I Condition on vector field to be a diffeomorphism.

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A condition for a vector field V(x) on R^2 to ensure that the mapping x^µ → x^µ + V^µ(x) is a diffeomorphism involves controlling the norm of the derivative of V. Specifically, if the supremum of the norm of the partial derivatives of V is sufficiently small, it can guarantee the diffeomorphic property. Monotonicity assumptions can also be beneficial, although they primarily apply in one-dimensional cases. There are multidimensional versions of monotonicity that may be relevant to this discussion. The problem of finding a generic condition for V to avoid collapsing points remains a complex and intriguing topic.
kroni
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Hi everybody,

Let V(x) a vector field on a manifold (R^2 in my case), i am looking for a condition on V(x) for which the function x^µ \rightarrow x^µ + V^µ(x) 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
 
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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
 
Monotonicity work only in 1D,
\sup_{x\in\mathbb{R}^2}\Big\|\frac{\partial V}{\partial x}\Big\|[\itex] is non local.<br /> I will look in the book you advise. I find this problem really interesting, may be treated and treated again, but interesting<br /> <br /> Thanks for your answer.<br /> <br /> Clément
 
kroni said:
Monotonicity work only in 1D,
there is multidimensional version of monotonicity, by the way:)
 
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