Condition on vector field to be a diffeomorphism.

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  • Thread starter kroni
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  • #1
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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
 

Answers and Replies

  • #2
wrobel
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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
 
  • #3
80
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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
 
  • #4
wrobel
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Monotonicity work only in 1D,
there is multidimensional version of monotonicity, by the way:)
 

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