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Solving large classes of nonlinear systems of PDEs. (English) Zbl 1130.35026

Summary: It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff-continuous functions. The usual Navier-Stokes equations, as well as their various modifications aiming at a realistic modeling, are included as particular cases. The same holds for the critically important constitutive relations in various branches of Continuum Mechanics. The solution method does not involve functional analysis, nor various Sobolev or other spaces of distributions or generalized functions. The general and type independent existence and regularity results regarding solutions presented here have recently been introduced in the literature.

MSC:

35G20 Nonlinear higher-order PDEs
35A25 Other special methods applied to PDEs
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[1] Oberguggenberger, M.B.; Rosinger, E.E., ()
[2] Anguelov, R., Dedekind order completion of \(C(X)\) by Hausdorff continuous functions, Quaestiones mathematicae, 27, 153-170, (2004) · Zbl 1062.54017
[3] MacNeille, H.M., Partially ordered sets, Transactions of the American mathematical society, 42, 416-460, (1937) · Zbl 0017.33904
[4] Forster, O., Analysis 3, () · Zbl 0549.32019
[5] Lewy, H., An example of smooth linear partial differential equation without solutions, Annals of mathematics, 66, 2, 155-158, (1957) · Zbl 0078.08104
[6] Rajagopal, K.R.; Wineman, A.S., On constitutive equations for branching of response with selectivity, International journal of nonlinear mechanics, 15, 83-91, (1980) · Zbl 0442.73002
[7] Rajagopal, K.R., On implicit constitutive theories, Application of mathematics, 28, 4, 279-319, (2003) · Zbl 1099.74009
[8] Rajagopal, K.R.; Srinivasa, A.R., On thermo – mechanical restrictions of continua, Proceedings of the royal society of London. series A. mathematical, physical and engineering sciences, 460, 631-651, (2004) · Zbl 1041.74002
[9] Arnold, V.I., Lectures on pdes, (2004), Springer Universitext
[10] Evans, L.C., ()
[11] Baire, R., Lecons sur LES fonctions discontinues, (1905), Collection Borel Paris · JFM 36.0438.01
[12] Sendov, B., Hausdorff approximations, (1990), Kluwer Dordrecht · Zbl 0167.04702
[13] ()
[14] Anguelov, R., An introduction to some spaces of interval functions · Zbl 1062.54017
[15] Anguelov, R.; Markov, S., Extended segment analysis, Freiburger intervall - berichte, 10, 1-63, (1981)
[16] Anguelov, R.; Markov, S.; Sendov, B., On the normed linear space of Hausdorff continuous functions, (), 281-288 · Zbl 1142.46316
[17] Anguelov, R.; Markov, S.; Sendov, B., The set of Hausdorff continuous functions — the largest linear space of interval functions, Reliable computing, 12, 337-363, (2006) · Zbl 1110.65036
[18] R. Anguelov, F. Minani, Interval viscosity solutions of Hamilton-Jacobi equations. Technical Report UPWT 2005/3, University of Pretoria · Zbl 1280.70011
[19] R. Anguelov, E.E. Rosinger, Dedekind order completion of \(\mathcal{M}(\Omega)\) by Hausdorff continuous functions (in press) · Zbl 1330.35080
[20] Anguelov, R.; Rosinger, E.E., Hausdorff continuous solutions of nonlinear PDEs through the order completion method, Quaestiones mathematicae, 28, 3, 271-285, (2005) · Zbl 1330.35080
[21] R. Anguelov, E.E. Rosinger, Solution of nonlinear PDEs by Hausdorff continuous functions (in press) · Zbl 1330.35080
[22] Nicolescu, M., Analiză matematică II, (1958), Editura Technică Bucureṣti · Zbl 0089.03101
[23] Oxtoby, J.C., Measure and category, (1971), Springer New York · Zbl 0217.09201
[24] Luxemburg, W.A.J.; Zaanen, A.C., Riesz spaces I, (1971), North Holland Amsterdam · Zbl 0231.46014
[25] Zaanen, A.C., The universal completion of an Archimedean Riesz space, Indagationes mathematicae, 45, 4, 435-441, (1983) · Zbl 0528.47005
[26] Dilworth, R.P., The normal completion of the lattice of continuous functions, Transactions of the American mathematical society, 68, 427-438, (1950) · Zbl 0037.20205
[27] Mack, J.E.; Johnson, D.G., The Dedekind completion of \(C(X)\), Pacific journal of mathematics, 20, 2, 231-243, (1967) · Zbl 0152.39802
[28] Markov, S., Calculus for interval functions of a real variable, Computing, 22, 325-337, (1979) · Zbl 0408.65026
[29] Markov, S., Extended interval arithmetic involving infinite intervals, Mathematica balkanica, 6, 269-304, (1992) · Zbl 0830.65034
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