×

zbMATH — the first resource for mathematics

New versions of the Colombeau algebras. (English) Zbl 1115.46035
This paper constitutes an important contribution in the growing field of theories of nonlinear generalized functions. Among other results, a new version of the simplified (or special) algebra of J.–F. Colombeau is constructed. This new algebra is a factor space of moderate elements (formed by sequences of families of smooth functions), modulo negligible ones. Let us remark that this construction can be reinterpreted as bi-parametric \((C,E,P)\)-algebra, as they were introduced by J.–A. Marti. The construction presented here shares the main properties of the Colombeau algebra. In addition, representation of generalized functions exists as weak asymptotic series whose coefficients are distributions, which is a new feature (albeit in the stream of previous works of the same author).
The author, furthermore, gives a construction of Colombeau type algebras generated by harmonic or polyharmonic regularizations of distributions connected with the half plane, the latter being, as far as the reviewer knows, a true novelty. In the same spirit, the author introduces Colombeau type algebras based on analytic representation of distributions, connected with an octant.
Some applications are given, such as finding asymptotic expressions for the product of distributions (for example, for \(\delta^2\), as already given earlier by Li Bang He and Li Ya–Qing). More interestingly, the approach given here seems to be very efficient for solving nonlinear hyperbolic systems, with the introduction, for example, of \(\delta\)-shock wave solutions.

