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# Blog - fluid flows and infinite-dimensional manifolds (part 1)

I hope John won't get angry with me for starting yet another blog post project without finishing the others first, but I just need to get some of the stuff out of the way by making a brain dump on some Wiki pages, and a blog stub seems to be a good start:

This will try to explain how fluid flow can be modelled by diffeomorphisms, how certain diffeomorphism groups can be seen as infinite dimensional Riemannian manifolds and how certain nonlinear partial differential equations arise as geodesic equations. This is just too cute to go unmentioned. Related pages are Analytical hydrodynamics and Burgers equation.

Maybe this will grow into more than one blog post. The end should explain that the Burgers equation describes breaking waves, with the breaking of the wave being a singularity of the solution, and what the paper

• Boris Khesin, Gerard Misiolek: "Shock waves for the Burgers equation and curvatures of diffeomorphism groups"

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edited February 2012

The version is ready for review. I hope it is interesting enough to publish it :-) I think I forgot how I am supposed to format references, but John will surely remind me :-)

Comment Source:The version is ready for review. I hope it is interesting enough to publish it :-) I think I forgot how I am supposed to format references, but John will surely remind me :-)
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edited February 2012

Thanks, Tim! It looks great. I'll review it, and if you allow me I'll just go ahead and edit it while I do. I have been dying to talk about that Khesin-Misiolek paper for a while now, so if it's okay with you, maybe I'll write a followup to this blog post of yours. Is that okay?

(In the title, I'll now change 'trough' to 'through'. A 'trough' is where pigs eat.)

Comment Source:Thanks, Tim! It looks great. I'll review it, and if you allow me I'll just go ahead and edit it while I do. I have been dying to talk about that Khesin-Misiolek paper for a while now, so if it's okay with you, maybe I'll write a followup to this blog post of yours. Is that okay? (In the title, I'll now change 'trough' to 'through'. A 'trough' is where pigs eat.)
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edited February 2012

By the way, I don't like blog articles to have a title that's a fun joke that doesn't explain what the article is about, followed by a subtitle that explains what's really going on. In various places on the web, the title is what people will see. So I favor boring straightforward descriptive titles, preferably short enough to easily fit in the skinny blog format.

How about: Fluid flows and infinite-dimensional manifolds?

Comment Source:By the way, I don't like blog articles to have a title that's a fun joke that doesn't explain what the article is about, followed by a subtitle that explains what's really going on. In various places on the web, the title is what people will see. So I favor boring straightforward descriptive titles, preferably short enough to easily fit in the skinny blog format. How about: Fluid flows and infinite-dimensional manifolds?
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edited February 2012

John wrote:

I'll review it, and if you allow me I'll just go ahead and edit it while I do.

Thanks and yes, of course.

I have been dying to talk about that Khesin-Misiolek paper for a while now, so if it's okay with you, maybe I'll write a followup to this blog post of yours. Is that okay?

Yes, of course! Maybe I'll be able to contribute a little bit to this, I'll try to learn more about it anyway. We have an interesting cross section of theoretical insight into singularities and the numerical and experimental phenomenon of breaking waves. I would still like to explore at least the numerical aspect a little bit more and implement the spectral approximation for the Burgers equation, but that will have to wait...

(In the title, I'll now change 'trough' to 'through'. A 'trough' is where pigs eat.)

At least I did run it through a spell checker...

So I favor boring straightforward descriptive titles, preferably short enough to easily fit in the skinny blog format. How about: Fluid flows and infinite-dimensional manifolds?

The title is the only original part of the post, so could we just put it into the subtitle? I think I'll go over there and do it, let's see...

Comment Source:John wrote: <blockquote> <p> I'll review it, and if you allow me I'll just go ahead and edit it while I do. </p> </blockquote> Thanks and yes, of course. <blockquote> <p> I have been dying to talk about that Khesin-Misiolek paper for a while now, so if it's okay with you, maybe I'll write a followup to this blog post of yours. Is that okay? </p> </blockquote> Yes, of course! Maybe I'll be able to contribute a little bit to this, I'll try to learn more about it anyway. We have an interesting cross section of theoretical insight into singularities and the numerical and experimental phenomenon of breaking waves. I would still like to explore at least the numerical aspect a little bit more and implement the spectral approximation for the Burgers equation, but that will have to wait... <blockquote> <p> (In the title, I'll now change 'trough' to 'through'. A 'trough' is where pigs eat.) </p> </blockquote> At least I did run it through a spell checker... <blockquote> <p> So I favor boring straightforward descriptive titles, preferably short enough to easily fit in the skinny blog format. How about: Fluid flows and infinite-dimensional manifolds? </p> </blockquote> The title is the only original part of the post, so could we just put it into the subtitle? I think I'll go over there and do it, let's see...
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I can tell that in another life you'd be a writer, and all your artistry is going into those titles of your posts... so okay, I will let you have your cute titles if that encourages you to blog more!

