Terminology: equation collapse

[nomadreid]

This is more a terminology than a conceptual question. I am proof-reading a paper (by a non-native speaker) in which the statement occurs:
" the solution of the Navier–Stokes equation collapses within a finite time into simple nonlinear waves"
Is this a possible phrasing? I am familiar with the wavefunction collapse, the collapse of large cardinals, transitive collapse (Mostowski isomorphism), the collapse of the economy, collapse of a building, and so forth, but can one say that a solution collapses? Is there a better phrasing?

 

[jedishrfu]

I suppose they could have said reduces to. Collapse to me implies something time related or perhaps limit related whereas reduces to implies some factor is removed or canceled out to give a simpler solution.

 

[nomadreid]

Thanks, jedihrfu. I like "reduces to." but in that statement he does connect it to a temporal process ("within a finite time"). So on one side there is the time (collapses), on the other side it is an algebraic derivation ("reduces to"), so I remain puzzled.
This brings up another question which I forgot to pose because I am not too familiar with Navier-Stokes: the author refers to "equation" in the singular, whereas I always see the plural "equations". The equation(s) in question is (are) indexed, so I would be inclined to change his singular to plural, but I am not sure whether one can refer to the following in the singular:

 

[fresh_42]

I don't think that collapse is a good word here, but I'm no native speaker either. In connection with differential equations, one usually speaks of the stability of a solution, attractors and repellers. The above sounds more like an unstable equilibrium, than a collapse.

 

[nomadreid]

Thanks, fresh_42. Assuming this represents an unstable equilibrium, how would you rephrase the author's idea that the solution in question is an equation which represents a situation which loses smoothness (by which I am not sure whether he means mathematical smoothness, i.e., differentiability, or physical smoothness) over a large interval of time?

 

[fresh_42]

I doubt that mathematical smoothness is meant, but I'm not sure what a simple nonlinear wave should be. Waves are pretty smooth, and I cannot see, where singularities should mathematically arise other than in asymptotic behavior. An unstable equilibrium is the same as in physics: the decay to lower energy states, which is no surprise, as physics and differential geometry is more or less the same.

I would say for 'the solution collapses over a finite amount of time into' as 'the solution is unstable and decays over a finite time into the stable ...'. At least this is what I expect to happen by what you've described, not quite sure whether this is what actually happens.

Another term are regular and abnormal solutions, I think. At least Tao used them here:
https://terrytao.wordpress.com/2019/01/08/255b-notes-2-onsagers-conjecture/

 

[nomadreid]

Super. Thanks for all that, fresh_42. That helps a lot.

 

[jedishrfu]

The NS equations are vector field equations and so they represent each coordinate in your space including time:

https://en.wikipedia.org/wiki/Navier–Stokes_equations

In vector notation, they would appear as one equation but to work with them you need to realize that the one vector equation represents a set of equations, one for each coordinate in whatever coordinate system you are using ie cartesian( xyz ), spherical( r theta phi) or cylindrical( r theta z) ... (your angles may vary :-) )

Here's a colorful introduction the the equations:



and more prosaic discussions:

 

[nomadreid]

Thanks, jedishrfu. So, clear, N-S equationS it is, then.
Also thanks for the three videos. They will help me get more acquainted with the equations. Videos like the "Stripped Down" video are my favourite kind of introduction to a field, and then the more prosaic ones become more palatable.

 

[nomadreid]

Update: When I questioned said author's use of the wod "collapse" as I have outlined here, he referred me to the frequent use of the word "collapse" by the physicist E.A. Kuznetsov in the same context, for example in https://arxiv.org/pdf/physics/0204080.pdf, as justification. Is Kuznetsov's use justified, or is it also questionable?

 

[fresh_42]

... then the vorticity blows up (or collapses) in a finite time ...

I would not call a Russian as witness to decide a linguistic question about English. That does not mean, that Kuznetsov cannot be understood, it only means, that a) English is very likely not his native language, and b) that common Russian is a lot more colorful than common English is. (You should hear them swear!)

In his abstract, Kuznetsov says

Blow-up of this quantity means that solution of the Hopf equation in 3D can not be continued in the Sobolev space $H^2 (\mathbb{R}^3 )$
for infinite time.

so it is at least defined - roughly. I think a debate on whether this is a suitable term for the behavior of a solution won't make much sense. I would have chosen another, but Kuznetsov's is a possibility. In this sense, the question will be: Does your friend use it in the same way, namely: the solution will leave $H^2 (\mathbb{R}^3 )$, or does he use it similarly handwaving?

I would stick with Tao, i.e. less colorful expressions. At least he grew up in English.

 

[nomadreid]

Thanks, fresh_42. Sorry about the delay in answering. Here is the author's explanation (translated from his Russian*)
"The term 'collapse of solution' means that at some point in time to this solution loses smoothness and its derivative with respect to the spatial variable tends to infinity. This is an instantaneous process, not an extension of time. That is, only when t < to, the solution is smooth."
I took your advice and suggested less colorful expressions, but he insisted, so I let him keep it. It's his paper.
(*Off-topic: Um, yes, I went through several years of listening to them swear. Although they have a very big set of expressions to choose from, however, they tend to stick to a choice few. Not as restricted as the Americans, but not as free as the Italians.)