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A variational principle for certain dissipative evolution equations Tzou, Leo

Abstract

We formulate a variational principle which models several first order parabolic Cauchy problems. Unlike the one proposed by Brezis-Ekeland in [2], this principle does not require the identification of the extremal value, which was a major hurdle for the applicability of their approach. Our method is based on a concept of "self-duality" inherent in many parabolic evolution equations, including those driven by the gradient of a convex energy functional. The new principle is used to provide global existence and uniqueness results of solutions for the heat equation (of course) but also for quasi-linear parabolic equations, porous media and fast diffusion equations.

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