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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.