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I have been reading the paper The Incompressible Non-Relativistic Navier-Stokes Equation from Gravity. In it they state,

"An instability, if it occurs, must necessarily break a symmetry ... that the background solution preserves."

There is also discussion about the non-linear terms contributing to instabilities. Although I have some thoughts about how this should be interpreted. I wanted to see if there is some standard discussion relating to non linear instabilities in this context.


In order to provide more specificity, in standard discussion for understanding string theory, weak field approximations for gravity involve linearized fluctuations around a Minkowski metric: $$g_{\mu\nu}(x) = \eta_{\mu\nu}+h_{\mu\nu}(x)$$

In stronger fields, the fluctuations become highly non-linear. The non-linearity is thus understood as being associated with strong gravitational fields.

The Navier-Stokes equations can be understood as a composition of two types of equations, the Heat equations and the Euler equations. In a crude sense, the unforced heat equations generally govern the diffusion of energy throughout an object, and the Euler equations govern the conservation of mass.

The incompressible Navier-Stokes equations in cartesian coordinates are:

$\partial_tu+u\partial_xu+v\partial_yu+w\partial_zu = -\partial_xp + \nu(\partial_{xx}u+\partial_{yy}u+\partial_{zz}u)$

$\partial_tv+u\partial_xv+v\partial_yv+w\partial_zv = -\partial_yp + \nu(\partial_{xx}v+\partial_{yy}v+\partial_{zz}v)$

$\partial_tw+u\partial_xw+v\partial_yw+w\partial_zw = -\partial_zp + \nu(\partial_{xx}w+\partial_{yy}w+\partial_{zz}w)$


Where one can identify the "heat equation terms" as (noting that in complex form these are the Schrodinger equations):

$\partial_tu = \nu(\partial_{xx}u+\partial_{yy}u+\partial_{zz}u)$

$\partial_tv = \nu(\partial_{xx}v+\partial_{yy}v+\partial_{zz}v)$

$\partial_tw = \nu(\partial_{xx}w+\partial_{yy}w+\partial_{zz}w)$

And one can identify the "Eulerian terms" as:

$u\partial_xu+v\partial_yu+w\partial_zu = -\partial_xp$

$u\partial_xv+v\partial_yv+w\partial_zv = -\partial_yp$

$u\partial_xw+v\partial_yw+w\partial_zw = -\partial_zp$

In the solution I provided in a previous question, the interesting characteristics is that solution satisfies both sets of equations (sans the pressure term for the heat equation). The pressure identified is also negative, and the solution is unbounded. This doesn't seem to be the type of space identified with the AdS in the above referenced paper or the related paper discussing the gravity-fluid dynamics relationship. There is one other thing that is of interest, and that is the vanishing of the non-linear and pressure terms with the given solution (related to the apparent separability of terms).

If I relate the Eulerian terms with gravity, then the solution in my mind suggest that there are possibly some set of solutions that can be shared in common with both the Heat equation terms and the Eulerian terms (and the number of these are unknown).

It seems that those solutions would have some sort of privileged status, or perhaps they are completely trivial and not interesting, but I am thinking that answer is context dependent. However, I am somewhat encouraged that those solutions are not trivial, since bifurcation is not generally considered a good thing in equations describing physical phenomenon.

So getting back to the original question, in the above referenced paper it is implied that instability at high Reynolds numbers ($\frac{1}{\nu} >> 1$) is associated with symmetry breaking, which is a necessary physical phenomenon that is not well understood. So I am trying to tie these thoughts together and was wondering what misconceptions I might have on this or how these are tied together in academic settings.

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Hi Hal - it doesn't seem quite clear what you're asking. Maybe it's just my lack of knowledge of the subject, but could you try stating your exact question a more explicitly? –  David Z Feb 1 '13 at 5:48
I will but it will have to wait til this evening –  Hal Swyers Feb 1 '13 at 11:06
Actually tomorrow... –  Hal Swyers Feb 2 '13 at 1:06
Nice question and the paper seems very interesting. –  Dilaton Feb 2 '13 at 20:04
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1 Answer

As I understand it from the statements in sections A.1 and A.2, this behaviour is due to the appearance of the constants in equation A.5. The equation is solved to linear order, i.e. for small perturbations of the solution. The terms in A.5 have to remain small in order for the solution to be valid. In those expressions, the Reynolds number enters in the form of the coefficient $a\propto R$, while momentum in the $x$-direction enters through the constant $c$. For large values of a, the expressions only remain small if $c$ is equal to zero, corresponding to a symmetry in the $x$-direction. Now if that symmetry is broken, the expressions are no longer small and therefore no longer solve the linear equations; one needs to take into account non-linear behaviour.

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