×

The order completion method for systems of nonlinear PDEs revisited. (English) Zbl 1177.35055

Summary: We present further developments regarding the enrichment of the basic theory of order completion as presented in M. B. Oberguggenberger and E. E. Rosinger [Solution of continuous nonlinear PDEs through order completion, Amsterdam: Elsevier Science (1994; Zbl 0821.35001)]. 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, 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.

MSC:

35G20 Nonlinear higher-order PDEs
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
06B30 Topological lattices
46E05 Lattices of continuous, differentiable or analytic functions
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)

Citations:

Zbl 0821.35001
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Anguelov, R.: Dedekind order completion of C(X) by Hausdorff continuous functions. Quaest. Math. 27, 153–170 (2004) · Zbl 1062.54017
[2] Anguelov, R., Rosinger, E.E.: Hausdorff continuous solutions of nonlinear PDEs through the order completion method. Quaest. Math. 28(3), 271–285 (2005) · Zbl 1330.35080
[3] Anguelov, R., van der Walt, J.H.: Order convergence on \(\mathcal{C}(X)\) . Quaest. Math. 28(4), 425–457 (2005) · Zbl 1094.46004
[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) · Zbl 1110.65036
[5] Arnold, V.I.: Lectures on PDEs. Springer Universitext (2004)
[6] Baire, R.: Lecons sur les fonctions discontinues. Collection Borel, Paris (1905) · JFM 36.0438.01
[7] Bartle, R.G.: The Elements of Real Analysis. Wiley, New York (1976) · Zbl 0309.26003
[8] Beattie, R., Butzmann, H.-P.: Convergence Structures and Applications to Functional Analysis. Kluwer Academic, Dordrecht (2002) · Zbl 1246.46003
[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) · Zbl 0509.35067
[10] Dilworth, R.P.: The normal completion of the lattice of continuous functions. Trans. Am. Math. Soc. 68, 427–438 (1950) · Zbl 0037.20205
[11] Forster, O.: Analysis 3, Integralrechnung im \(\mathbb{R}\) n mit Anwendungen. Vieweg, Wiesbaden (1981) · Zbl 0479.26010
[12] Lin, F.-H.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Commun. Pure Appl. Math. 51, 241–257 (1998) · Zbl 0958.35102
[13] Luxemburg, W.A., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971) · Zbl 0231.46014
[14] Micheal, E.: Continuous selections I. Ann. Math. 63, 361–382 (1956) · Zbl 0071.15902
[15] Neuberger, J.W.: Sobolev Gradients and Differential Equations. Springer Lecture Notes in Mathematics, vol. 1670. Springer, Berlin (1997) · Zbl 0935.35002
[16] Neuberger, J.W.: Continuous Newton’s method for polynomials. Math. Intell. 21, 18–23 (1999) · Zbl 1052.30502
[17] Neuberger, J.W.: A near minimal hypothesis Nash-Moser theorem. Int. J. Pure Appl. Math. 4, 269–280 (2003) · Zbl 1021.47043
[18] Neuberger, J.W.: Prospects of a central theory of partial differential equations. Math. Intell. 27(3), 47–55 (2005) · Zbl 1090.35011
[19] Oberguggenberger, M.B., Rosinger, E.E.: Solution of Continuous Nonlinear PDEs through Order Completion. North-Holland, Amsterdam (1994) · Zbl 0821.35001
[20] Oxtoby, J.C.: Measure and Category, 2nd edn. Springer, New York (1980) · Zbl 0435.28011
[21] Sobolev, S.L.: Le probleme de Cauchy dans l’espace des functionelles. Dokl. Acad. Sci. URSS 7(3), 291–294 (1935) · Zbl 0012.40603
[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)
[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) · Zbl 1145.54003
[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) · Zbl 1154.35328
[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) · Zbl 0207.52603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.