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## Turbulent fluid dynamics on a graph as an idea

Incompressible fluid dynamics is usually studied in physics in the mathematical setting space and time, $(x,t)$ where velocity $u(x,t)$ and pressure $p(x,t)$ follow a set of transport equations, which are $div(u)=0$ and

$(\partial_t - \Delta)u = - R\langle u, \nabla u\rangle - \partial_t p(x,t)$

which we can simplify with $p==0$.  Here $R$ is the Reynold’s number.  The physicists’ intuition is that laminar flow is possible when $R$ is small but the solutions are turbulent when $R>>1$.

Weak solutions to Navier-Stokes on R^3 were proved to exist in 1933 (attached) and there is also a recent paper.  As a curious novice in this direction, I am still confused about these weak solutions, especially with observed phase transition to turbulence.  In turbulent flow, there does not seem to be any chance of C^infinity type behavior for the fluid and the physics literature provides many metrics.  I know something about elliptic regularity in related context when weak smooth solutions of elliptic equations bootstrap to smoothness.  But what are weak solutions with regularity that are turbulent?  Apologies for airing my confusions.  I will try to clarify for myself what a weak solution is exactly mathematically.  Clearly this is a delicate issue worth investigating.  John Nash, may he rest in peace, had always emphasized physical intuition even in his paper on parabolic kernels of uniformly elliptic operators and as a novice I would like to sharpen mine.

2 Attachments

Preview attachment weak-solutions-navier-stokes-1933.pdf

weak-solutions-navier-stokes-1933.pdf

Preview attachment regularity-weak-solutions-navier-stokes.pdf

regularity-weak-solutions-navier-stokes.pdf

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

11:32 AM (1 minute ago)

 to David

Historically, Leray’s introduction to his seminal paper of 1933 mentions that he believed in the irregularity development in finite time for Navier’s equations he calls the equations for velocity and pressure which I had been mangling for simplicity.  These are actually equations for velocity u and also a pressure term

(d/dt u – Laplacian(u)) = – <u,grad(u)> – d/dt pressure + (external forces set to zero)

div(u) == 0  (incompressible)

The useful way according to Chorin to rewrite these is to introduce the ‘Reynold’s number’ R directly and put

d/dt u + R <u,grad u> = Laplacian(u) – d/dt pressure

div(u) = 0

We know generally that turbulence in this system can happen when R is much higher than 1.  If we just set pressure to zero, we have the form that I was using for my exploratory thinking about the case of volatility on a graph instead of u:

(d/dt – Laplacian) u = – R<u,grad u> =: F(u)

In the intuition of physicists, laminar flow should result when R is small and there can be a phase transition to turbulence when R crosses some threshold.  By early 1940s Onsager was using Navier-Stokes with high Reynold’s number to explain the formation of vortices, so I am trying to place the lack of singularity formation examples that Leray alluded to in his 1933 paper to the Onsager model around eight years later.

Navier-Stokes is a favourite of numerical analysis by computers for engineers, so I can look at this problem directly as well.  The motivation for me ultimately is to understand financial market volatility with an exact model of this type if possible.  I feel comfortable that the graph of the market will serve well as an x-variable for this problem now.

Attachments area

Preview attachment chorin-numerical-navier-stokes.pdf

chorin-numerical-navier-stokes.pdf