INITIAL CONDITIONS Schwartz

E Konstantinos

Kyritsis
email: ckiritsi@uoi.gr

Konstantinos E Kyritsis
email: c_kyritsis@yahoo.com

On

INITIAL CONDITIONS Konstantinos E Schwartz

ON THE SOLUTION OF THE 4TH CLAY MILLENNIUM PROBLEM ABOUT THE NAVIER-STOKES EQUATIONS. EQUIVALENCE OF SMOOTH COMPACT SUPPORT AND SMOOTH

Keywords: Incompressible flows, regularity, blow-up, Navier-Stokes equations, Euler equations, Clay millennium problem Mathematical Subject Classification: 76A02

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Clay millennium problem about the Navier-Stokes equations is one of the 7 famous problem of mathematics that the Clay Mathematical Institute has set a high monetary award for its solution. It is considered a difficult problem as it has resisted solving it for almost a whole century. The Navier-Stokes equations are the equations that are considered to govern the flow of fluids, and had been formulated long ago in mathematical physics before it was known that matter consists from atoms. So actually they formulate the old infinite divisible material fluids. Although it is known that under its assumptions of the millennium problem the Navier-Stokes equations have a unique smooth and local in time solution, it was not known if this solution can be extended smoothly and globally for all times, which would be called the regularity of the Navier-Stokes equations in 3 dimensions. The corresponding case of regularity in 2 dimensions has long ago been proved to hold but the 3dimensions had resisted proving it. Of course the natural outcome would be that regularity holds also in 3-Dimensions. Many people felt that this difficulty hides our lack of understanding of the laws of 3-dimensional flow of the incompressible fluids.

In this paper is presented a very short solution to the Clay Millennium problem about the Navier-Stokes equations. The solution proves that there cannot be a blow up in finite or infinite time, and the local in time smooth solutions can be extended for all times, thus regularity. This happily is proved not only for the Navier-Stokes equations but also for the inviscid case of the Euler equations both for the periodic or non-periodic formulation and without external forcing (homogeneous case). But it is also indicated that once the hypotheses of external forcing of the millennium problem allow 1) for the existence of a unique smooth solution local in time and 2) the criterion of the accumulation of the vorticity as in Theorem 2.2 still holds with external forcing as in the formulation of the Millennium problem, then the same result of regularity (no blow up) holds also for this

The formulation of the millennium problem and the 4 sub-problems (A) , (B), (C), (D)

In this paragraph we highlight the basic parts of the standard formulation of the 4 th Clay millennium problem as in [START_REF] Fefferman | Existence and smoothness of the Navier-Stokes equation[END_REF] Fefferman C.L. 2006.

The Navier-Stokes equations are given by (by R we denote the field of the real numbers, ν>0 is the density normalized viscosity coefficient ) 3 , t>=0 , n=3) (eq.2.1 )

i i j i n j j i u x p x u u u t               1 (x∈R
0 1       n i i i x u divu (x∈R 3 , t>=0 , n=3) (eq.2.2 )
with initial conditions u(x,0)=u 0 (x) x∈R 3 and u 0 (x) ∈ C ∞ divergence-free vector field on R 3 (eq.2.

3) If ν=0 then we are taking about the Euler equations and inviscid case.

      n i i x 1 2 2
is the Laplacian operator .The Euler equations are (eq2.1), (eq2.2), (eq2.3) when ν=0.

It is reminded to the reader, that in the equations of Navier-Stokes, as in (eq. 2.1) the density ρ, is constant, it is custom to normalized to 1 and omit it.

For physically meaningful solutions we want to make sure that u 0 (x) does not grow large as |x|. This is set by defining u 0 (x) , and f(x,t) and called in this paper Schwartz initial conditions , to satify

K K a a x x C x u     ) 1 ( ) ( , 0
on R 3 for any α and K (eq.2.4 )

(Schwartz used such functions to define the space of Schwartz distributions)

Remark 2.1. It is important to realize that smooth Schwartz initial velocities after (eq 2.4) will give that the initial vorticity ω0 =curl(u 0 ) , in its supremum norm, is bounded over all 3-space.

(, ) ≤ ,, (1 + + ) -on R 3 × , + ∞ for any α,m,K (eq.2.5 )

We accept as physical meaningful solutions only if it satisfies p, u C  (R 3 [0,)) (eq.2.6 ) and ℜ 3 (, )

2 < for all t>=0 (Bounded or finite energy) (eq.2.7 )

Remark 2.2 It is important to realize that smooth external force (densities) with the Schwartz property as in (eq.2.5) , have not only a rule for upper bounded spatial partial derivatives but also the same rule for time upper bounded partial derivatives.

Remark 2.3

We must stress here that imposing smoothness of the coordinate functions of velocities and external forces of the initial t=0 data and later time t data in the Cartesian coordinates plus and Schwartz condition as in (eq 2.5) is not equivalent with imposing similar such smoothness of the coordinate functions and conditions in the cylindrical or spherical coordinates. We will give in the paragraph 4, remark 4.5 an example of a strange blowup, where at any time t>0 the coordinates of the velocities are smooth and bounded in all space as functions in the polar coordinates and still the vorticity has infinite singularity at zero.

Alternatively, to rule out problems at infinity, we may look for spatially periodic solutions of (2.1), (2.2), (2.3). Thus we assume that u 0 (x) , and f(x,t) satisfy u 0 (x+ej)= u 0 (x), f(x+ej,t)= f(x,t), p(x+ej ,0)=p(x,0), for 1<=j<=3 (eq.2.8 )

(ej is the jth unit vector in R 3 )

In place of (2.4) and (2.5), we assume that u 0 (x), is smooth and that (, ) ≤ ,, (1 + ) -on R 3 × , + ∞ for any α,m,K (eq.2.9 )

We then accept a solution of (2. In the next paragraphs we may also write u0 instead of u 0 for the initial data velocity. Take ν>0 and n=3. Let u0(x) be any smooth, divergent-free vector field satisfying [START_REF] Kyritsis | A solution of the 4 th Clay Millennium problem about the Navier-Stokes equations[END_REF]. Take f(x,t) to be identically zero. Then there exist smooth functions p(x,t) , u(x,t) on R 3 x[0,+∞) that satisfy (2.1), (2.2), (2.3) , (2.6) , (2.7).

(Conjecture B) Existence and smoothness of Navier-Stokes solution on R 3 /Z 3 .

Take ν>0 and n=3. Let u0(x) be any smooth, divergent-free vector field satisfying [START_REF] Kyritsis | Review of the solutions of the Clay millennium problem about P ≠ NP =exptime[END_REF]; we take f(x,t) to be identically zero. Then there exist smooth functions p(x,t) , u(x,t) on R 3 x[0,+∞) that satisfy (2.1), (2.2), (2.3) , (2.10) , (2.11).

(Conjecture C) Breakdown of Navier-Stokes solution on R 3

Take ν>0 and n=3. Then there exist a smooth, divergent-free vector field u0(x) on R 3 and a smooth f(x,t) on R 3 x[0,+∞) satisfying ( 4), [START_REF] Kyritsis | Β Solution of the Clay Millennium Problem about the Navier-Stokes Equations: A Recent Approach Chapter 4 in the book New Visions in[END_REF] for which there exist no smooth solution (p(x,t) ,u(x,t)) of ( Here, in the formulation of the millennium problem the hypotheses of smooth with Schwartz condition initial velocities satisfies this condition therefore we have the existence and uniqueness of smooth solution locally in time, both in the non-periodic and the periodic setting without external forcing (homogeneous case).

The existence and uniqueness of a smooth solutions locally in time is stated in the formulation by C.L. Fefferman [START_REF] Fefferman | Existence and smoothness of the Navier-Stokes equation[END_REF] for the homogeneous cases and conjectures (A), (B). When a smooth Schwartz condition external force is added (inhomogeneous case) , it is natural to expect that also there should exist a local in time unique sooth solution. But this I did not find to be stated in the A.J. Majda-A.L. Bertozzi [START_REF] Majda | Vorticity and incompressible flow Cambridge University texts in applied mathematics[END_REF], so I will avoid assuming it.

We state here also two, very well-known criteria of no blow-up and regularity.

In this theorem the || ||m is the corresponding Sobolev spaces norm and . We denote by V m ={u in H m (R n ) and divu=0} where H m (R n ) are the Sobolev spaces with the L 2 norm. Proof: See A.J. Majda-A.L. Bertozzi [START_REF] Majda | Vorticity and incompressible flow Cambridge University texts in applied mathematics[END_REF], Corollary 3.2 pp 112). QED Remark 2.5 Obviously this proposition covers the periodic case too.

Theorem 2.2 Supremum of vorticity sufficient condition of regularity

Let the initial velocity u0 ∈ V m m>= [3/2]

3.

What is that we do not understand with the Navier-Stokes equations? The need for more consciousness for interpretations. Why we chose the geometric calculus approach for the solution?

It has been written in the initial formulation of the problem, that our difficulty of solving this millennium problem shows that there are several things that we do not understand very well in the Navier-Stokes equations.

In this paragraph we will investigate this issue. We will explain also why the rather elementary geometric calculus approach is better so as to solve the millennium problem, compared to more advanced functional analysis.

1) One primary point, known but often forgotten is the next. The Euler and the Navier-Stokes equations are the equations that are considered to govern the flow of fluids, and had been formulated long ago in mathematical physics before it was known that matter consists from atoms. So actually they formulated the old infinite divisible material fluids. After L. Boltzmann and the discovery of material atoms, the truer model is that of statistical mechanics. We may consider that the two different types of matter, a) infinite divisible b) made from finite atoms, behave the same as far as flows in fluid dynamics, and certainly there are many common properties but ultimately are mathematically and logically different. One example of the difference is that in the atomic structured material fluid model, the angular velocity of the spin e.g. of electrons, protons, neutrons which is about 1 terahertz (infrared range) can vary increase or decrease, independently from the vorticity, which is only the part of the angular velocity which is "geared to the environmental" rotation of the fluid. In the classical Weierstrass calculus of infinite divisible material fluids (Euler and Navier-Stokes equations) this distinction does not exist and all the angular velocity of a point is due to the vorticity. In [START_REF] Muriel | An Exact Solution of the 3-D Navier-Stokes Equation Department of Electrical Engineering[END_REF] Muriel, A. 2000 a corresponding to the millennium problem in statistical mechanics has been solved in the direction of regularity. Similarly, in [START_REF] Kyritsis | On the 4 th Clay Millennium problem: Proof of the regularity of the solutions of the Euler and Navier-Stokes equations, based on the conservation of particles[END_REF] Kyritsis, K. November 2017 a solution of the current millennium problem has been proved in the direction of regularity, but only if adding an additional hypothesis to the initial formulation, that of existence of finite atomic particles that are conserved during the flow. Strictly speaking a mathematical model of the material fluids and their flow which will have a high degree of exactness should take in to account that matter consists of atoms, (the electron range of magnitudes is of the order 10 -15 meters) and this suggests that we should avoid utilizing concepts of continuity and smoothness that use ε>0 δ>0 in their definition smaller than 10 -15 meters. To address this difficulty of our current (Weierstrass) calculus the author developed the Democritus digital and finite decimal differential calculus (see [START_REF] Kyritsis | An introduction to the natural differential and integral calculus, without the infinite[END_REF] Kyritsis K. 2019b , [START_REF] Kyritsis | B An Axiomatic System for a Physical or Digital but Continuous 3-Dimensional Euclidean Geometry, Without Infinite Many Points[END_REF] Kyritsis K. 2017 B , [21] Kyritsis K. 2022) In this finite calculus, we define concepts, of seemingly infinitesimal numbers (they are finite), seemingly infinite numbers (they are finite) and feasible finite numbers, so as to develop a differential and integral calculus up to decimal numbers with only a fixed finite number of decimals (decimal density of level of precision). Different levels of precision give different definitions of continuity and smoothness. These multi-precision levels Democritus calculi is what an applied mathematician is doing when applying the Newton-Leibniz and Weierstrass calculus with the infinite (and infinitesimals). The Democritus calculus strictly speaking is not logically equivalent to the Newton-Leibniz calculus or to the Weierstrass calculus. E.g. classical Weisstrass calculus continuity corresponds in the Democritus calculus of being continuous not only to a single precision level but to all possible precision levels. Because in the Democritus calculus continuity and smoothness is only up to a precision level, the turbulence can be defined in a way that in Weierstrass calculus cannot be defined. In a turbulent flow, the flow in the Democritus calculus may be smooth relative to a precision level but non-smooth relative to a coarser precision level (or the opposite) in the Weierstrass calculus this is impossible. Furthermore, now when a computer scientist is experimenting with computers to discover if in a flow there will be a blow up or not in finite time, within the Democritus calculus and its Navier-Stokes equations he will have an absolute proof and criterion. If the vorticity will become seemingly infinite (still finite) in a feasible finite time interval there is a blow up. If it becomes only feasible finite in any feasible finite time interval, there is no blow up. Of course blow-up in the Democritus calculus is not equivalent with a blow up in the Weisstrass calculus. Finally, with the Democritus calculus the applied mathematician acquires the subjective quality of congruence. In other words, what he thinks, sais and writes is what he acts and applies. With the infinite in the ontology of calculus this is not possible and it is unavoidable the incongruence, because infinite cannot be acted in the applications in a material reality where all are finite.

2) It is known that when the calculus (which is used in modeling the fluids) was discovered by Newton and Leibniz, the original mathematical ontology was utilizing infinitesimals, smaller than any positive real numbers but not zero. Then later with Weierstrass calculus this ontology was abandoned, we restricted ourselves to the real numbers only, and we utilized limits and convergence. So when we take the law of force (momentum conservation) of Newton F=m*γ on a solid finite particle and then take the limit by shrinking it to a point to derive the Euler and Navier-Stokes equations, we must not forget, that originally the limit was not to a point but to an infinitesimal solid body particle. This is not the same! In [21] Kyritsis K. 2022, I have restored with strict mathematics the original ontology of infinitesimals of Newton-Leibniz , utilizing algebra of intervals (or inverses of ordinal numbers as J. H Conway has also done with the surreal numbers see [START_REF] Conway | On numbers and games (Sureal numbers[END_REF] J H. Conway and [23] K Kyritsis ordinal real numbers 1,2,3). Then we have a two-density calculus with two different linearly ordered fields, a) the real numbers b) a larger such field of Newton-Leibniz fluxions, with infinitesimal, finite and infinite numbers. The topologies of convergence of a solid finite particle by shrinking it to a point or to an infinitesimal solid particle are different! And this affects the issue of vorticity and angular velocity of infinitesimal particle. When I was a University student, and I was learning about the equations of Navier-Stokes, I was satisfied to see that the simple law of force (momentum conservation) of Newton F=m*γ was converted to the Navier-Stokes equations, but I was shocked to realize, that the rest of the independent information about the motion of the solid finite particle, namely its rotational momentum, was not shrunk to an angular velocity ω of the infinitesimal solid particle. So we see now that this is not reasonable in the Weisstrass calculus, which shrinks to a point, while it is possible in the older Newton-Leibniz calculus which shrinks to an infinitesimal solid body, and would lead to a different model of flows of fluids, with independent initial data of angular velocities, besides linear velocities and besides the derived from them vorticity.

3) In the current solution of the millennium problem, we may observe a 20%-80% Pareto rule. In other words, more than 80% of the equations utilized as governing equations of the flow, are those derived from fundamental theorem of the calculus, (in the form of Stokes theorem, divergence theorem, green theorem, Helmholtz-kelvin theorem, fundamental theorem of calculus etc.) and less that 30% the PDE of the Navier-Stokes equations. So I might say that the main equations governing the phenomenon of flow is the machinery of exterior differential algebra (wedge product) differentiation (differential forms) etc. rather than simply PDE equations. For reasons of simplicity and because we are restricted here to only 3 spatial dimensions, we do not utilize the symbolism of the wedge products and differential forms, but only the Stokes theorem, divergence theorem etc. 4) These versions of the fundamental theorem of the calculus (Stokes theorem etc.) lead to an extension of the law of momentum conservation of 3D fluid parts to a law of 1D line density (rotatory) momentum conservation (Theorem 4.1) and law of 2D surface density (rotatory) momentum conservation (Theorem 4.2). These laws are very valuable for infinite divisible fluids so valuable as the existence of finite atoms in the atomics structured fluids. Without these extra laws of momentum density conservation, we would have no hope to solve the millennium problem. As T. Tao had remarked, only an integral of 3D energy conservation and an integral of 3D momentum conservation is not adequate to derive that momentum point densities ρ•u, or energy point densities (1/2)ρ•u 2 will not blow up. 5) Besides the forgotten conservation law of finite particles, which unfortunately we cannot utilize in the case of infinite divisible fluids to solve the millennium problem, there are two more forgotten laws of conservation or invariants. The first of them is the obvious that during the flow, the physical measuring units dimensions (dimensional analysis) of the involved physical quantities (mass density, velocity, vorticity, momentum, energy, force point density, pressure, etc.) are conserved. It is not very wise to eliminate the physical magnitudes interpretation and their dimensional analysis when trying to solve the millennium problem, because the dimensional analysis is a very simple and powerful interlink of the involved quantities and leads with the physical interpretation, to a transcendental shortcut to symbolic calculations. By eliminating the dimensional analysis, we lose part of the map to reach our goal.

6) The 2 nd forgotten conservation law or invariant, is related to the viscosity (friction). Because we do know that at each point (pointwise), the viscosity is only subtracting kinetic energy, with an irreversible way, and converting it to thermal energy, (negative energy point density), and this is preserved in the flow, (it can never convert thermal energy to macroscopic kinetic energy), we know that its sign does not change too, it is a flow invariant, so the integrated 1D or 2D work density is always of the same sign (negative) and as sign, an invariant of the flow. The conservation or invariance of the sign of work density by the viscosity (friction) is summarized in the lemma 3.1 below.

7) Finally we must not understate the elementary fact that the force densities Fp due to the pressures p, =-∇ are conservative, irrotational vector field, and they do not contribute to the increase or decrease of the rotatory momentum and vorticity of the fluid during the flow. Because of this we get that the conserved 1D and 2D densities of momentum in Theorems 4.1 and 4.2 are only of the rotatory type. 8) Anyone who has spent time to try to prove existence of Blow up or regularity in the various physical quantities of the fluid like velocity, vorticity, acceleration, force density, momentum, angular momentum, energy etc. he will observe that in the arguments the regularity and uniform in time boundedness propagates easily from derivatives to lower order of differentiation, while the blowup arguments propagate easily from the magnitudes to their derivatives. The converses are hard in proving. This is due to the usual properties of the calculus derivatives and integrals. The hard part of the proofs, must utilize forms of the fundamental theorem of the calculus like Stokes theorem, divergence theorem etc. 9) Based on the above 8 remarks about what is not very well understood with Navier-Stokes equations I decided that elementary geometric calculus should be the appropriate context to solve the millennium problem, and this I did indeed.

Lemma 3.1 The viscosity sign forgotten invariant.

If we integrate the force point density of the viscosity, over a line (1D work density) or surface (2D work density) or a volume (work) its sign will remain the same during the flow.

Proof: Because we do know that pointwise, the viscosity is only subtracting kinetic energy, with an irreversible way, and converting it to thermal energy, (negative energy point density), and this is preserved in the flow, (it can never convert thermal energy to macroscopic kinetic energy), we deduce that its sign does not change too it is a flow invariant , so the integrated 1D or 2D work density is always of the same sign (negative) and as sign an invariant of the flow. QED.

The Helmholtz-Kelvin-Stokes theorem in the case of viscous flows. New monotone semi-invariants of viscous flows with the interpretation of average rotational momentum axial 1-D line densities.

Here we apply the idea that the most valuable equations that govern he flow of the fluid are not literally the Navier-Stokes equations but the invariants or semi-invariant properties of the flow, derived from the abstract multi-dimensional fundamental theorems of calculus, in the forms of divergence theorems, Stokes theorems, Greens theorems etc. Actually this is the mechanism of wedge-products and abstract algebra of differential forms which is beyond classical partial differential equations. We do not utilize though definitions and symbolism of wedge-products and differential forms in his paper so as to keep it elementary and easy to read. The main discovery of this paragraph is the Helmholtz-Kelvin-Stokes theorem 4.3 in the case of viscous flows and the resulting general no-blow-up theorem 4.4 for the viscous flows without external forcing. A blow-up when it occurs, it will occur at least as blow-up of the vorticity, or of ρ•ω. If we discover average value invariants of the flow with physical units dimensions ρ•ω, that in the limit can give also the point value of the ρ•ω, and that are invariants independent from the size of averaging, it is reasonable that we can deduce conclusions, if the point densities can blow-up or not.

Theorem 4.1 The Helmholtz-Kelvin-Stokes theorem in the case of inviscid Euler equations flows without external force or homogeneous case. (Α 1D line density of rotatory momentum, conservation law).

Let initial data in R 3 so that they guarantee the existence of a unique smooth solution to the Euler equation in a local time interval [0,T]. Then at any time t ∈ [0,T] the circulation Γ(c) of the velocities on a closed smooth loop is equal to the flux of the vorticity on smooth surface S with boundary the loop c, and is constant and preserved as both loop

and surface flow with the fluid. In symbols (ρ=1 is the density of the incompressible fluid) A blow-up when it occurs, it will occur at least as blow-up of the vorticity, or of ρ•ω. If we discover bounded average value invariants of the flow with physical units dimensions ρ•ω, that in the limit can give also the point value of the ρ•ω, and that are invariants and bounded independent from the size of averaging, it is reasonable that we can deduce conclusions, if the point densities can blow-up or not.

() = = = ρ • (eq.
Here we convert the surface vorticity flux invariant of Helmholtz-Κelvin-Stokes to one with 3D integration which will be more convenient in the arguments as the volumes are preserved by incompressible flows and most important, the integration is 3-dimensional which can be utilized to define average values of the vorticity (flux) on 3D finite particles..

We will prove at first a lemma about the 3D volume integral of Theorem 4.2, and convergence of average values of vorticity, based on this 3D integral, to point values to vorticity.

We define an average value for the volume 3D integral of vorticity flux.

The numerator of this average value of vorticity has also the interpretation of rotational momentum average axial density on the ball B and relative to the axis a. A reason for this is that the physical dimensions of measuring units of this magnitude is that of rotational momentum line density. This is because the rotational momentum point density has physical dimensions [moment_of_inertia]*[ω]*[s]^(-3)=[m][s]^(-1)[t]^(-1), where [m] for mass, [s] for distance, [t] for time, and this magnitude has physical units dimensions, ([ρ][ω][s]^3 )=([m][s]^(-1)[t]^(-1))[s]^(1), thus rotational momentum point density integrated on 1-d line axial density. And the full quotient therefore has physical units dimensions [m][s]^(-3)[t]^(-1) )=[ρ][ω].

Lemma 4.1 Let a ball B of radius r and center x, and the average vorticity in it as in the Definition 4.1 so that its axis a that defines the average vorticity is also the axis of the point vorticity ωx at the center x of the ball. By taking the limit of shrinking the ball to its center x , (r->0), the average vorticity converges to the point vorticity ωx . In symbols →0 = ωx . If the axis a of the ball to estimate the average vorticity is not the axis of the point vorticity, then the limit of the average vorticity will be equal to the projection component ωa(x,t) of the point vorticity ω(x,t) on the axis a.

Proof:

We simply apply an appropriate 3-dimensional version, with iterated integrals of the 1-dimensional fundamental theorem of the calculus. QED.

Remark 4.1. Such a limit of 3D body to a point is the same as the limit that from the Newton equation of force F=mγ, we derive the Navier-Stokes equations.

Since the flow of a fluid under the Euler or Navier-Stokes equations, with or without smooth Schwartz external force is a smooth and continuous mapping F , then such a limit will be conserved to still be a valid limit during the flow. In other words F(

→0 )= ( ()→0
) and B->0 , implies Ft(B)->0. We define of course in an obvious appropriate way the average vorticity Ft( ) as in definition 4.1, for the flow-image of a ball B after time t. Simply the disc surfaces will no longer be flat, and the loop no longer perfect circle. But the integrals in the definition will be the same. Constancy of the average vorticity on such surfaces will only be, up to its Euclidean norm and vertical angle to the surface. We must notice though that although a relation F(

→0 )= ( →0
) would hold , the value of this limit will not be the vorticity ωF(x) at the flowed point! Unfortunately, the Lemma 4.2 holds not on arbitrary 3D shapes and arbitrary integration parametrization on it, but only when we start with standard 3D shapes like a sphere, a cylinder a cube etc. and the normal parametrization on them. The reason is that we need to take in to account in a normal way the average vorticity around a point in an unbiased way, that an arbitrary shape will not give.

Another important conservation point is that the relation of the vorticity ωx being tangent to an axis a (or general curve) is conserved during inviscid Euler flows. It is the conservation of vorticity lines (See [START_REF] Majda | Vorticity and incompressible flow Cambridge University texts in applied mathematics[END_REF] Majda, A. J. -Bertozzi, A. L. 2002, Proposition 1.9 in page 21). Therefore for inviscid (and incompressible) flows the axis of the initial point vorticity ω(0) , which is also the axis to estimate the average vorticity on the ball B, will still be after the flow and at time t, tangent to the point vorticity ω(t). But for general viscous flows this will not be so. Notice that such limits of average values would not work for the circulation of the velocity on a loop, as in the application of the iterated 1-dimensional fundamental theorem of the calculus would require boundaries of the integration.

Lemma 4.2 Let the Euler or Navier-Stokes equations of incompressible fluids in the non-periodic or periodic setting, with smooth initial data and we assume that the initial data in the periodic or non-periodic case, are so that the supremum of the vorticity is finite denoted by Fω on all 3-space at time t=0. Let the average vorticity, or average rotational momentum density, defined as in Definition 4.1 but with integration parametrization one any smooth 3D shape B of any size, that of course both as a diffeomorphic image of a spherical ball with its spherical coordinates integration parametrization. Then the average vorticity or average rotational momentum density is also upper bounded by the Fω. In symbols This equation is derived after applying as in Theorem 4.1 the loop integral of the circulation at the Navier-Stokes equations instead at the Euler equations taking the material-flow derivative outside the integral, and eliminating the conservative, irrotational part of the pressure forces as gradient of the pressure. Here the viscosity is not zero thus the left hand of the equations is not zero as in the case of Euler equations, where it is conserved. The right hand side is nothing else than the loop work density of the point density of the force of viscosity at any time t. And as the viscosity always subtracts energy, this right hand side work density is always negative during the flow. We notice after the Lemma 3.1 that the viscosity force point density keeps constant sign on the trajectory path as orbital component during the flow and relative to the velocity on the trajectory. It is always as orbital component opposite to the motion and represents the always irreversible energy absorption and linear momentum and angular momentum decrease. Similarly, for any rotation of the fluid e.g. with axis the trajectory path. The viscosity force point density as component on the loop is always opposite to the rotation, it never converts thermal energy to add to linear or angular momentum. This opposite to motion monotonicity of the viscosity force density applies to the Navier-Stokes equations but also as opposite to rotation monotonicity in the vorticity equation = * ∇ u+ν∆ (see [START_REF] Majda | Vorticity and incompressible flow Cambridge University texts in applied mathematics[END_REF] Majda, A.J-Bertozzi, A. L. 2002, (eq 1.33 ) and (eq 1.50 ) in pages 13 and 20 ) . So if we choose a direction of the loop so that the circulation integral on the right hand side is positive then this will have the same sign during the flow (although different absolute value), and will make the left hand side of the (eq. 4.9) always negative during the flow. But this means from the left-hand side of the equation that the circulation of the velocity on the loop is always decreasing during the flow.

|| || = | 0 • || | ≤ Fω (eq.
() < 0 for any t in [0,T] (eq. 4.10) Thus (eq. 4.6) is proved, and (eq. 4.7) is direct consequence.

To prove the equation 4.

() = () = -() +ρ () <= ρ () Similarly • =-() + • <= • <=|S| M0
since as in the proof of Theorem 4.3 the friction circulation term is always negative and due to the Schwartz conditions on the external force in space and time the constant M0 is independent from space and time and the size of the surface of the loop in the integration. |S| is the area of the flux integration.

Then from smoothness and elementary 1-dimensional calculus we would get an inequality like

| 0 • -()•| <=|S| M1(Fext , t)
where again due to the Schwartz conditions on the external force in space and time the constant M1 is independent from space and the trajectories paths and depends only on the time and the external force.

Similarly by dividing the first equation by |B| which does not change by time and integrating for 3D ball , we can result similarly to an inequality like

| 0 • -0 (0)• |(0)| | ≤ 1 ( , ) = 1 ( , )
where again the constant M2 is independent from space and the size of the ball and depends only on the time t. Let the Euler or Navier-Stokes equations of incompressible fluids in the non-periodic or periodic setting (homogeneous case with no external forces), with a) smooth initial data and whatever else hypothesis is necessary so as, also to guarantee the existence and uniqueness of smooth solutions to the equations locally in time [0, T).

b) Furthermore we assume that the initial data in the periodic or non-periodic case, are such that the supremum of the vorticity, denoted by Fω , is finite at t=0. (In the periodic case, smoothness of the initial velocities is adequate to derive it, while in the non-periodic setting smooth Schwartz initial velocities is adequate to derive it)

Then it holds that there cannot exist any finite or infinite time blow-up at the point vorticities during the flow.

Proof: The proof will by contradiction. The main idea of the proof is to utilize that in the case of a blow-up the vorticity will converge to infinite, so it will become larger than an arbitrary lower bound M+Fω , Μ>0 , Fω >0 and by approximating it with average flux vorticity of a 3D spherical particle, and tracing it back at the initial conditions where all is bounded by Fω ,utilizing the semi-invariance of the average vorticity that we have proved, we will get that Fω > M+Fω .

So let us assume that there is a blow up, in a finite time or infinite time T * , with the hypotheses of the theorem 4.2. Then from the Theorem 2.2 and (eq. 2.12 ) which is the well-known result of the control of regularity or blow up by the vorticity we get that , lim t→T * 0 T |ω . , τ | L ∞ dτ=+∞ (eq. 2.12)

We conclude that there will exist an infinite sequence of points {xtn , n natural number, 0<tn<T*, lim n→∞ = * } so that the point vorticity ( ) blows-up, or equivalently lim n→∞ ( ) =+ ∞ . We do not need to assume them on the same trajectory. Therefore, for every positive arbitrary large real number M0 , there is a n0 such that for all natural numbers n> n0 , it holds that ω(xtn)>M0 . We choose M0=M00 +Fω , for an arbitrary large positive number M00 . So (eq. 4.12)

Epilogue

In this paper I solved the millennium problem about the Navier-Stokes equations in the homogeneous case without external forcing, and proved that there cannot be a blowup in finite or infinite time (regularity) both in the periodic and non-periodic setting without external forcing (homogeneous case). But it is also indicated how to prove that once the hypotheses of external forcing of the millennium problem allow 1) to have the existence and uniqueness of a smooth solution local in time and 2) the criterion of the accumulation of the vorticity as in Theorem 2.2 continuous to hold with external forcing as in the formulation of the Millennium problem, then that the same result of regularity (no blow up) holds also for this inhomogeneous case with external forcing. Furthermore, I proved also the by far more difficult same result for the Euler inviscid flows. I did so by utilizing (e.g. in in the inviscid case) that not only the momentum is conserved but also rotatory versions of the momentum 1D line and 2D surface densities are conserved. Then I extended the conservation in the case of viscous Navier-Stokes flows to monotone semi invariants, in other words that these densities are monotonously decreasing due to friction. This allowed me to prove with elementary geometric calculus that there cannot be any blow up (regularity). The solution of this millennium problem gave the opportunity to discover 2 new monotone semi invariants (1D and 2D densities of (rotatory type) momentum) for the viscous Navier-Stokes equations.

  We denote Euclidean balls by (, ): = { ∈ 3 : | -| ≤ }, where ||x|| is the Euclidean norm. The 4 sub-problems or conjectures of the millennium problem are the next: (Conjecture A) Existence and smoothness of Navier-Stokes solution on R 3 .

Theorem 2 . 1

 21 Velocities Sobolev norm sufficient condition of regularity. Given an initial condition u0 ∈ V m m>=[3/2]+2=3.5 e.g. m=4 , then for any viscosity ν>=0 . there exists a maximal time T* (possibly infinite) of existence of a unique smooth solution u ∈ C([0,T*];V m ) ᴖ C 1 ([0,T*];V m-2 ) to the Euler or Navier-Stokes equation. Moreover, if T*<+∞ then necessarily limt->T* ||u(. , t)||m =+∞.

