Weak solutions for gradient flows under monotonicity constraints

TL;DR

This paper develops a framework for weak solutions to gradient flows with monotonicity constraints, establishing existence, uniqueness, and properties, and linking constrained solutions to obstacle problems.

Contribution

Introduces a new notion of weak solutions for constrained gradient flows, proving existence, uniqueness, and connecting to obstacle problems.

Findings

Existence and uniqueness of weak solutions under monotonicity constraints.

Energy identity as a selection criterion for non-unique solutions.

Constrained solutions coincide with obstacle problem solutions for autonomous energies.

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.