Slow motion of gradient flows

By Felix Otto, Maria G. Reznikoff. 2007

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Abstract We present sufficient conditions on an energy landscape in order for the associated gradient flow to exhibit slow motion or “dynamic metastability.” The first condition is a weak form of convexity transverse to the so-called slow manifold, N . The second condition is that the energy restricted to N is Lipschitz with a constant δ ≪ 1 . One feature of the abstract result that makes it of broader interest is that it does not rely on maximum principles. As an application, we give a new proof of the exponentially slow motion of transition layers in the one-dimensional Allen–Cahn equation. The analysis is more nonlinear than previous work: It relies on the nonlinear convexity condition or “energy–energy-dissipation inequality.” (Although we do use the maximum principle for convenience in the application, we believe it may be removed with additional work.) Our result demonstrates that a broad class of initial data relaxes with an exponential rate into a δ-neighborhood of the slow manifold, where it is then trapped for an exponentially long time.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/c05b6f86be4daca4b4d0537be48dfae551a8e163