# 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 $\R \times M \to \R$, where:

• $t$ denotes the time parameter, and varies in $\R$
• $x$ 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 $\R$.

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.