Flow equation for a differential operator

From Diffgeom
Jump to: navigation, search

Definition

Basic definition for Euclidean space

Suppose we have a differential equation F(y) = 0 where y: \R^n \to \R is the unknown function.

Then consider the differential equation:

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

Here:

  • t \in \R, called the time parameter, and x \in \R^n, called the space parameter, are the two independent variables
  • u: \R \times \R^n \to \R is the dependent variable, written as u(t,x)
  • F(u) on the right side actually means F(y) where y is the map sending x to u(x,t) (that is, y is the map u at a fixed time)

This is called the flow equation for F. The solution curves for this (in the space of all possible functions y define the flow of the differential equation).

Definition for a differential manifold

Note that \R^n can be replaced by any open set in \R^n, or even more generally, by any differential manifold M. In the general case, y becomes a map from M to \R, and F is a differential operator on the manifold.

The flow equation becomes:

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

Here:

  • t \in \R, called the time parameter, and x \in M, called the space parameter, are the two independent variables
  • u: \R \times M \to \R is the dependent variable, written as u(t,x)
  • F(u) on the right side actually means F(y) where y is the map sending x to u(t,x) (that is, y is the map u at a fixed time)

Fixed points and trajectories

Fixed points are solutions of the original equation

A point y is a fixed point of the flow if and only if it solves the original equation F(y) = 0. This tells us the following thing:

If the limits of the flow are well-defined at \infty or -\infty,, then the limiting points are fixed points of the flow, hence they are solutions for F

Trajectories of the flow

To understand the flow, we can look at the trajectory at each point. Note that by the way the flow is defined, the trajectory is invariant under time-translation of the original differential equation.

The trajectories at a point could be of the following types:

  • They could be defined over [0,T) \times M. This phenomenon is called short-time existence of the solution.
  • They could be defined over \R \times M. This phenomenon is called global existence of the solution.

Further, those that satisfy global existence, could satisfy any of the following:

  • There exists a limiting point at \infty, which is hence a solution to the original differential equation
  • Fill this in later