×

A global theory of algebras of generalized functions. (English) Zbl 0995.46054

A geometric approach is used to construct the Colombeau algebra of distributions on a smooth manifold which includes the space of classical distributions on that manifold. The algebra is derived by intrinsic construction based on differential calculus so that it is a differential algebra in which its elements possess Lie derivatives with respect to arbitrary smooth vector fields. The algebra is furthermore shown to retain all the distinguishing properties of the local theory in a global context.

MSC:

46T30 Distributions and generalized functions on nonlinear spaces
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aragona, J.; Biagioni, H. A., Intrinsic definition of the Colombeau algebra of generalized functions, Anal. Math., 17, 75-132 (1991) · Zbl 0765.46019
[2] Balasin, H., Geodesics for impulsive gravitational waves and the multiplication of distributions, Classical Quantum Gravity, 14, 455-462 (1997) · Zbl 0868.53062
[3] Biagioni, H. A.; Oberguggenberger, M., Generalized solutions to the Korteweg-de Vries and the regularized long-wave equations, SIAM J. Math. Anal., 23, 923-940 (1992) · Zbl 0757.35068
[4] Biagioni, H. A.; Oberguggenberger, M., Generalized solutions to Burgers’ equation, J. Differential Equations, 97, 263-287 (1992) · Zbl 0777.35071
[5] Clarke, C. J.S.; Vickers, J. A.; Wilson, J., Generalised functions and distributional curvature of cosmic strings, Classical Quantum Gravity, 13, 2485-2498 (1996) · Zbl 0859.53074
[6] Colombeau, J. F., New Generalized Functions and Multiplication of Distributions (1984), North-Holland: North-Holland Amsterdam · Zbl 0761.46021
[7] Colombeau, J. F., Elementary Introduction to New Generalized Functions (1985), North-Holland: North-Holland Amsterdam · Zbl 0627.46049
[8] Colombeau, J. F., Multiplication of distributions, Bull. Amer. Math. Soc. (N.S.), 23, 251-268 (1990) · Zbl 0731.46023
[9] Colombeau, J. F.; Meril, A., Generalized functions and multiplication of distributions on \(C^∞\) manifolds, J. Math. Anal. Appl., 186, 357-364 (1994) · Zbl 0819.46026
[10] Colombeau, J. F.; Heibig, A.; Oberguggenberger, M., Le problème de Cauchy dans un espace de fonctions généralisées I, C. R. Acad. Sci. Paris Sér. I Math., 317, 851-855 (1993) · Zbl 0790.35015
[11] Colombeau, J. F.; Heibig, A.; Oberguggenberger, M., Le problème de Cauchy dans un espace de fonctions généralisées II, C. R. Acad. Sci. Paris Sér. I Math., 319, 1179-1183 (1994) · Zbl 0826.46028
[12] Colombeau, J. F.; Oberguggenberger, M., On a hyperbolic system with a compatible quadratic term: Generalized solutions, delta waves, and multiplication of distributions, Comm. Partial Differential Equations, 15, 905-938 (1990) · Zbl 0711.35028
[13] Damsma, M.; de Roever, J. W., Colombeau algebras on a \(C^∞\)-manifold, Indag. Math. N.S., 3, 341-358 (1991) · Zbl 0761.46022
[14] de Rham, G., Differentiable Manifolds (1984), Springer: Springer Berlin
[15] Dieudonné, J., Eléments d’Analyse (1974), Gauthier-Villars: Gauthier-Villars Paris
[16] Grosser, M.; Farkas, E.; Kunzinger, M.; Steinbauer, R., On the foundations of nonlinear generalized functions, I & II, Mem. Amer. Math. Soc., 153 (2001) · Zbl 0985.46026
[17] Hörmander, L., The Analysis of Linear Partial Differential Operators, I. The Analysis of Linear Partial Differential Operators, I, Grundlehren der Mathematischen Wissenschaften, 256 (1990), Springer: Springer Berlin
[18] Jelínek, J., An intrinsic definition of the Colombeau generalized functions, Comment. Math. Univ. Carolinae, 40, 71-95 (1999) · Zbl 1060.46513
[19] Kriegl, A.; Michor, P. W., The Convenient Setting of Global Analysis. The Convenient Setting of Global Analysis, American Mathematical Society Mathematics Surveys and Monographs, 53 (1997), American Mathematical Society: American Mathematical Society Providence · Zbl 0889.58001
[20] Kunzinger, M.; Steinbauer, R., A rigorous solution concept for geodesic equations in impulsive gravitational waves, J. Math. Phys., 40, 1479-1489 (1999) · Zbl 0947.83020
[21] Marsden, J. E., Generalized Hamiltonian mechanics, Arch. Rational Mech. Anal., 28, 323-361 (1968) · Zbl 0155.51302
[22] Oberguggenberger, M., Multiplication of Distributions and Applications to Partial Differential Equations. Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics, 259 (1992), Longman: Longman Harlow · Zbl 0818.46036
[23] Oberguggenberger, M., Nonlinear theories of generalized functions, (Albeverio, S.; Luxemburg, W. A.J.; Wolff, M. P.H., Advances in Analysis, Probability, and Mathematical Physics—Contributions from Nonstandard Analysis (1994), Kluwer: Kluwer Dordrecht) · Zbl 1042.46510
[24] Schwartz, L., Sur L’impossibilité de la Multiplication des Distributions, C. R. Acad. Sci. Paris, 239, 847-848 (1954) · Zbl 0056.10602
[25] Vickers, J. A.; Wilson, J., Invariance of the distributional curvature of the cone under smooth diffeomorphisms, Classical Quantum Gravity, 16, 579-588 (1999) · Zbl 0933.83033
[26] Vickers, J. A., Nonlinear generalised functions in general relativity, (Grosser, M.; Hörmann, G.; Kunzinger, M.; Oberguggenberger, M., Nonlinear Theory of Generalized Functions (1999), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton), 275-290 · Zbl 0929.35163
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.