Scalar weak maximum principle

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Template:Flow equation property

Definition

Basic definition

Let M be a differential manifold and F be a differential operator that acts on functions M \to \R. Consider the flow equation associated with f, namely the equation for u:\R \times M \to \R given as:

\frac{\partial u}{\partial t} = F(u)

An initial value problem corresponding to this differential equation is a specification of u(0,x) for each x \in M.

The differential operator F is said to satisfy the scalar weak maximum principle if whenever u is a solution for which there are constants C_1 and C_2 such that C_1 \le u(0,x) \le C_2 for all x \in M, then C_1 \le u(t,x) \le C_2 for all x \in M, t \in \R^+.

In other words, any bounded set in which the range of u(0,x) lies also contains the image of u(t,x) for all t.

Definition in terms of trajectory properties

A differential operator F is said to satisfy the scalar weak maximum principle if the trajectories of the corresponding flow equation are all bound-narrowing.

Relation with other properties=

One-sided maximum principles

  • The flow equation is said to satisfy a one-sided scalar weak maximum principle if the trajectories of the flow equation are only min-increasing