Renormalised solutions of nonlinear parabolic problems with \(L^1\) data: Existence and uniqueness. (English) Zbl 0895.35050

The authors prove an existence and uniqueness theorem for renormalized solutions to \[ {\partial\over \partial t} u-\text{div} a(t,x,Du) =f\text{ in } \Omega \times (0,t), \]
\[ u|_{t=0} =u_0 \text{ in }\Omega, \quad u=0 \text{ on } \partial\Omega \times (0,T), \] where \(f\) and \(u_0\) are \(L^1\) functions on their domains of definition, and \(a:(0,T) \times \Omega \times \mathbb{R}^N\to \mathbb{R}^N\) is a monotone (but not strictly monotone) Carathéodory function which defines a bounded, coercive, continuous operator on \(L^p(0,T; W^{1,p}_0 (\Omega))\). A renormalized solution is a function \(u\in C^0 ([0,T]\); \(L^1 (\Omega))\) such that its truncates \(T_K(u)\in L^p(0,T; W_0^{1,p} (\Omega))\), \[ \lim_{K\to \infty} \int_{K\leq | u|\leq K+1} | Du|^p dxdt=0, \] and it satisfies the differential equation in a generalized sense.


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35R05 PDEs with low regular coefficients and/or low regular data
35D05 Existence of generalized solutions of PDE (MSC2000)
Full Text: DOI


[1] DOI: 10.1080/03605309208820857 · Zbl 0812.35043 · doi:10.1080/03605309208820857
[2] DOI: 10.1006/jdeq.1993.1106 · Zbl 0803.35046 · doi:10.1006/jdeq.1993.1106
[3] DOI: 10.1016/0362-546X(93)90120-H · Zbl 0796.35077 · doi:10.1016/0362-546X(93)90120-H
[4] Bénilan, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 pp 241– (1995)
[5] DOI: 10.1016/S0362-546X(96)00030-2 · Zbl 0909.35075 · doi:10.1016/S0362-546X(96)00030-2
[6] Bènilan, C. R. Acad. Sci. Paris 324 (1997)
[7] Serrin, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 pp 385– (1964)
[8] Prignet, Rend. Mat. 15 pp 321– (1995)
[9] Murat, Comptes rendus du 26ème Congrès national d’analyse numerique, Les Karellis, France, 1994 1 (1994)
[10] Murat, Solutiones renormalizadas de EDP elipticas non lineares (1993)
[11] Lions, Quelques methodes de résolution des problèmes aux limites non linéaires (1969)
[12] Leray, Bull. Soc. Math. 93 pp 97– (1965)
[13] DOI: 10.2307/1971423 · Zbl 0698.45010 · doi:10.2307/1971423
[14] Boccardo, Nonlinear Anal. Th. Math. Appl. 13 pp 376– (1989)
[15] DOI: 10.1016/0022-1236(89)90005-0 · Zbl 0707.35060 · doi:10.1016/0022-1236(89)90005-0
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.