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Renormalised solutions of nonlinear parabolic problems with L1 data: existence and uniqueness

Published online by Cambridge University Press:  14 November 2011

D. Blanchard  and
F. Murat
D. Blanchard
Affiliation:
URA-CNRS 1378—Analyse et Modèles Stochastiques, Université de Rouen, 76821 Mont Saint Aignan cedex, France
F. Murat
Affiliation:
URA-CNRS 189—Laboratoire d'Analyse Numérique, Tour 55–65, Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France
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Extract

In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem

where the data f and u0 belong to L1(Ω × (0, T)) and L1 (Ω), and where the function a:(0, T) × Ω × ℝN → ℝN is monotone (but not necessarily strictly monotone) and defines a bounded coercive continuous operator from the space into its dual space. The renormalised solution is an element of C0 ([ 0, T] L1 (Ω)) such that its truncates TK(u) belong to with

this solution satisfies the equation formally obtained by using in the equation the test function S(u)φ, where φ belongs to and where S belongs to C∞(ℝ) with

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 6 , 1997 , pp. 1137 - 1152
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

References

1Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M. and Vazquez, J. L.. An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995), 241–73.Google Scholar
2Blanchard, D.. Truncations and monotonicity methods for parabolic equations. Nonlinear Anal. 21 (1993), 725–43.CrossRefGoogle Scholar
3Boccardo, L. and Gallouët, T.. On some nonlinear elliptic and parabolic equations involving measure data. J. Fund. Anal. 87 (1989), 149–69.CrossRefGoogle Scholar
4Boccardo, L. and Gallouët, T.. Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differential Equations 17 (1992), 641–55.CrossRefGoogle Scholar
5Boccardo, L., Giachetti, D., Diaz, J. I. and Murat, F.. Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Differential Equations 106 (1993), 215–37.CrossRefGoogle Scholar
6Boccardo, L., Murat, F. and Puel, J.- P.. Existence results for some quasilinear parabolic equations. Nonlinear Anal. Th. Math. Appl. 13 (1989), 376–92.Google Scholar
7DiPerna, R. J. and Lions, P.-L.. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. 130 (1989), 321–66.CrossRefGoogle Scholar
8Leray, J. and Lions, J.- L.. Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les methodes de Minty et Browder. Bull. Soc. Math. 93 (1965), 97–107.Google Scholar
9Lions, J.- L.. Quelques methodes de résolution des problèmes aux limites non linéaires (Paris: Dunod et Gauthier-Villars, 1969).Google Scholar
10Lions, P.- L. and Murat, F.. Renormalized solutions of nonlinear elliptic equations (to appear).Google Scholar
11Murat, F.. Solutiones renormalizadas de EDP elipticas non lineares (Technical report R93023, Laboratoire d'Analyse Numérique, Paris VI, France, 1993).Google Scholar
12Murat, F.. Équations elliptiques non linéaires avec second membre L1 ou mesure. In Comptes rendus du 26ème Congrès national d'analyse numerique, Les Karellis, France, 1994 (Universite de Lyon 1, 1994).Google Scholar
13Prignet, A.. Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures. Rend. Mat. 15 (1995), 321–37.Google Scholar
14Serrin, J.. Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964), 385–7.Google Scholar
15Andreu, F., Mazón, J. M., Segura de León, S. and Toledo, J.. Existence and uniqueness for a degenerate parabolic equation with L1 data, to appear.Google Scholar
16Bènilan, P. and Bouhsiss, F.. Une remarque sur l'unicitè des solutions pour l'opérateur de Serrin. C. R. Acad. Sci. Paris 324 (1997), to appear.Google Scholar
17Prignet, A.. Existence and uniqueness of entropy solutions of parabolic problems with L1 data. Nonlinear Analysis Th. Math. Appl. 28 (1997), 1943–54.CrossRefGoogle Scholar
15Andreu, F., Mazón, J. M., Segura de León, S. and Toledo, J.. Existence and uniqueness for a degenerate parabolic equation with L1 data, to appear.Google Scholar
16Bènilan, P. and Bouhsiss, F.. Une remarque sur l'unicitè des solutions pour l'opérateur de Serrin. C. R. Acad. Sci. Paris 324 (1997), to appear.Google Scholar
17Prignet, A.. Existence and uniqueness of entropy solutions of parabolic problems with L1 data. Nonlinear Analysis Th. Math. Appl. 28 (1997), 1943–54.CrossRefGoogle Scholar