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I was watching this video on the Navier-Stokes Equation (https://www.youtube.com/watch?v=XoefjJdFq6k) and had the following questions:

  • @ 0:53 Prove that for any given condition, the Navier-Stokes Equation will spit out a solution that will last for all time : Does this mean that solving the Navier-Stokes Equation will mean that it will become possible to perfectly predict how a fluid will behave at any time in the future based on its current condition - or does it mean that this would allow to predict how a fluid will behave with a range of possible behaviors?

  • @ 4:09 What does it mean for a function to be bigger than another function? Can someone please elaborate on this?

  • @ 4:19 A series of intricate inequalities to bound that term by a series of linear terms : Does this mean that for 2 dimensions, the behavior of a fluid at any future time can be mathematically bounded?

  • @ 4:33 The argument collapsed in 3 dimensions : Can someone give an example of a fluid system in 2 dimensions vs. a fluid system in 3 dimensions? What do "dimensions" correspond to in the Navier-Stokes Equation?

  • @ 4:44 They tried solving an easier version of the equation called the Weak Navier-Stokes Equation : Is this what is currently being used in real life engineering applications?

In short, I can understand the main idea - we can not prove that the Navier Stokes Equation will provide a solution that is accurate for all future conditions. But at the same time, I think that the Navier-Stokes Equation is extensively being used to analyze complex real world fluid systems (note: I do not have an engineering background). Does this mean that:

  • The Navier-Stokes Equation currently is able to well predict the behavior of any fluid at any time in practice, but we can not mathematically prove this fact?

  • As the video implied, a simplified version of the Navier-Stokes Equation is currently used in complex real world situations - this simplified Navier-Stokes Equation is solvable, but in return does not provide a fully accurate solution?

Can someone please help me understand these ideas behind the Navier-Stokes Equation?

Thanks!

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    $\begingroup$ Can you predict the path of one molecule of fluid in turbulent flow? $\endgroup$
    – Solar Mike
    Commented Feb 13, 2022 at 8:10

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@ 0:53 Prove that for any given condition, the Navier-Stokes Equation will spit out a solution that will last for all time

No. It just says it can keep predicting something like the right result. No infinities in result, which is already a good achievement. Your bathtub doesnt explode occasionally, water behaves 'normally'. We want that property, some sort of simulation that doesnt produce infinities.

@ 4:09 What does it mean for a function to be bigger than another function? 

@ 4:19 Does this mean that for 2 dimensions, the behavior of a fluid at any future time can be mathematically bounded?

More like it is restricted to some range of values. Navier stokes is not a perfect simulation, it consist of a lot of averaging. If you restricted conditions are broad enough and there is not much turbulence, then probably.

@ 4:33 The argument collapsed in 3 dimensions : Can someone give an example of a fluid system in 2 dimensions vs. a fluid system in 3 dimensions?

What do "dimensions" correspond to in the Navier-Stokes Equation? - Meaning is number of directions or gradients around the target. Math is in slightly changed formulas to account for more variables.

@ 4:44 They tried solving an easier version of the equation called the Weak Navier-Stokes Equation : Is this what is currently being used in real life engineering applications?

Rarely. What used in real life is very far from it. And very specific to a fluid that is being simulated and even to a hardware. GPU can better solve cells, so we use those. For example this https://en.m.wikipedia.org/wiki/Lattice_Boltzmann_methods

The Navier-Stokes Equation currently is able to well predict the behavior of any fluid at any time in practice, but we can not mathematically prove this fact? - No. It is just an approximation. It is nowhere near close to be able to predict turbulences, and nothing can. It can make similarly behaving simulation, with vortexes of similar size and frequency of appearing, which is good enough usually. Without vortexes it can be solved quite precise.

As the video implied, a simplified version of the Navier-Stokes Equation is currently used in complex real world situations - this simplified Navier-Stokes Equation is solvable, but in return does not provide a fully accurate solution? - Result is better as we improve resolution in time and space and use tricks. We dont use navier stokes directly because other methods in the same time can provide better precision through better time or space resolution or more useful tricks. The more further you go away from a general solution, the more benefits for your particular task you can get.

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  • $\begingroup$ Thank you so much for your answer! Do you have any idea for @4:09? $\endgroup$
    – stats_noob
    Commented Feb 14, 2022 at 17:13
  • $\begingroup$ @stats_noob dont know. It seems some sort of 'importance', or how broad the thing is. They talk about solving navier stokes in more restricted cases like 2d or slow versions. So, they are less important, less broad. I dont like the video in general, they add a truckload of mystery to a math that is not that useful. Stuff that changes the world is the stuff we can calculate quickly and in most cases. So, im biased $\endgroup$
    – Surprised Seagull
    Commented Feb 14, 2022 at 17:28

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