Bound-narrowing trajectory

This article defines a property that can be evaluated for a trajectory on the space of functions on a manifold

Definition

Let $M$ be a manifold and $u=u(t,x)$ be a function $\mathbb {R} \times M\to \mathbb {R}$, where:

• $t$ denotes the time parameter, and varies in $\mathbb {R}$
• $x$ denotes the spatial parameter, and varies in $M$

In other words, $u$ is a trajectory (or path) in the space of all functions from $M$ to $\mathbb {R}$.

Then, $u$ is said to be bound-narrowing if it is both max-decreasing and min-increasing or equivalently if it satisfies the following condition:

$C_{1}\leq u(0,x)\leq C_{2}\forall x\in M\implies C_{1}\leq u(t,x)\leq C_{2}\forall x\in M$