I'll edit the post a bit and let you know before I actually post it. I'm also eager to post "Quantropy (Part 3)", and Jacob Biamonte has also written something to post... so that's all good.

Comment Source:I can tell that in another life you'd be a writer, and all your artistry is going into those titles of your posts... so okay, I will let you have your cute titles if that encourages you to blog more! <img src = "http://math.ucr.edu/home/baez/emoticons/tongue2.gif" alt = ""/> I'll edit the post a bit and let you know before I actually post it. I'm also eager to post "Quantropy (Part 3)", and Jacob Biamonte has also written something to post... so that's all good.
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There surely is no rush to publish it, so you could keep it up your sleeve for a slow day.

I have found a classic, Whitham: "Linear and Nonlinear Waves", which has an extensive discussion of the Burgers equation and more generally equations of the form $u_t + f(u) u_x = 0$, and how to deduce general information about wavy solutions. Some part of that material would be a good crossover to some numerical results.

Comment Source:There surely is no rush to publish it, so you could keep it up your sleeve for a slow day. I have found a classic, Whitham: "Linear and Nonlinear Waves", which has an extensive discussion of the Burgers equation and more generally equations of the form $u_t + f(u) u_x = 0$, and how to deduce general information about wavy solutions. Some part of that material would be a good crossover to some numerical results.
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7.

There's a funny typo:

If you take a Lie group M of the shelf,

The Edit button didn't work (redirected to wiki home).

Comment Source:There's a funny typo: > If you take a Lie group M of the shelf, The Edit button didn't work (redirected to wiki home).
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edited February 2012

It'll take a while for me to edit and post this blog article, but not too long. I just got finished with an insanely busy round of giving talks; next week I'll be in Chiang Mai, but I can probably do a little blogging from there. I've always loved the idea of fluid flow as geodesic motion on an infinite-dimensional Lie group, ever since I read about it in V. Arnol'd's book, so it'll be fun to get a chance to think and talk about that!

Comment Source:It'll take a while for me to edit and post this blog article, but not too long. I just got finished with an insanely busy round of giving talks; next week I'll be in Chiang Mai, but I can probably do a little blogging from there. I've always loved the idea of fluid flow as geodesic motion on an infinite-dimensional Lie group, ever since I read about it in V. Arnol'd's book, so it'll be fun to get a chance to think and talk about that!
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Martin wrote:

The Edit button didn't work (redirected to wiki home).

Yep, same over here, I started a technical thread for that.

There's a funny typo:

Is the typo "of" instead of "off"? Or is it something else? It has often been helpful for me to structure my thoughts by imagining handling mathematical gadgets like something you can hold in your hand, that's why I use this metaphor (take a Lie group off the shelf, now you have a Lie group in your hand, what can you do with it etc.)

It'll take a while for me to edit and post this blog article, but not too long.

No problem, I'm not busy waiting.

I've always loved the idea of fluid flow as geodesic motion on an infinite-dimensional Lie group, ever since I read about it in V. Arnol'd's book, so it'll be fun to get a chance to think and talk about that!

Yes, it is cute, but besides that, it also has a stairway to the lower realms of the ivory tower (PDEs that describe breaking waves = singularities).

Comment Source:Martin wrote: <blockquote> <p> The Edit button didn't work (redirected to wiki home). </p> </blockquote> Yep, same over here, I started a technical thread for that. <blockquote> <p> There's a funny typo: </p> </blockquote> Is the typo "of" instead of "off"? Or is it something else? It has often been helpful for me to structure my thoughts by imagining handling mathematical gadgets like something you can hold in your hand, that's why I use this metaphor (take a Lie group off the shelf, now you have a Lie group in your hand, what can you do with it etc.) <blockquote> <p> It'll take a while for me to edit and post this blog article, but not too long. </p> </blockquote> No problem, I'm not busy waiting. <blockquote> <p> I've always loved the idea of fluid flow as geodesic motion on an infinite-dimensional Lie group, ever since I read about it in V. Arnol'd's book, so it'll be fun to get a chance to think and talk about that! </p> </blockquote> Yes, it is cute, but besides that, it also has a stairway to the lower realms of the ivory tower (PDEs that describe breaking waves = singularities).
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Tim wrote:

Is the typo "of" instead of "off"? Or is it something else?

