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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.

    Thanks

    Clément
     
  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.

    Clément
     
  5. Jun 5, 2016 #4
    there is multidimensional version of monotonicity, by the way:)
     
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