Re: How to prove advection flows are topology preserving?

From: Jonathan Dursi (ljdursi_at_eugenia.asci.uchicago.edu)
Date: 07/18/04 

Date: Sun, 18 Jul 2004 21:43:41 GMT

In article <40FAAF48.3080706@cauchy.math.missouri.edu>, Stephen
Montgomery-Smith wrote:
> manuel gamito wrote:
>> Hello all,
>>
>> My intuition tells me that if I have an initial set S and I advect it
>> through some continuous flow field V(x) with:
>>
>> dx/dt = V(x) for every particle x such that x(0) \in S
>>
>> then the topology of the deformed set S after advection will not
>> change. Specifically, the set will not break apart and will not merge
>> (if initially it had disconnected parts). It will also not
>> self-intersect, if the flow V(x) is single-valued.
>>
>> Can anyone tell me where I can find a reference where this is proven?
>> Or else, if my intuition is wrong after all, what other conditions
>> have to be imposed on V(x) in order to enforce topology preservation?

An excellent book on this sort of problem is

The Kinematics of Mixing : Stretching, Chaos, and Transport (Cambridge
Texts in Applied Mathematics) by J. M. Ottino, ISBN: 0521368782.

If what you wrote were true, there would be no way to mix two fluids
besides waiting for diffusive processes to occur. Even in the
incompressible limit, if your set were (for instance) to flow towards
two opposing vorticies, it would be torn into two sections.

One criteria for whether a flow is mixing or not in the sense that
your intial set gets mixed into the larger flow is whether the Lyaponov
exponents of the flow field are ever positive.

        - Jonathan 

-- 
ljdursi@flash.uchicago.edu

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