It should read "off" methinks. (The symmetry group of my shelf is the permutations :-) )

It has often been helpful for me to structure my thoughts by imagining handling mathematical gadgets like something you can hold in your hand, that's why I use this metaphor (take a Lie group off the shelf, now you have a Lie group in your hand, what can you do with it etc.)

Yeah. I like that, too.

Comment Source:Tim wrote: > Is the typo "of" instead of "off"? Or is it something else? It should read "off" methinks. (The symmetry group **of** my shelf is the permutations :-) ) > It has often been helpful for me to structure my thoughts by imagining handling mathematical gadgets like something you can hold in your hand, that's why I use this metaphor (take a Lie group off the shelf, now you have a Lie group in your hand, what can you do with it etc.) Yeah. I like that, too.
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Interesting article.

I cleaned up a couple of bits of formatting: iTeX treats sequences of letters with no spaces as defining the name of an operator. This is a convenience to avoid having to implement \newcommand-like stuff (which would be Wrong). This works for you for the adjoint operator, $ad$, but against you for tangent spaces and diffeomorphism groups which you sometimes wrote as $TM$ and $DM$ instead of $T M$ and $D M$ (see source of this comment). Also, \coloneqq is a valid iTeX command and $a \coloneqq b$ looks better than $a := b$.

You'll notice that I'm deliberately not mentioning all those Wikipedia links that could have gone to the nLab ...

Comment Source:Interesting article. I cleaned up a couple of bits of formatting: iTeX treats sequences of letters with no spaces as defining the name of an operator. This is a convenience to avoid having to implement \newcommand-like stuff (which would be Wrong). This works for you for the adjoint operator, $ad$, but against you for tangent spaces and diffeomorphism groups which you sometimes wrote as $TM$ and $DM$ instead of $T M$ and $D M$ (see source of this comment). Also, \coloneqq is a valid iTeX command and $a \coloneqq b$ looks better than $a := b$. You'll notice that I'm deliberately *not* mentioning all those Wikipedia links that could have gone to the nLab ...
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edited February 2012

I think for the Azimuth crowd Azimuth should link to Wikipedia, and Wikipedia should link to the nLab, since Wikipedia is more like the general introduction while the nLab is more specialized.

In fact I rather think I should not link to either Wikipedia or the nLab and let everybody make up their own minds where they like to look up math buzz words :-)

(But if you decide to participate in writing a blog post, you could link to the nLab as often as you like, of course)

Comment Source:I think for the Azimuth crowd Azimuth should link to Wikipedia, and Wikipedia should link to the nLab, since Wikipedia is more like the general introduction while the nLab is more specialized. In fact I rather think I should not link to either Wikipedia or the nLab and let everybody make up their own minds where they like to look up math buzz words :-) (But if you decide to participate in writing a blog post, you could link to the nLab as often as you like, of course)
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Thanks, Andrew, for fixing those mistakes related to this:

iTeX treats sequences of letters with no spaces as defining the name of an operator.

I find this feature quite annoying, since the time it saves is small, and it means I have to think about whether I'm typing in TeX or iTeX - as well as fix the mistakes that amateur iTeX users always make when they trip over this feature!

You'll notice that I'm deliberately not mentioning all those Wikipedia links that could have gone to the nLab ...

Thanks! That way I won't need to point out that for most Azimuth readers, Wikipedia links are more useful than nLab links.

Comment Source:Thanks, Andrew, for fixing those mistakes related to this: > iTeX treats sequences of letters with no spaces as defining the name of an operator. I find this feature quite annoying, since the time it saves is small, and it means I have to think about whether I'm typing in TeX or iTeX - as well as fix the mistakes that amateur iTeX users always make when they trip over this feature! > You'll notice that I'm deliberately not mentioning all those Wikipedia links that could have gone to the nLab ... Thanks! That way I won't need to point out that for most Azimuth readers, Wikipedia links are more useful than nLab links.
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I have to think about whether I'm typing in TeX or iTeX

Actually, you don't. If you always space stuff out that shouldn't go together then it will look right in both TeX and iTeX. I started doing this anyway to avoid my spell-checker complaining about things like dx, but it's also good TeX style to separated things that ought to be separated. In maths mode, TeX ignores spaces that you put in because it knows better where to put spaces and how big those spaces should be. That's why you have to use commands like \; and so on to override TeX's spacing algorithms.