MSC:
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46F10 Operations with distributions and generalized functions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bang-He, Acta Scientia Sinica 21 pp 561– (1978)
[2] Bang-He, Acta Scientia Sinica 28 pp 923– (1985)
[3] Bang-He, Conference Proceedings 29 pp 431– (2000)
[4] and , Relativistic Quantum Mechanics (McGraw-Hill Book Company, 1964).
[5] Boie, Comment. Math. Univ. Carolin. 39 pp 309– (1998)
[6] Distributions, Complex Variables, and Fourier Transforms (Addison-Wesley Publ. Comp., Reading, Massachusetts, 1965). · Zbl 0151.18102
[7] Carmichael, J. Elisha Mitchell Sci. Soc. 93 pp 115– (1977)
[8] Elementary Introduction to New Generalized Functions, North-Holland Mathematics Studies Vol. 113 (North-Holland, Amsterdam, 1985).
[9] Danilov, Theoret. and Math. Phys. 114 pp 1– (1998)
[10] Danilov, Integral Transforms Spec. Funct. 6 pp 137– (1997)
[11] and , Propagation and interaction of nonlinear waves to quasilinear equations, in: Proceedings of the Eighth International Conference in Hyperbolic Problems: Theory, Numerics, Applications, Magdeburg, February/March 2000, Vol. I, International Series of Numerical Mathematics Vol. 140 (Birkhäuser Verlag Basel, 2001), pp. 267-276.
[12] Danilov, Nonlinear Stud. 8 pp 135– (2001)
[13] and , Propagation and interaction of delta-shock waves of a hyperbolic system of conservation laws, in: Hyperbolic Problems: Theory, Numerics, Applications, edited by Thomas Y. Hou and Eitan Tadmor, Proceedings of the Ninth International Conference on Hyperbolic Problems held in CalTech, Pasadena, March 25-29, 2002, (Springer-Verlag, 2003), pp. 483-492.
[14] Danilov, Amer. Math. Soc. Transl. Ser. 2 208 pp 33– (2003) · doi:10.1090/trans2/208/02
[15] Danilov, Dokl. Ross. Akad. Nauk 39469 pp 10– (2004)
[16] Danilov, J. Differential Equations 211 pp 333– (2005)
[17] Egorov, Russian Math. Surveys 45 pp 1– (1990)
[18] and , Generalized Functions. Vol 1: Properties and Operations (Academic Press, New York, 1964).
[19] , , and , Geometric Theory of Generalized Functions with Applications to General Relativity (Kluver Academic Publ., Dordrecht, 2001).
[20] Itano, J. Sci. Hyroshima Univ. Ser. A-1 29 pp 51– (1965)
[21] Ivanov, Izv. Vyssh. Uchebn. Zaved. Mat. 3 pp 10– (1972)
[22] Ivanov, Izv. Vyssh. Uchebn. Zaved. Mat. 1 pp 19– (1981)
[23] and , On the multiplication of Schwartz’s distributions, Trudy Inst. Math. and Mech., Sverdlovsk, 3-10 (1974) (in Russian).
[24] and , On the multiplication of tempered distributions from S ’(Rn ), n 2, Manuscript No. 1224-75, deposited at VINITI (1975) (in Russian).
[25] Kakichev, Math. Notes 27 pp 899– (1980) · Zbl 0449.46037 · doi:10.1007/BF01145431
[26] Lee Keyfitz, J. Differential Equations 118 pp 420– (1995)
[27] Khrennikov, Dokl. Ross. Akad. Nauk 38365 pp 28– (2002)
[28] Khrennikov, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 pp 1– (2002)
[29] , and , An associative algebra of vector-valued distributions and singular solutions of nonlinear equations, in: Proceedings of the conference ”Mathematical Modelling ofWave Phenomena”, Växjö, November 2002, edited by B. Nilsson and L. Fishman, Mathematical Modelling in Physics, Engineering and Cognitive Sciences Vol. 7 (University Press, 2004), pp. 191-205.
[30] Kytmanov, Izv. Vyssh. Uchebn. Zaved. Mat. 122 pp 36– (1978)
[31] The Bochner-Martinelli Integral and Its Applications (Birkhäuser Verlag, Basel - Boston - Berlin, 1995).
[32] Laugwitz, J. Reine Angew. Math. 207 pp 53– (1961)
[33] Laugwitz, J. Reine Angew. Math. 208 pp 22– (1961)
[34] Livchak, Mat. Zametki VI pp 29– (1968)
[35] Livchak, Mat. Zametki VI pp 38– (1968)
[36] To the theory of generalized functions, in: Proceedings of the Riga Algebraic Seminar, Izd. Rizhsk. Gos. Univ., Riga 1969, Trudy Rizhskogo Algebr. Seminar, pp. 98-164 (in Russian).
[37] Majda, Mem. Amer. Math. Soc. 41 pp 1– (1983)
[38] Majda, Mem. Amer. Math. Soc. 43 pp 1– (1983)
[39] Maslov, Uspekhi Mat. Nauk 35 pp 252– (1980)
[40] Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics Vol. 259 (Longman, Harlow, 1992).
[41] Oberguggenberger, Nonlinear Anal. 47 pp 5029– (2001)
[42] Oberguggenberger, Math. Nachr. 203 pp 147– (1999) · Zbl 0935.46041 · doi:10.1002/mana.1999.3212030110
[43] Functional Analysis (McGraw-Hill Book Company, New York - St. Louis - San Francisco - Sydney - Toronto, 1973).
[44] Schmieden, Math. Z. 69 pp 1– (1958)
[45] Shelkovich, Math. Notes 63 pp 313– (1998)
[46] Shelkovich, Dokl. Akad. Nauk SSSR 314 pp 159– (1990)
[47] Shelkovich, Izv. Vuzov Ser. Mat. 4 pp 70– (1991)
[48] Shelkovich, Math. Notes 57 pp 536– (1995)
[49] Delta-shock waves of a class of hyperbolic systems of conservation laws, in: Patterns and Waves, edited by A. Abramian, S. Vakulenko and V. Volpert (Akadem Print, St. Petersburg, 2003), pp. 155-168.
[50] A specific hyperbolic system of conservation laws admitting delta-shock wave type solutions, Preprint 2003-059 at the url: http://www.math.ntnu.no/conservation/2003/059.html
[51] Tillman, Math. Z. 77 pp 106– (1961)
[52] Generalized functions, in: Functional Analysis, edited by S. G. Krein (Nauka, Moscow, 1972), pp. 455-531 (in Russian)
[53] Verjbalovich, Izv. Vyssh. Uchebn. Zaved. Mat. 2 pp 17– (1975)
[54] Generalized Functions in Mathematical Physics (Nauka, Moscow, 1979) (in Russian).
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.