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What does the Navier-Stokes equation solution according to the Clay Math Institute look like in real life? As in how do you visualize $\mathbb{R}^3$ and $\mathbb{T}^3$ without the math?

I actually wanted an illustration of a real life situation, say a bowl with water. How would a problem look like and its corresponding solution in R^3 and T^3 instances.

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    $\begingroup$ Welcome to Physics Stack Exchange! Please include all relevant information in the post. Your readers have no idea what the solution from the Clay Math Institute looks like, nor do they know what you mean by $R^3$ and $T^3$. $\endgroup$
    – DanielSank
    Commented Oct 13, 2015 at 5:32
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    $\begingroup$ @DanielSank The solutions are purely hypothetical and the Clay Institute has no idea (and I think is still, 9/2015, offering a million dollars to people that know) if they exist and whether they are even smooth if they do exist. And $\mathbb R^3$ is regular boring Euclidean 3d space and $\mathbb T^3$ is the same except with periodic solutions that repeat if you go a unit in the x, y, or z directions. $\endgroup$
    – Timaeus
    Commented Oct 13, 2015 at 5:44
  • $\begingroup$ $\mathbb{R}^3$ can be visualized as the ordinary three dimensional space we experience in the real world, extended without limit and without any curvature. $\mathbb{T}^3$ is the three-dimensional version of the torus $\mathbb{T}^2$. $\endgroup$
    – Meer Ashwinkumar
    Commented Oct 13, 2015 at 5:51
  • $\begingroup$ So what is the relation of all this with Navier-Stokes ? What is the question, indeed ? $\endgroup$
    – Fabrice NEYRET
    Commented Oct 30, 2015 at 16:08

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$\mathbb{R}^3$ and $\mathbb{T}^3$ are both manifolds. These have a complicated mathematical definition, but for us physicists they are simple to visualise. $\mathbb{R}^3$ is regular three dimensional flat space - to see what this is like just look around you.

$\mathbb{T}^3$ is the three dimensional torus. This is a bit harder to understand, but it just means that if you travel in a straight line in any direction you end up back where you started. This isn't a realistic model of the universe (or at least we don't think so) but in many cases it makes maths easier because the universe represented by $\mathbb{T}^3$ is finite in size and some of the problems caused by dealing with an infinite universe no longer apply.

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  • $\begingroup$ If you travel in a line with an irrational slope on a unit torus then you don't end up back where you started. $\endgroup$
    – Timaeus
    Commented Oct 13, 2015 at 5:58
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    $\begingroup$ @Timaeus: OK, you explain a three-torus without using any maths! $\endgroup$
    – John Rennie
    Commented Oct 13, 2015 at 5:59
  • $\begingroup$ I think the whole question is asking about the solutions. So the torus solutions are periodic, you jump to the left or right or up or down or forwards or back a "wavelength" and it looks the same. But these solutions are purely hypothetical. And we now that they are wrong unphysical equations anyway as they aren't the correct limit of the actual particle dynamics. So it's hypothetical math about unphysical equations. $\endgroup$
    – Timaeus
    Commented Oct 13, 2015 at 6:02
  • $\begingroup$ @Timaeus How about: "if you travel in a straight line in some directions, you end up back where you began. Moreover, no two points are greater than some finite, bounding distance $d_{max}$ apart". I agree with John though it is very hard to word the description so as to account for the pathological irrational slope line and yet also leave the impression of a compact object without some detailed maths, so you're statement is probably just as good as any in this context. In Peppa Pig, beloved by my son, a person who cites the irrational slope line in this context is known as a "clever cloggs" :) $\endgroup$
    – Selene Routley
    Commented Oct 13, 2015 at 12:15

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