Comment Source:> I have to think about whether I'm typing in TeX or iTeX Actually, you don't. If you always space stuff out that shouldn't go together then it will look right in both TeX and iTeX. I started doing this anyway to avoid my spell-checker complaining about things like dx, but it's also good TeX style to separated things that ought to be separated. In maths mode, TeX **ignores spaces** that you put in because it knows better where to put spaces and how big those spaces should be. That's why you have to use commands like \; and so on to override TeX's spacing algorithms.
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15.

Tim: in your blog article, you wrote:

We will take as a model of the domain of the fluid flow a compact Riemannian manifold $M$ that is also a Lie group.

I really don't think we need $M$ to be a Lie group. What we need is for the diffomorphism group $\mathrm{Diff}(M)$ to be an infinite-dimensional Lie group - and it always will be if $M$ is a compact Riemannian manifold. So I'm going to delete that last bit.

I'm also going to change your $D M$ to $\mathrm{Diff}(M)$ because the latter is standard and you have another very important thing called $D$ in this blog entry!

Comment Source:Tim: in your blog article, you wrote: > We will take as a model of the domain of the fluid flow a compact Riemannian manifold $M$ that is also a Lie group. I really don't think we need $M$ to be a Lie group. What we need is for the diffomorphism group $\mathrm{Diff}(M)$ to be an infinite-dimensional Lie group - and it always will be if $M$ is a compact Riemannian manifold. So I'm going to delete that last bit. I'm also going to change your $D M$ to $\mathrm{Diff}(M)$ because the latter is standard and you have another very important thing called $D$ in this blog entry!
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Tim wrote:

We will take as a model of the domain of the fluid flow a compact Riemannian manifold ...

This might come as a bit of a shock to some, since the simplest 3d example is a 3-torus, and normal folks don't think about water flowing around in a 3-torus! We are saying goodbye to ordinary reality and embracing the joys of math here. Can you add a sentence or 2 to calm the reader?

It's technically a bit more tricky, but perfectly fine, to work with a compact Riemannian manifold with boundary. Then we can study water in a bottle.

Comment Source:Tim wrote: > We will take as a model of the domain of the fluid flow a compact Riemannian manifold ... This might come as a bit of a shock to some, since the simplest 3d example is a 3-torus, and normal folks don't think about water flowing around in a 3-torus! We are saying goodbye to ordinary reality and embracing the joys of math here. Can you add a sentence or 2 to calm the reader? It's technically a bit more tricky, but perfectly fine, to work with a compact Riemannian manifold _with boundary_. Then we can study water in a bottle.
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17.

Tim wrote:

We call a function between manifolds smooth if it maps smooth curves to smooth curves...

This is not the usual definition of a smooth function between Fréchet spaces (or Fréchet manifolds); the usual definition is here.

I believe the definition you're proposing is weaker (except for $\mathbb{R}^n$, where its a theorem in Michor and Kriegl's book that it's equivalent to the usual one). I'm not 100% sure it's weaker! But it's certainly not the one people usually use in the study of Fréchet manifolds.

So, we should fix this or find the mistake I'm making now...

Comment Source:Tim wrote: > We call a function between manifolds <b>smooth</b> if it maps smooth curves to smooth curves... This is not the usual definition of a smooth function between Fr&eacute;chet spaces (or Fr&eacute;chet manifolds); the usual definition is <a href = "http://en.wikipedia.org/wiki/Fr%C3%A9chet_space#Properties_and_further_notions">here</a>. I believe the definition you're proposing is weaker (except for $\mathbb{R}^n$, where its a theorem in Michor and Kriegl's book that it's equivalent to the usual one). I'm not 100% sure it's weaker! But it's certainly not the one people usually use in the study of Fr&eacute;chet manifolds. So, we should fix this or find the mistake I'm making now...
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Tim wrote:

We call a function between manifolds smooth if it maps smooth curves to smooth curves; in this way we get the definition of a smooth Fréchet manifold.

