We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraints. First, we provide a notion of weak solution, inspired by the theory of curves of maximal slope, and then we prove existence (employing time-discrete schemes with different implementations of the constraint), uniqueness, power and energy identity, comparison principle and continuous dependence. As a by-product, we show that the energy identity gives a selection criterion for the (non-unique) evolutions obtained by other notions of solutions. Finally, we show that for autonomous energies the evolution obtained with the monotonicity constraint actually coincides with the evolution obtained by replacing the constraint with a fixed obstacle, given by the initial datum.

我们考虑在单调性约束下二次非自治能量的梯度流。首先，我们受到最大斜率曲线理论的启发，提供了弱解的概念，然后我们证明了存在性（使用具有不同约束实现的时间离散方案）、唯一性、功率和能量恒等式、比较原理和连续依赖性。作为副产品，我们表明能量恒等式为其他解的概念获得的（非唯一）演化提供了一个选择标准。最后，我们表明，对于自主能量，用单调性约束获得的进化实际上与通过用初始数据给出的固定障碍替换约束获得的进化重合。