Year: 2019
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Negri, Matteo, Kimura, Masato
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2970952811