I also don't like this sentence because it seems to use the concept of smooth Fréchet manifold to define the concept of smooth Fréchet manifold! What we really want is to define smooth functions between (open subsets of) Fréchet spaces, and then use that to define smooth Fréchet manifolds, by requiring that their transition functions be smooth.

Comment Source:Tim wrote: > We call a function between manifolds <b>smooth</b> if it maps smooth curves to smooth curves; in this way we get the definition of a smooth Fréchet manifold. I also don't like this sentence because it seems to use the concept of smooth Fréchet manifold to define the concept of smooth Fréchet manifold! What we really want is to define smooth functions between (open subsets of) Fréchet spaces, and then use that to define smooth Fréchet manifolds, by requiring that their transition functions be smooth.
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Tim wrote:

The dual space will still be a Banach space, but a Banach space does not need to be isomorphic to its dual (those who are are called reflexive).

No, a Banach space $V$ is reflexive if the canonical map $i : V \to V^{**}$ is an isomorphism. So for example $L^p(X)$ is reflexive for $1 \lt p \lt \infty$ when $X$ is a $\sigma$-finite measure space, but it's only isomorphic to its dual when $p = 2$.

Comment Source:Tim wrote: > The dual space will still be a Banach space, but a Banach space does not need to be isomorphic to its dual (those who are are called reflexive). No, a Banach space $V$ is [reflexive](http://en.wikipedia.org/wiki/Reflexive_space) if the canonical map $i : V \to V^{**}$ is an isomorphism. So for example $L^p(X)$ is reflexive for $1 \lt p \lt \infty$ when $X$ is a $\sigma$-finite measure space, but it's only isomorphic to its dual when $p = 2$.
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edited March 2012

John wrote:

No, a Banach space V is reflexive if the canonical map i:V→V ** is an isomorphism

Whoops!

I also don't like this sentence because it seems to use the concept of smooth Fréchet manifold to define the concept of smooth Fréchet manifold! What we really want is to define smooth functions between (open subsets of) Fréchet spaces, and then use that to define smooth Fréchet manifolds, by requiring that their transition functions be smooth.

Right, in my head I thought about the model space as example of topological manifolds, then the definition of smooth maps between them etc.

I believe the definition you're proposing is weaker (except for ℝ n, where its a theorem in Michor and Kriegl's book that it's equivalent to the usual one). I'm not 100% sure it's weaker! But it's certainly not the one people usually use in the study of Fréchet manifolds.

I don't know either, but we could present the usual definition and then this - possibly - more general one and ask this as a question in the blog post.

Comment Source:John wrote: <blockquote> <p> No, a Banach space V is reflexive if the canonical map i:V→V ** is an isomorphism </p> </blockquote> Whoops! <blockquote> <p> I also don't like this sentence because it seems to use the concept of smooth Fréchet manifold to define the concept of smooth Fréchet manifold! What we really want is to define smooth functions between (open subsets of) Fréchet spaces, and then use that to define smooth Fréchet manifolds, by requiring that their transition functions be smooth. </p> </blockquote> Right, in my head I thought about the model space as example of topological manifolds, then the definition of smooth maps between them etc. <blockquote> <p> I believe the definition you're proposing is weaker (except for ℝ n, where its a theorem in Michor and Kriegl's book that it's equivalent to the usual one). I'm not 100% sure it's weaker! But it's certainly not the one people usually use in the study of Fréchet manifolds. </p> </blockquote> I don't know either, but we could present the usual definition and then this - possibly - more general one and ask this as a question in the blog post.
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John wrote:

This might come as a bit of a shock to some, since the simplest 3d example is a 3-torus, and normal folks don't think about water flowing around in a 3-torus! We are saying goodbye to ordinary reality and embracing the joys of math here. Can you add a sentence or 2 to calm the reader? It's technically a bit more tricky, but perfectly fine, to work with a compact Riemannian manifold with boundary. Then we can study water in a bottle.

I wasn't thinking about readers who would like to think in three dimensions, I only thought about one and two dimensional examples. So that's clearly a hole in the exposition.

