Max-decreasing 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 max-decreasing if the function:

$t\mapsto {sup}_{x\in M}u(t,x)$

is a monotone decreasing function. (The function defined above is termed the timewise-max function for $u$).

The corresponding notion is of a min-increasing trajectory -- viz a trajectory where the minimum (or infimum) keeps increasing.