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A variational principle for gradient flows

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Abstract.

We verify – after appropriate modifications – an old conjecture of Brezis-Ekeland ([3], [4]) concerning the feasibility of a global variational approach to the problems of existence and uniqueness of gradient flows for convex energy functionals. Our approach is based on a concept of ‘‘self-duality’’ inherent in many parabolic evolution equations, and motivated by Bolza-type problems in the classical calculus of variations. The modified principle allows to identify the extremal value –which was the missing ingredient in [3]– and so it can now be used to give variational proofs for the existence and uniqueness of solutions for the heat equation (of course) but also for quasi-linear parabolic equations, porous media, fast diffusion and more general dissipative evolution equations.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver BC, Canada, V6T 1Z2

    N. Ghoussoub

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    L. Tzou

Authors
  1. N. Ghoussoub
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  2. L. Tzou
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Additional information

Both authors were partially supported by a grant from the Natural Science and Engineering Research Council of Canada.

This paper is part of this author’s Master’s thesis under the supervision of the first named author.

Revised version: 31 March 2004

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Ghoussoub, N., Tzou, L. A variational principle for gradient flows. Math. Ann. 330, 519–549 (2004). https://doi.org/10.1007/s00208-004-0558-6

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  • DOI: https://doi.org/10.1007/s00208-004-0558-6

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