With regard to

Could you give me a heads-up when you are done with your first sweep? I'd like to avoid racing conditions :-)

Comment Source:John wrote: <blockquote> <p> This might come as a bit of a shock to some, since the simplest 3d example is a 3-torus, and normal folks don't think about water flowing around in a 3-torus! We are saying goodbye to ordinary reality and embracing the joys of math here. Can you add a sentence or 2 to calm the reader? It's technically a bit more tricky, but perfectly fine, to work with a compact Riemannian manifold with boundary. Then we can study water in a bottle. </p> </blockquote> I wasn't thinking about readers who would like to think in three dimensions, I only thought about one and two dimensional examples. So that's clearly a hole in the exposition. With regard to <blockquote> <p> Can you add a sentence or 2 to calm the reader? </p> </blockquote> Could you give me a heads-up when you are done with your first sweep? I'd like to avoid racing conditions :-)
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I'm not too keen on using that Wikipedia page as the source for the "usual definition" of smoothness for Frechet manifolds. There's a lot of details missing there. I usually quote the "Historical Remarks" at the end of the first chapter of Kriegl and Michor's book where they note that there are several inequivalent notions of what $C^\infty$ means even for Frechet spaces. That's sort of the point of their work: rather than seeing "smooth" as $C^\infty$ (ie the limit of $C^k$), they say "smooth is as smooth does" and start from what we would expect smooth maps to do. Explained right, this can actually be more intuitive and so easier to understand than "locally diffeomorphic to an open subset of a Frechet space".

Comment Source:I'm not too keen on using that Wikipedia page as the source for the "usual definition" of smoothness for Frechet manifolds. There's a lot of details missing there. I usually quote the "Historical Remarks" at the end of the first chapter of Kriegl and Michor's book where they note that there are several inequivalent notions of what $C^\infty$ means even for Frechet spaces. That's sort of the point of their work: rather than seeing "smooth" as $C^\infty$ (ie the limit of $C^k$), they say "smooth is as smooth does" and start from what we would expect smooth maps to do. Explained right, this can actually be more intuitive and so easier to understand than "locally diffeomorphic to an open subset of a Frechet space".
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23.

Tim wrote:

Could you give me a heads-up when you are done with your first sweep?

I'm done with my first sweep.

I'm not too keen on using that Wikipedia page as the source for the "usual definition" of smoothness for Frechet manifolds. There's a lot of details missing there.

Maybe I'll add them, or maybe we should do what you suggest:

I usually quote the "Historical Remarks" at the end of the first chapter of Kriegl and Michor's book.

The 'standard' theory of Fréchet manifolds as in Palais' book is different from Kriegl and Michor's new approach, and Tim should probably pick one of those and talk only about that, instead of scaring my readers (who presumably are interested in fluid flow more than this foundational stuff) with two approaches.

In fact, as far as I know all the actual hard work on fluid dynamics uses Hilbert manifolds rather than Fréchet manifolds! So personally I would have stuck with Hilbert manifolds (which requires thinking about Sobolev spaces, but are otherwise more like ordinary finite-dimensional Riemannian manifolds) rather than talk about Fréchet manifolds. But of course Tim gets to do what he wants!

Comment Source:Tim wrote: > Could you give me a heads-up when you are done with your first sweep? I'm done with my first sweep. > I'm not too keen on using that Wikipedia page as the source for the "usual definition" of smoothness for Frechet manifolds. There's a lot of details missing there. Maybe I'll add them, or maybe we should do what you suggest: > I usually quote the "Historical Remarks" at the end of the first chapter of Kriegl and Michor's book. The 'standard' theory of Fr&eacute;chet manifolds as in Palais' book is different from Kriegl and Michor's new approach, and Tim should probably pick one of those and talk only about that, instead of scaring my readers (who presumably are interested in fluid flow more than this foundational stuff) with two approaches. In fact, as far as I know all the actual hard work on fluid dynamics uses Hilbert manifolds rather than Fr&eacute;chet manifolds! So personally I would have stuck with Hilbert manifolds (which requires thinking about Sobolev spaces, but are otherwise more like ordinary finite-dimensional Riemannian manifolds) rather than talk about Fr&eacute;chet manifolds. But of course Tim gets to do what he wants!
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24.

Let me know when you're ready, Tim.

Comment Source:Let me know when you're ready, Tim.
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25.

I removed the definition of smoothness of Kriegl and Michor and mention the way Hamilton does it in the paper I cite only. I hope that clears up matters for the Azimuth readers. Besides this point, is there anything else to do? If not, I'm ready.

BTW: If you (anyone) has a comment, it would be easier for me to find it if you use a specific prefix, like "TODO". That's what programmers like and what the integrated development environment "eclipse" supports, for example (it automatically complies a list of all comments containing the string "TODO").

