Weak solutions for gradient flows under monotonicity constraints

Matteo Negri; Masato Kimura

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arXiv · 1908.10111

Weak solutions for gradient flows under monotonicity constraints

Matteo Negri·Masato Kimura

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Abstract

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraint in time and natural regularity assumptions. We provide first a notion of weak solution, inspired by the theory of curves of maximal slope, and then existence (employing time-discrete schemes with different "implementations" of the constraint), uniqueness, power and energy identity, comparison principle and continuous dependence. As a byproduct, we show that the energy identity gives a selection criterion for the (non-unique) evolutions obtained by other notions of solutions. We finally show that, for autonomous energies, the solutions obtained with the monotonicity constraint actually coincide with those obtained with a fixed obstacle, given by the initial datum.

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Matteo Negri, Masato Kimura. 2019-08-27. Weak solutions for gradient flows under monotonicity constraints. https://arxiv.org/abs/1908.10111

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