1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Condition on vector field to be a diffeomorphism.

  1. Jun 5, 2016 #1
    Hi everybody,

    Let [itex] V(x) [/itex] a vector field on a manifold ([itex] R^2 [/itex] in my case), i am looking for a condition on [itex] V(x) [/itex] for which the function [itex] x^µ \rightarrow x^µ + V^µ(x) [/itex] is a diffeomorphism. I read some document speaking about the flow, integral curve for ODE solving but i fail to find a generic condition to avoid V to send two point on the same coordinate. I think about the generator of the diffeomorphism group but it's only defined infinitesimaly.


  2. jcsd
  3. Jun 5, 2016 #2
    there are a lot of possible approaches to sufficient conditions. For example, if ##\sup_{x\in\mathbb{R}^2}\Big\|\frac{\partial V}{\partial x}\Big\|## is small enough then it is a diffeomorphism. monotonicity assumption can also help. Perhaps the book Topics in Nonlinear Functional Analysis by L Nirenberg would be of use
  4. Jun 5, 2016 #3
    Monotonicity work only in 1D,
    [itex]\sup_{x\in\mathbb{R}^2}\Big\|\frac{\partial V}{\partial x}\Big\|[\itex] is non local.
    I will look in the book you advise. I find this problem really interesting, may be treated and treated again, but interesting

    Thanks for your answer.

  5. Jun 5, 2016 #4
    there is multidimensional version of monotonicity, by the way:)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted