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## TOWARD EXPLICIT SOLUTIONS OF NAVIER STOKES EQUATION ON S4

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### zulfikar.ahmed@gmail.com<zulfikar.ahmed@gmail.com>

1:44 AM (0 minutes ago)

 to harrington, jharris, jhp, jhricko_4, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira, josesoto, josue, jpadgett, jpbalz
Around October of 2016 when I was working on stochastic volatility models (which were quite successful empirically) I had the idea of using Navier-Stokes equations to model FINANCIAL MARKET VOLATILITY because at the time it seemed that turbulence in the financial markets might have the non-linearity of the Navier-Stokes equation.  In the intervening time my views have certainly developed as I had been focused on establishing S4 physics for a Scientific Revolution where the world can see through new eyes the universe a static cosmos eternal and unchanging with galaxies relatively still governed by the single geometry of S4 and macroscopically the single force of electromagnetism that I pronounced on July 4 2018.  In the process I have been developing my ideas about how to replace Einstein’s gravitational field equations properly and I stumbled upon the realization that the Navier-Stokes equation is the dynamic system for gravity.  This is not new per se, since Navier-Stokes equations are the standard macroscopic fluid model for classical equilibrium statistical mechanics.  However this time around revisiting the Navier-Stokes equations I realized that the scale invariance of Navier-Stokes solved in the spherical harmonics basis is dictated by the Clebsch-Gordan coefficients of spherical harmonics on S4 (i.e. phi_m phi_n = sum_{m,n} c^{mn}_j phi_j) where the phi_j are spherical harmonics of S4.

These are universal constants c^{mn}_j which appear in the nonlinear mixing terms of the solution of the Navier-Stokes equations in the eigenbasis of S4.  These are scale invariant in the sense that scaling the eigenvalues by k should produce:

c^{mn}_p = c^{(km)(kn)}_{kp}

This does not seem like a hard problem since scaling the sphere by (1/k) should do this.  I will actually spend some time to work this out at some point — it just looks a bit messy and besides 2D spherical harmonics the 4D case is not used so it’s still research territory but this would be a breakthrough in our understanding of Nature in terms of turbulence phenomena.  When Andrei Kolmogorov in 1941 came up with his 5/3 scaling law it was a breakthrough in our understanding of turbulence.  That underneath the hood the source of the scaling might be the geometry of the entire universe and thus scale invariance in all the quadratic coupling problems would instantly demystify stochasticity, turbulence, and source of fractal geometry of Nature instantly.

In other words, I claim that it is the CLEBSCH-GORDON COEFFICIENTS in S4 spherical HARMONICS AND THEIR SCALE INVARIANCE that is the geometric explanation of all the fractal phenomena that we observe in Nature empirically.  Note that no such explanation is natural to the euclidean geometry.  You need compactness of space to have any expansion at all.  Therefore SPHERICAL geometry of space is crucial to explain scale invariance phenomena. Thus my S4 model radically alters human understanding of the universe in which we live by clarifying the mathematical structure behind scale invariance and fractal geometry of Nature.  Benoit Mandelbrot had gone far into modeling of fractal phenomena with a long history going back to Cantor but no one before me has completely solved this mystery of Nature with a concrete mathematical solution that can connect the classical atom, the scattering amplitude computations and the turbulence in hydrodynamics in one shot.  This is indeed a joyous day for humanity.

Thanks,
Zulf

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## TURBULENT HYDRODYNAMICS EXTENSION TO FINANCIAL VOLATILITY: EXTENDING GIGA’S RESULTS

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### zulfikar.ahmed@gmail.com<zulfikar.ahmed@gmail.com>

10/1/16

 to renkel, jhansen, jharrington, jharris, jhp, jhricko_4, jhs, jhutasoi, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira
Intuition suggests that if we replace the hydrodynamic Navier-Stokes equation in dimension d>=3,
d/dt u – nu*Laplacian(u) = (u, grad(u))
u(0,x) = u0(x)
to the situation I claim in financial volatility, the Laplacian will be replaced by a SYMMETRIC FELLER GENERATOR, an integro-differential operator.  The Navier-Stokes theory tells us that smoothness of solutions is a hard problem.  Weak solutions always exist and satisfy an equality like
d/dt ||u|| + \int_t || grad(u)||_2^2 ds = || u(0)||_2^2
and it’s a famous open problem to determine if there can be smooth solutions or singularities develop in finite time.  If they do develop, singularities must have one-dimensional Hausdorff-meansure zero (Cafarelli-Kohn-Nirenberg).  Since now I have produced statistical evidence that this (u,grad(u)) type nonlinearity exists for volatility a natural question is whether we can cheaply push through the Navier-Stokes theory to our volatility models.  As luck would have it, Giga’s work (for long time existence for Navier-Stokes with small data initial conditions etc.) is nicely phrased in terms of a more general operator than Laplacian.  His basic setup is
d/dt u + Au = Fu
where A say a self-adjoint operator that satisfies || exp(-tA) f||_p <= C || f ||_s / t^sigma for some constant sigma.  Now even though the types of Feller generators A we are interested in such as a stable process etc. satisfy this type of bound because there are heat kernel bounds of the form p(t,x,y) <= C t^{-d/beta} ( 1 + d(x,y)/t^{1/beta})^{-(d+beta)}.  So we can use Giga’s theory to claim some parts of the Navier-Stokes theory to our volatility setting such as small initial data existence — although this is more of a schematic proposal.  If we use a more general operator A instead of Laplacian, we have scaling different so the Hausdorff dimension of the singular set should change slightly.  Otherwise, we should have all the results of turbulent solutions of Navier-Stokes hold for volatility as well.  So this is a slightly more precise step toward making VOLATILITY STORMS a significant rigorous concept that will put the link between turbulent hydrodynamics and financial volatility in firm footing.

## NAVIER-STOKES EQUATIONS ARE NOTORIOUSLY DIFFICULT TO SOLVE IN R^3 BUT THEY ARE EASY SOLVE IN S4

Terry Tao have a very good blog on the Navier-Stokes equation from which I am learning some of the basic issues about how to think about them.  What he does to think about these is use a dyadic decomposition of frequencies.  He analyses the energy $E(t) = \int u(t.x)^2 dx$ of the solution of the Navier-Stokes equation by a dyadic decomposition and considering the pieces of energy in the dyadic basis:

$\partial_t E_ = \Pi_{N,N_1,N_2} - D_N + F_N$

where $D_N$ is the energy dissipation rate and $F_N$ is the energy injection rate and $\Pi_{N,N_1,N_2}$ is the ‘energy flow rate’.  I am not interested in the Navier-Stokes equation with a forcing term so I ignore the energy injection rate but Terry defines the three types of regimes of fluid flow in terms of the size of the terms on the right: when $F_N$ dominates, ‘injection regime’, when $D_N$ dominates, ‘dissipation regime’; then $\Pi_{N,N_1,N_2}$ dominates ‘energy flow regime’.  I will explain in a moment how we can use the same concepts and solve the Navier-Stokes equations exactly in the S4 model which is not very easy to do on $latex\mathbf{R}^3$ and I am stealing these concepts from Terry’s analysis because I am interested in not inventing anything new that I don’t have to.  In my case I will be expanding things in an eigenspinor basis rather than a dyadic basis and the analogy of the mixing terms being the ‘energy flow’ term and the diffusion term being the dissipation regime is perfectly sane and sensible.  We want to use this description to understand what Kolmogorov was after in turbulence and then we want to make an identification between this problem and the classical atom problem and note that technically the issues are almost identical and therefore claim that in fact the source of the turbulence is at the source of the Dirac equation for electromagnetism rather than in the Navier-Stokes macroscopic model.  This link is the most important thing that I can say that Kolmogorov was not looking for.

Terry’s heuristic is $\Pi_{N,N_1,N_2} = O( NE_N^{3/2})$ using the form of the nonlinear term inner-producted with the solution for the energy $N$ while the dissipation term is $D_N = \nu N^2E_N$.  Therefore Terry expects dissipation regime — and I won’t bother using the precise symbol for dealing with the constants and just use greater or smaller with the understanding that constants don’t matter for order of magnitude estimates and Terry has a precise blog about this — for $N \ge \nu^{-1} E_N^{1/2}$ and he expects energy flow regime when $1 \le N \le \nu^{-1} E_N^{1/2}$.

This is beautiful because we won’t change a word of this analysis which is extremely valuable to me.  The ideas here go back to Andrei Kolmogorov and that is what is interesting.  The important issue is Kolomogorov’s 5/3 law which Terry gets to using the power law asymptotic $E_N \sim AN^{-\alpha}$.  Read his blog because he does a fabulous explanation for analytic argument for the Kolmogorov’s argument.  I will just use the whole setup as a program and try to see whether in the exact solution all of the pieces Terry is describing can be proven.  I certainly have the sort of simplicity where I can try to prove these.

The structural constants $latec c_{N,N_1,N_2}$ in Terry’s setup will be related to the Clebsch-Gordan coefficients of expanding products of spherical harmonics in terms of spherical harmonics of $S^4$.

Are these constants scale-invariant?  In other words are $c_{\lambda N, \lambda N_1, \lambda N_2}$ scale invariant if they are the Clebsch-Gordan coefficients?  The first idea that pops to my mind is that they should be because the sphere of radius 1 is being compared with the sphere of a radius $1/\lambda>0$.  I like to check these simple intuitions because I hate getting $1/\lambda$ and $\lambda$ mixed up.  So if you consider $e^{i \lambda \theta}$ and $\lambda$ is an integer then that is the $\lambda$-th Fourier mode of a circle.  Let’s say $\lambda >> 1$ high frequency.  Then you can shrink the circle to radius $1/\lambda$ and get it as the first Fourier mode.  Right so the Clebsch-Gordon coefficients of $S^4$ will be scale invariant in this way although there may be complications for proving a theorem that I am evading but at the moment we are more interested in understanding the source of the scale invariance for the Navier-Stokes in the S4 model where it should be coming from the Clebsch-Gordon coefficients rather than being pedantic.  So in the S4 model we would be explaining the SCALE INVARIANCE phenomena in Navier-Stokes in Nature directly from the Clebsch-Gordon scale invariance which is mathematically a relatively trivial observation say compared to Fermat’s Last Theorem or something hard.  So this is mathematically not so interesting in difficulty but it is tremendously insightful about why we can be assured of scale invariance in turbulence in Nature from the S4 model.  In other words Kolmogorov was not interested, I don’t think, in turbulence because he wanted to be an applied mathematician but rather because he was attempting to get to the heart of Nature through the turbulence problem.  And here Nature is bared before us in a transparent manner — scale invariance of turbulence can be described through a beautiful geometry of S4.

## GRAVITATIONAL EVOLUTION EQUATIONS SHOULD REDUCE TO NAVIER-STOKES ON S4

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### zulfikar.ahmed@gmail.com<zulfikar.ahmed@gmail.com>

4:52 PM (12 minutes ago)

 to harrington, jharris, jhp, jhricko_4, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira, josesoto, josue, jpadgett, jpbalz
I am glad I took a step back from considering a direct N=10^{82} atom model on the equator S3 of S4 of radius 3246 Mpc.  Since JOSIAH WILLARD GIBBS established foundations of equilibrium statistical mechanics 1902 it seems that there is strong feeling that thermodynamic models for large atoms in such a situation should be macroscopically described by the NAVIER-STOKES equation, which I had began thinking about several years ago in the context of models for finance but now I can see that they are models for fundamental stochasticity in Nature in S4 physics. Ebin and Marsden proved short time existence for these for closed manifolds in the late 1960s.  These equations are nonlinear parabolic equations and on riemannian manifolds with Levi-Civita connection D, they have the form
dv/dt – nu*Laplacian(v) + D_v(v) =  -grad(p)
div(v) = 0
v=v(x,t)
v(x,0)=v0(x)
Consider the fluid restricted to a tubular neighbordhood of a hypersurface and solve this equation by extending the v0(x) to all of S4 and using spherical harmonics decomposition after translating the equation to the differential form duals.  The nonlinear term D_v(v) in local coordinates is quadratic with Christoffel symbol coefficients in v = v^i d/dx^i so the translation to 1-forms produces a quadratic nonlinearity  Q(v^#) as well where v^# denotes the metric-dual 1-form of v.  Then use the spherical harmonics basis where we get a system of ordinary differential equations with eigenspaces being mixed in a quadratic manner but these can be solved by the SAME strategy that I employed for the classical atom problem and I will have to make this a bit more rigorous but essentially nu will be an analog to the coupling constant for EM interactions.  Formally the solution is almost identical and we get transition in the S4 model smooth bounded solutions to turbulence when the nu parameter (reynold’s constant is sufficiently high).

