. Jan Harm van der Walt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Harm van der Walt.

Rights and permissions

Reprints and permissions

About this article

Cite this article

van der Walt, J.H. The Order Completion Method for Systems of Nonlinear PDEs Revisited. Acta Appl Math 106, 149–176 (2009). https://doi.org/10.1007/s10440-008-9287-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-008-9287-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation