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edited Apr 13, 2017 at 12:58

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(This is mostly an extended comment to answer this comment this comment.)

Yes, you are right to worry about the curves. If you initial data curve is real analytic, then you can find a local, real analytic change of variables, so that the proof using CK goes through identically.

When your initial data curve is only smooth, but not real analytic, you will then need to construct a sequence of approximate change of variables and show that the appropriate limit converges. The sequence of approximate change of variables will be based on first approximating your data curve using something like Weierstrass approximation by a sequence of real analytic curves, and then straighten out these approximate curves using a real analytic change of variables. The fact that hyperbolic equations have finite speeds of propagation means that the inevitable spatial localization in this procedure is relatively harmless.

Unfortunately I am not aware of any place where this argument is written down in full detail. It is one of those folk theorems that "everyone knows is true", and many subparts or special cases of it can be found in many places.

(This is mostly an extended comment to answer this comment.)

Yes, you are right to worry about the curves. If you initial data curve is real analytic, then you can find a local, real analytic change of variables, so that the proof using CK goes through identically.

When your initial data curve is only smooth, but not real analytic, you will then need to construct a sequence of approximate change of variables and show that the appropriate limit converges. The sequence of approximate change of variables will be based on first approximating your data curve using something like Weierstrass approximation by a sequence of real analytic curves, and then straighten out these approximate curves using a real analytic change of variables. The fact that hyperbolic equations have finite speeds of propagation means that the inevitable spatial localization in this procedure is relatively harmless.

Unfortunately I am not aware of any place where this argument is written down in full detail. It is one of those folk theorems that "everyone knows is true", and many subparts or special cases of it can be found in many places.

(This is mostly an extended comment to answer this comment.)

Yes, you are right to worry about the curves. If you initial data curve is real analytic, then you can find a local, real analytic change of variables, so that the proof using CK goes through identically.

When your initial data curve is only smooth, but not real analytic, you will then need to construct a sequence of approximate change of variables and show that the appropriate limit converges. The sequence of approximate change of variables will be based on first approximating your data curve using something like Weierstrass approximation by a sequence of real analytic curves, and then straighten out these approximate curves using a real analytic change of variables. The fact that hyperbolic equations have finite speeds of propagation means that the inevitable spatial localization in this procedure is relatively harmless.

Unfortunately I am not aware of any place where this argument is written down in full detail. It is one of those folk theorems that "everyone knows is true", and many subparts or special cases of it can be found in many places.

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created May 8, 2016 at 3:45

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(This is mostly an extended comment to answer this comment.)

Yes, you are right to worry about the curves. If you initial data curve is real analytic, then you can find a local, real analytic change of variables, so that the proof using CK goes through identically.

When your initial data curve is only smooth, but not real analytic, you will then need to construct a sequence of approximate change of variables and show that the appropriate limit converges. The sequence of approximate change of variables will be based on first approximating your data curve using something like Weierstrass approximation by a sequence of real analytic curves, and then straighten out these approximate curves using a real analytic change of variables. The fact that hyperbolic equations have finite speeds of propagation means that the inevitable spatial localization in this procedure is relatively harmless.

Unfortunately I am not aware of any place where this argument is written down in full detail. It is one of those folk theorems that "everyone knows is true", and many subparts or special cases of it can be found in many places.