The technical solution is sufficiently easy in this case that the important issue is to note that the Navier-Stokes equation — so long as it describes a fluid composed of atoms interacting by Lennard-Jones potentials governed by my S4 matter and electromagnetic equations (which I have not established but trust that the equilibrium statistical mechanics theory provides justification for the link since it seems to have a universal character to such interacting systems) implies that Navier-Stokes equation governs the hypersurface evolution.

This is an extremely important re-iteration of ANDREI KOLMOGOROV’S obsession with the question of stochasticity in the Navier-Stokes equation.  In fact this is the dynamics governing gravitational evolution in our universe and is the source of fractal geometry of Nature literally for this is the macroscopic description of microscopic interacting systems in equilibrium and scale invariance is encoded in the equation.  This is providing the link between Richard Hamilton’s Ricci flow type geometric evolution and Benoit Mandelbrot’s view of Nature’s geometry as fractal. And in the S4 model this can be solved exactly.

Note that without the S4 model this unity between perfect symmetry of the four-sphere and the stochasticity that we observe in Nature cannot be understood with a concrete realization of the solution in the eigenbases where these fundamental equations lose their analytic sophistication and can be solved easily in an explicit basis.

As I had promised, here is the stochasticity of Nature explained with a crutch — that crutch being the leap from a per atom model to the macroscopic fluid dynamics model.  But my classical atom model can be repeated here almost exactly. The missing link in my own understanding then reduces to the issue of classical statistical mechanics of interacting particles.  This is of course on top of other wonderful properties of this model like no need for renormalizability for scattering amplitude computations.

Thanks,

Zulf

## TOY MODEL FOR S4 HYPERSURFACE EVOLUTION: 19 JULY 2018 REPRISE

This is a test run not an attempt at any final model.  Consider an interaction Hamiltonian in classical mechanics on $latex\mathbf{R}^4$.  This $\mathbf{R}^4$ will represent the tangent bundle of $S^4$ of radius 3246 Mpc.  We want to see what to do on the flat 4-space following this Hamiltonian and then consider the geometry later.  We would like to take many iterations to examine different aspects of this problem until we get a model that is right for S4 physics.

$H = \sum_{i=1}^N p_i^2/2m + U(r_1,\dots,r_N)$

The aspect that is interesting right now is setting things up so that the distribution of directions of the particles is uniform.  Ok so this is a problem of equilibrium statistical mechanics for inhomogeneous fluids.  The established equation for fluid dynamics is the Navier-Stokes equations which is the macroscopic dynamical equation for the movement of the ensemble and Vladimir Arnold has produced a hydrodynamic theory for riemannian manifolds.  This is TopologicalMethodsHydrodynamics.

What we need assuming for the moment that we make the leap from microdynamics to macrodynamics, is the fluid motion equation on $S^4$.  We’ll stick in the diffusion term to make it Navier-Stokes.

## CHECK OF FIRST HIGHLY NONTRIVIAL MATHEMATICAL CONJECTURE BASED ON PHYSICAL INTUITION FOR S4

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### zulfikar.ahmed@gmail.com<zulfikar.ahmed@gmail.com>

12:56 PM (1 minute ago)

