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On the validity of the Picard algorithm for nonlinear parabolic equations

Published online by Cambridge University Press:  12 July 2007

Marco Cannone
Affiliation:
Laboratoire d'Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, Cité Descartes-5, bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France (cannone@math.univ-mlv.fr)
Grzegorz Karch
Affiliation:
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland and Institute of Mathematics, Polish Academy of Sciences, PO Box 21, ul. Sniadeckich 8, 00-956 Warszawa 10, Poland. (karch@math.uni.wroc.pl)
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Abstract

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We discuss some ill-posedness results for solutions arising from the Picard iterations algorithm (i.e. the Banach fixed-point theorem) in the case of the nonlinear heat equation, the viscous Hamilton–Jacobi equation, the convection–diffusion equation and the incompressible Navier–Stokes system.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2005