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As a student of meteorology, I wonder why Navier-Stokes equations (NSE) are still not understood in terms of whether or not there are unique solutions. In atmospheric dynamics, NSE is used as a basic equation of motion governing the dynamic behavior of air parcels, taught from the first lecture on and included (in direct or modified form) in numerical prediction algorithms.

Assuming the atmosphere can be modeled by NSE, we know there ARE solutions because when we look out the window there is SOME weather. What I mean: IF a physical solution exists for all times and for the whole field, there IS a solution, even if we don't know it.

So what is the mathematical problem in practical terms? From a naïve point of view, I am tempted to assume that atmospheric motion for some spherical Earth model can be predicted spatially and for all time with arbitrary precision if we know the starting conditions with arbitrary precision. Is there a point at which a numerical prediction based on some initial solution necessarily fails or suddenly becomes ambiguous?

I've never tried: but does it really mean we have not one known solutions for $u(x,t)$ and a given $p(x,t)$ and $\nu>0$? Not even some trivial ones? Hard to believe; can someone shed some light on it without going into deep maths?

Kyle Kanos
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MichaelW
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    Note that real gases are made of individual atoms while the hypothetical Navier-Stokes fluid is infinitely divisible. Thus existence could fail because the solution develops arbitrarily fine structure, and at some point it stops being a good model of a physical gas. – eyeballfrog Oct 14 '23 at 15:13
  • "Assuming the atmosphere can be modeled by NSE, we know there ARE solutions because when we look out the window there is SOME weather." You assumed x & x followed. So what? This tells you nothing about the equation or the weather or their combination. – philipxy Oct 15 '23 at 20:37
  • @eyeballfrog you are over-egging that one a bit. The physical fluid need not be continuous, and indeed, continuum mechanics was assembled by people who were firm believers in the atomic nature of matter and is consistent with that belief. The Navier-Stokes existence and uniqueness problems exist within the continuous domain – Phil Sweet Oct 15 '23 at 23:59

2 Answers2

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I've never tried: but does it really mean we have not one known solutions for $u(x,t)$ and a given $p(x,t)$ and $>0$? Not even some trivial ones? Hard to believe... can someone shed some light on it without going into deep maths?

There are several, well-known analytic solutions to the NSE for specific functions of pressure and initial conditions $u^\circ$. This (closed) question on this site contains one such list of exact solutions.

So the problem isn't that one wants solutions for specific functions of pressure and velocity, it's for arbitrary functions (well, any $C^\infty$ functions whose domain is $\mathbb{R}^n$ for $t>0$) that satisfy not only the NSE & initial conditions, but some additional constraints (e.g., bounded energy: $\int\vert u\vert^2 \,\mathrm{d}x<E$).

Additionally, the proofs ought to be mathematical proofs (i.e., relying on deduction and logic). Results of numerical model are (a) going to be specific to the inputs (rather than generic $u(x,t)$, $p(x,t)$) and (b) could considered a datapoint in a statistical proof (which probably wouldn't be valid anyway).

You may also want to read What do mathematicians mean by Navier Stokes existence and smoothness problem?

Kyle Kanos
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    Indeed, and it should be mentioned also that general solution(s) to Navier-Stokes equations should be able to describe compressible flows and turbulence behavior on general, not just only for specific geometries/shapes and conditions. What's good to know that some progress has been made for deciphering turbulence. – Agnius Vasiliauskas Oct 13 '23 at 20:01
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I will add to Kyle Kanos' answer.

Assuming the atmosphere can be modeled by NSE, we know there ARE solutions because when we look out the window there is SOME weather. What I mean: IF a physical solution exists for all times and for the whole field, there IS a solution, even if we don't know it.

A mathematical model, no matter how sophisticated, is at best a close approximation of a physical system; there is never an exact correspondence between the two. Therefore, one cannot use the continued existence and uniqueness of the physical system to infer the existence and uniqueness of mathematical solutions to the given model.

When I was an undergraduate student, I participated in a research project with a friend in which we formulated a model for a physical system. We explored existence and uniqueness of solutions to the model. We claimed to our peers that "of course we know solutions must exist because the corresponding physical system exists. We just need to mathematically formalize it." Now, several years later, it seems the height of arrogance to assume that just because we wrote down a naive model, the physical system must conform to it.

Now I know that it is up to the scientist to show that their mathematical model closely resembles reality, not the reverse. Since the physical system exists and is unique, one way for the scientist to justify their model is to prove that its solutions also exist and are unique.

Plutoro
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