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Generalized solutions to a characteristic Cauchy problem

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Języki publikacji
EN
Abstrakty
EN
We give a meaning to the nonlinear characteristic Cauchy problem for the wave equation in base form by replacing it by a family of non-characteristic ones. This leads to a well-formulated problem in an appropriate algebra of generalized functions. We prove existence of a solution and we precise how it depends on the choice made. We also check that in the classical case (non-characteristic) our new solution coincides with the classical one.
Wydawca
Rocznik
Strony
1--29
Opis fizyczny
Bibliogr. 27 poz.
Twórcy
autor
  • Equipe Analyse Algébrique Non Linéaire, Université des Antilles et de la Guyane, Laboratoire CEREGMIA, Campus de Schoelcher, BP 7209, 97275 Schoelcher Cedex, Martinique
autor
  • Equipe Analyse Algébrique Non Linéaire, Université des Antilles et de la Guyane, Laboratoire CEREGMIA, Campus de Schoelcher, BP 7209, 97275 Schoelcher Cedex, Martinique
Bibliografia
  • [1] J.-F. Colombeau, Elementary Introduction to New Generalized Functions, North-Holland Math. Stud. 113, North-Holland, Amsterdam, 1984.
  • [2] J.-F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland, Amsterdam, 1984.
  • [3] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, vol. 3: Transformations, Sobolev, opérateurs, Masson, Pairs, 1987.
  • [4] A. Delcroix, Remarks on the embedding of spaces of distributions into spaces of Colombeau generalized functions, Novi Sad J. Math. 35 (2005), 27-40.
  • [5] A. Delcroix, Some properties of (C, E, P)-algebras: Overgeneration and 0-order estimates, preprint (2008), http://hal.archives-ouvertes.fr/hal-00326671/fr/.
  • [6] A. Delcroix and D. Scarpalézos, Topology on asymptotic algebras of generalized functions and applications, Monatsh. Math. 129 (2000), 1-14.
  • [7] A. Delcroix, V. Dévoué and J.-A. Marti, Generalized solutions of singular differential problems. Relationship with classical solutions, J. Math. Anal. Appl. 353 (2009), 386-402.
  • [8] A. Delcroix, V. Dévoué and J.-A. Marti, Well posed problems in algebras of generalized functions, Appl. Anal. 90 (2011), 1747-1761.
  • [9] V. Dévoué, On generalized solutions to the wave equation in canonical form, Disser-tationes Math. 443 (2007), 1-69.
  • [10] V. Dévoué, Generalized solutions to a non-Lipschitz Cauchy problem, J. Appl. Anal. 15 (2009), 1-32.
  • [11] V. Dévoué, Generalized solutions to a non-Lipschitz Goursat problem, Differ. Equ. Appl. 1 (2009), 153-178.
  • [12] Y. V. Egorov, On the solubility of differential equations with simple characteristics, Russ. Math. Surv. 26 (1971), 113-130.
  • [13] Y. V. Egorov and M. A. Shubin, Partial Differential Equations, Springer, 1993.
  • [14] P. R. Garabedian, Partial Differential Equations, Wiley, 1964.
  • [15] L. Gårding, T. Kotake and J. Leray, Uniformisation et développement asymptotique de la solution du problème de Cauchy linéaire, à données holomorphes; analogie avec la théorie des ondes asymptotiques et approchées. (Problème de Cauchy I bis et VI), Bull. Soc. Math. France 92 (1964), 263-361.
  • [16] M. Grosser, M. Kunzinger, M. Oberguggenberger and R. Steinbauer, Geometrie Theory of Generalized Functions with Applications to General Relativity, Kluwer Academic Press, 2001.
  • [17] Y Hamada, Problème de Cauchy analytique, Publ. RIMS Kyoto Univ. 39 (2003), 601-624.
  • [18] L. Hörmander, On the characteristic Cauchy problem, Ann. Math. 88 (1968), 341-370.
  • [19] J.-A. Marti, Fundamental structures and asymptotic microlocalization in sheaves of generalized functions, Integral Transforms Spec. Fund. 6 (1998), 223-228.
  • [20] J.-A. Marti, (C, E, P)-sheaf structure and applications, in: Nonlinear Theory of Generalized Functions (Vienna 1977), Chapman & Hall/CRC Res. Notes Math. 401, Chapman & Hall, Boca Raton (1999), 175-186.
  • [21] J.-A. Marti, Non-linear algebraic analysis of delta shock wave to Burgers’ equation, Pacific J. Math. 210 (2003), 165-187.
  • [22] J.-A. Marti, Multiparametric algebras and characteristic Cauchy problem, in: Non-Linear Algebraic Analysis and Applications, Cambridge Sci. Publ. Ltd., Cambridge (2004), 181-192.
  • [23] M. Nedeljkov, M. Oberguggenberger and S. Pilipovic, Generalized solution to a semilinear wave equation, Nonlinear Anal. 61 (2005), 461-475.
  • [24] M. Nedeljkov, S. Pilipovic and D. Scarpalézos, The Linear Theory of Colombeau Generalized Functions, Pitman Res. Notes Math. Ser. 385, Longman Sci. Tech., Harlow, 1998.
  • [25] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Longman Sci. Tech., Harlow, 1992.
  • [26] D. Scarpalézos, Colombeau’s generalized functions: Topological structures; Microlocal properties. A simplified point of view. Part I, Bull. Cl. Sci. Math. Nat. Sci. Math. 25(2000), 89-114.
  • [27] D. Scarpalézos, Colombeau’s generalized functions: Topological structures; Microlocal properties. A simplified point of view. Part II, Publ. Inst. Math. Novi Sad 76(2004), 111-125.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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