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

  • D. Blanchard (a1) and F. Murat (a2)

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

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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.
2Blanchard D.. Truncations and monotonicity methods for parabolic equations. Nonlinear Anal. 21 (1993), 725–43.
3Boccardo L. and Gallouët T.. On some nonlinear elliptic and parabolic equations involving measure data. J. Fund. Anal. 87 (1989), 149–69.
4Boccardo L. and Gallouët T.. Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differential Equations 17 (1992), 641–55.
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.
6Boccardo L., Murat F. and Puel J.- P.. Existence results for some quasilinear parabolic equations. Nonlinear Anal. Th. Math. Appl. 13 (1989), 376–92.
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.
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), 97107.
9Lions J.- L.. Quelques methodes de résolution des problèmes aux limites non linéaires (Paris: Dunod et Gauthier-Villars, 1969).
10Lions P.- L. and Murat F.. Renormalized solutions of nonlinear elliptic equations (to appear).
11Murat F.. Solutiones renormalizadas de EDP elipticas non lineares (Technical report R93023, Laboratoire d'Analyse Numérique, Paris VI, France, 1993).
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).
13Prignet A.. Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures. Rend. Mat. 15 (1995), 321–37.
14Serrin J.. Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964), 385–7.
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.
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.
17Prignet A.. Existence and uniqueness of entropy solutions of parabolic problems with L1 data. Nonlinear Analysis Th. Math. Appl. 28 (1997), 1943–54.
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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