 to harrington, jharris, jhp, jhricko_4, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira, josesoto, josue, jpadgett, jpbalz
I am pleased to announce that a highly nontrivial mathematical conjecture was checked by physical intuition only.  It is that unlike MINIMAL hypersurfaces of spheres where it is a hard problem of continuous research interest for many many decades top mathematicians like Jim Simons and Shing-Tung Yau and Shing-Sheng Chern and Manfredo do Carmo and their collaborators and students had contributed, and all the theorems that try to understand the length of the second fundamental form restrictions to get rigid answers for minimal hypersurfaces close to the equator and they all have to stop to collect CLIFFORD TORI when the length squared of the second fundamental form  of the hypersurfaces exceeds a certain value which are precisely described by Alencar and do Carmo for constant curvature hypersurfaces.   The point here is that for some natural GEOMETRIC questions like minimal hypersurfaces one is considering critical points of functionals that are are similar to the Einstein-Hilbert action but they have critical points that are arbitrarily complicated but these arbitrary complicated critical points should not arise for the Einstein-Hilbert action for the same sort of physical reasoning that were used by Hilbert and Einstein.  So here we are employing physical intuition to make a mathematical conjecture in a setting where deep mathematics had been employed to try to sort out the rigidity of the solution that comes from mathematical intuition and I checked that the first level of ‘pathology’, the Clifford tori solutions, are NOT critical points of the Einstein-Hilbert action.  This could be a good mathematics paper if someone worked through proving the conjecture in full detail but I am overjoyed because this is the first case of truly surprising geometric result that occurs from a physical model that goes against the most established models of this age, i.e quantum field theory and general relativity buttressing my confidence that S4 model is the correct model.

PHYSICAL INTUITION — and this is unique in the PLANET today until S4 is accepted physics because no one else can claim to have physical intuition about S4 hypersurfaces — the Einstein-Hilbert action does NOT have critical points at these Clifford tori.  This is physical intuition for S4 that is verified by simple computation of the first variation.  But the conjecture is that NO critical point of Einstein-Hilbert action on S4 should exist that is not the equator BECAUSE that should be empty space without matter.

https://zulfahmed.wordpress.com/2018/07/19/removing-clifford-tori-from-possible-critical-points-of-einstein-hilbert-action-on-s4/

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## SEIBERG-WITTEN INVARIANTS REMINDER

Let $(X,g)$ be a smooth closed oriented four-manifold.  Let $\mu\in\Omega^2(X)$ with $*\mu=\mu$.  Let $s\in S_X$ be a $Spin^c$ structure and let $S_+,S_-, L$ be the corresponding vector bundles.  If $A$ is a connection on $L$ and $\psi$ is a section of $S_{+}$ then $(A,\psi)$ satisfy the Seiberg-Witten equations when

$F(A,\psi) = ( D_A\psi , F_A^{+} - q(\psi) - i\mu)$

vanishes.  The space of solutions $(A,\psi)$ do not depend on $g$ or $\mu$.  This information constitutes the Seiberg-Witten invariants of $X$.  Let $\mathcal{M} = \{ (A,\psi): F(A,\psi)=0\}/\mathcal{G}$

where $\mathcal{G}$ is the group of gauge transformations.  Then a Theorem says $\mathcal{M}$ is always compact.  Fixing $g$ for generic $\mu$ the space is a smooth finite dimensional manifold with a smooth circle action.  The dimension is $\dim(\mathcal{M}) = b^1 - 1 + b^{2+} + \frac{c_1^2-\tau}{4}$ where $\tau = b^{2+}-b^{2-}$ is the signature of $X$.  Here $c_1=c_1(L)$.

The Seiberg-Witten invariant $SW_X$ is defined as follows on spin-c structures over $X$.  First for topological reasons the dimension $b^1-1+b^{2+}+\frac{c_1^2-\tau}{4}$ is even call it $2d$.

(a)  If $b^{2+} - b_1$ is even $SW_X=0$.

(b)  If $d=0$  then $\mathcal{M}$ is a finite set of points then $SW_X(s) = \sum_{\mathcal{M}} \pm 1$.

(c) If $d>0$ then $SW_X(s) = \int_{\mathcal{M}} e^d$ where $e\in H^2(\mathcal{M},\mathbf{Z})$ represents the first Chern class of a bundle $\mathcal{M}^0\rightarrow\mathcal{M}$.

Consider the K3 surface $z_1^4 + z_2^4 + z_3^4 + z_4^4=0$.  This four-manifold has $b^{2+} =3$ and $b^{2-}=19$.  There is a unique spin-c structure with $c_1(s)=0$.  For this spin-c structure $SW_{K3}=\pm 1$ and for all others it is zero.  Freedman’s theorem tells us $K3#\bar{\mathbf{CP}^2}$ is homeomorphic to $#_3 \mathbf{CP}^2 #_{20}\bar{\mathbf{CP}^2}$ but Seiberg-Witten is nonzero for the first and zero for the second.

(These are quick notes from Clifford Taubes’ Park City Lectures to refresh my mind).