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Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces


Author: Laurent Véron
Journal: Math. Comp. 39 (1982), 325-337
MSC: Primary 47H15; Secondary 34A45
DOI: https://doi.org/10.1090/S0025-5718-1982-0669633-9
MathSciNet review: 669633
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Abstract: Let $\partial \Phi$ be the subdifferential of some lower semicontinuous convex function $\Phi$ of a real Hilbert space H, $f \in {L^2}(0,T;H)$ and ${u_n}$ a continouous piecewise linear approximate solution of $du/dt + \partial \Phi (u) \ni f$, obtained by an implicit scheme. If ${u_0} \in \operatorname {Dom} (\Phi )$, then $d{u_n}/dt$ converges to $du/dt$ in ${L^2}(0,T;H)$. Moreover, if ${u_0} \in \overline {\operatorname {Dom} (\partial \Phi )}$, we construct a step function ${\eta _n}(t)$ approximating t such that ${\lim _{n \to + \infty }}\smallint _0^T{\eta _n}|d{u_n}/dt - du/dt{|^2}\;dt = 0$. When $\Phi$ is inf-compact and when the sequence of approximation of f is weakly convergent to f, then ${u_n}$ converges to u in $C([0,T];H)$ and ${\eta _n}d{u_n}/dt$ is weakly convergent to $tdu/dt$.


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  • Hédy Attouch, Convergence de fonctionnelles convexes, Journées d’Analyse Non Linéaire (Proc. Conf., Besançon, 1977), Lecture Notes in Math., vol. 665, Springer, Berlin, 1978, pp. 1–40 (French). MR 519420
  • Pierre Baras, Compacité de l’opérateur $f\mapsto u$ solution d’une équation non linéaire $(du/dt)+Au\ni f$, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 23, A1113–A1116 (French, with English summary). MR 493554
  • Ph. Benilan, Equations d’Evolution dans un Espace de Banach Quelconque et Applications, Thèse, Université de Paris XI, Orsay, Paris, 1972.
  • H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
  • Haïm Brézis, Asymptotic behavior of some evolution systems, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977) Publ. Math. Res. Center Univ. Wisconsin, vol. 40, Academic Press, New York-London, 1978, pp. 141–154. MR 513816
  • H. Brezis, Propriétés régularisantes de certains semi-groupes non linéaires, Israel J. Math. 9 (1971), 513–534 (French). MR 283635, DOI https://doi.org/10.1007/BF02771467
  • Michael G. Crandall and L. C. Evans, On the relation of the operator $\partial /\partial s+\partial /\partial \tau $ to evolution governed by accretive operators, Israel J. Math. 21 (1975), no. 4, 261–278. MR 390853, DOI https://doi.org/10.1007/BF02757989
  • M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298. MR 287357, DOI https://doi.org/10.2307/2373376

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Article copyright: © Copyright 1982 American Mathematical Society