John: I found your comment on the definition of smooth for functions on Fréchet spaces and also the missing page numbers for Hamilton's paper.

Comment Source:I removed the definition of smoothness of Kriegl and Michor and mention the way Hamilton does it in the paper I cite only. I hope that clears up matters for the Azimuth readers. Besides this point, is there anything else to do? If not, I'm ready. BTW: If you (anyone) has a comment, it would be easier for me to find it if you use a specific prefix, like "TODO". That's what programmers like and what the integrated development environment "eclipse" supports, for example (it automatically complies a list of all comments containing the string "TODO"). John: I found your comment on the definition of smooth for functions on Fréchet spaces and also the missing page numbers for Hamilton's paper.
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26.

I think your blog post may be ready to go, Tim. I'll try to transfer it to the Azimuth Blog and see how it looks, making small changes if needed. Then with luck I'll post it.

Thanks!!!

Comment Source:I think your blog post may be ready to go, Tim. I'll try to transfer it to the Azimuth Blog and see how it looks, making small changes if needed. Then with luck I'll post it. Thanks!!!
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edited March 2012

Okay, it's here! It's very nice, I think. If someone sees typos, let me know.

I added a remark about how 'adjoint' is being used in two completely different meanings in the same sentence.

Comment Source:Okay, it's [here](http://johncarlosbaez.wordpress.com/2012/03/12/fluid-flows-and-infinite-dimensional-manifolds/)! It's very nice, I think. If someone sees typos, let me know. I added a remark about how 'adjoint' is being used in two completely different meanings in the same sentence.
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28.

Yes, saw that :-) The "adjoint representation" has always bugged me since I first learned about Lie algebras. We could start to think about a follow-up.

Comment Source:Yes, saw that :-) The "adjoint representation" has always bugged me since I first learned about Lie algebras. We could start to think about a follow-up.
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29.

It's interesting that 'conjugate' has two meanings which are somewhat similar to the two meanings of 'adjoint'. Weird, huh? And there's even a word 'adjugate' - but I forget what that means.

I'll start by writing a new revised to-do list.

I would like to write an article about some aspects of this paper that you didn't cover:

Or maybe you'd like to?

I really like the 'free particle picture' for solving the equations. I talked about it here, and this picture makes it very clear why we get shocks.

It would also be fun to talk about chaos in the incompressible Euler equations, following Arnol'd. That's a bit closer to 'chaos in weather', which everyone likes.

Comment Source:It's interesting that 'conjugate' has two meanings which are somewhat similar to the two meanings of 'adjoint'. Weird, huh? And there's even a word 'adjugate' - but I forget what that means. I'll start by writing a new revised to-do list. I would like to write an article about some aspects of this paper that you didn't cover: * B. Khesina and G. Misiolek, [Shock waves for the Burgers equation and curvatures of diffeomorphism groups](http://arxiv.org/abs/math/0702196). Or maybe you'd like to? I really like the 'free particle picture' for solving the equations. I talked about it [here](http://forum.azimuthproject.org/discussion/577/burgers-equation/?Focus=5497#Comment_5497), and this picture makes it very clear why we get shocks. It would also be fun to talk about chaos in the incompressible Euler equations, following Arnol'd. That's a bit closer to 'chaos in weather', which everyone likes.
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30.

Or maybe you'd like to?

I haven't had time yet to read and understand it - so we could make a little race and whoever has enough time and energy starts it, and we'll see how we go on from there ;-)

Comment Source:<blockquote> <p> Or maybe you'd like to? </p> </blockquote> I haven't had time yet to read and understand it - so we could make a little race and whoever has enough time and energy starts it, and we'll see how we go on from there ;-)
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31.

Okay. I think the whole point of the inviscid Burgers' equation is that it describes a 'gas' of tiny particles, one at each point of spacetime, where each particle moves at constant velocity until two of them collide.

Comment Source:Okay. I think the whole point of the inviscid Burgers' equation is that it describes a 'gas' of tiny particles, one at each point of spacetime, where each particle moves at constant velocity until two of them collide.
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32.
edited May 2012

I've changed the title of this blog article and corresponding wiki page to

Blog - fluid flows and infinite-dimensional manifolds (part 1)

since now it's part 1 of a multi-part series.

Comment Source:I've changed the title of this blog article and corresponding wiki page to [[Blog - fluid flows and infinite-dimensional manifolds (part 1)]] since now it's part 1 of a multi-part series.