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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 127, Issue 6
  • January 1997, pp. 1137-1152

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

  • D. Blanchard (a1) and F. Murat (a2)
  • DOI: http://dx.doi.org/10.1017/S0308210500026986
  • Published online: 14 November 2011
Abstract

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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2D. Blanchard . Truncations and monotonicity methods for parabolic equations. Nonlinear Anal. 21 (1993), 725–43.

3L. Boccardo and T. Gallouët . On some nonlinear elliptic and parabolic equations involving measure data. J. Fund. Anal. 87 (1989), 149–69.

4L. Boccardo and T. Gallouët . Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differential Equations 17 (1992), 641–55.

5L. Boccardo , D. Giachetti , J. I. Diaz and F. Murat . Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Differential Equations 106 (1993), 215–37.

7R. J. DiPerna and P.-L. Lions . On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. 130 (1989), 321–66.

17A. Prignet . 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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