4 . 1 )

 41 Proof: See[START_REF] Majda | Vorticity and incompressible flow Cambridge University texts in applied mathematics[END_REF] Majda, A.J-Bertozzi, A. L. 2002, Proposition 1.11 and Corollary 1.3 , in page 23. The proof is carried actually by integrating the Euler equations on a loop c and utilizing that the integral of the pressure forces (densities) defined as -∇p are zero as it is a conservative (irrotational) field of force (densities). Then by applying also the Stokes theorem that makes the circulation of the velocity on a loop equal to the flux of the vorticity on a smooth surface with boundary the loop (see e.g. Wikipedia Stokes theorem https://en.wikipedia.org/wiki/Stokes%27_theorem) the claim is obtained. QED.We may notice that this circulation and surface vorticity flux has physical measuring units[ρ]*[ω]*[s]^2=[m]*[s]^(-3)*[t]^(-1)[s]^2=[m]*[s]^(-1)*[t]^(-1) =[moment_of_inertia]*[ω]*[s]^(-3)thus angular momentum point density. While the ρ*ω has physical measuring units dimensions [ρ]*[ω]=[m]*[s]^(-3]*[t]^(-1) =[moment_of_inertia]*[ω]*[s]^(-2) thus 2 nd spatial derivative of rotational momentum of point density .

Definition 4 . 1 • = ρ 0 •

 410 We define as average value on ball in of the vorticity ω , denoted by ,the unique constant value of the vorticity on the interior of the ball that would give the same 3D flux of vorticity on the ball, ρ 0 . The integration on the surfaces S for the flux of the vorticity are in parallel circular discs in the ball. This average value of the vorticity is of course the|| || its direction is that of the vertical axis of the ball ΒWhere |Β|=(4/3)*π*r 3 is the volume of the ball B, of radius r, and | | || is the Euclidean norm of the vector. A more detailed symbolism of the average vorticity is the ( , )(,)

4 . 3 )Theorem 4 . 2 A 3 -Figure. 1 Proof:Theorem 4 . 3 . 2 for|

 434231432 Figure.1

Remark 4 . 2 .

 42 8 we notice that due to incompressibility, the flow is volume preserving, thus |B(x(t))|=|B(x(0)| , and by dividing both sides of the equation 4.7 , and after the definition || || = | 0 • || | it holds also for t ∈ [0,T] || ω B(0) || > || ω B(t) || (eq. 4.8) QED. We can extend the results of the theorems 4.1, 4.3 with Euler or Navier-Stokes equations to similar ones in the inhomogeneous case with external forces Fext., provided of course 1) we have the existence and uniqueness of a smooth solution local in time and 2) the criterion of the accumulation of the vorticity as in Theorem 2.2 continuous to hold with external forcing as in the formulation of the Millennium problem . We would start from an equation

Theorem 4 . 4

 44 The no blow-up theorem in finite or infinite time in the Euler, Navier-Stokes, periodic or non-periodic and homogeneous cases.

  ω(xtn)> M00 +Fω (eq. 4.11) Now we approximate this point vorticity with an average flux vorticity on a 3D particle after Definition 4.1 , theorem 4.2 and Lemma 4.1. Let a spherical ball particle B(r, xtn,) as in theorem 4.2. with center xtn and radius r>0. After Definition 4.1 ,theorem 4.2 and Lemma 4.1. we have that lim r→0 ω B = ω x(tn) , with || ω B || = | 0 2π S ω•ds |B(r, x(tn))| | (eq. 4.2)Therefore for arbitrary small positive number ε>0 , there is radius R, with

  It is stated in the same formal formulation of the Clay millennium problem by C. L. Fefferman see[START_REF] Fefferman | Existence and smoothness of the Navier-Stokes equation[END_REF] Fefferman C.L. 2006 (see page 2nd line 5 from below) that the conjecture (A) has been proved to holds locally. "..if the time internal [0,  ), is replaced by a small time interval [0,T), with T depending on the initial data....". In other words there is >T>0, such that there exists a unique and smooth solution u(x,t)C  (R 3 [0,T)). See also[START_REF] Majda | Vorticity and incompressible flow Cambridge University texts in applied mathematics[END_REF] A.J. Majda-A.L. Bertozzi ,Theorem 3.4 pp 104. In this paper, as it is standard almost everywhere, the term smooth refers to the space C  In the next the || ||m is the corresponding Sobolev spaces norm and . We denote by V m ={u in H m (R n ) and divu=0} where H m (R n ) are the Sobolev spaces with the L 2 norm.

	We must mention that in A.J. Majda-A.L. Bertozzi [16] ,Theorem 3.4 pp 104, Local in Time existence of Solutions
	to the Euler and Navier-Stokes equations it is proved that indeed if the initial velocities belong to V m
	m>=[3/2]+2 there exist unique smooth solutions locally in time [0,t].

Take ν>0 and n=3. Then there exist a smooth, divergent-free vector field u0(x) on R

3

and a smooth f(x,t) on R 3 x[0,+∞) satisfying (2.8), (2.9) for which there exist no smooth solution (p(x,t) ,u(x,t)) of (2.1), (2.2),

(2.3)

, (2.10) , (2.11) on R 3 x[0,+∞). Remark 2.4.

Thus after (eq. 4.11) 0 2 • B R, x tn > 00 + -ε (eq. 4.13)

Now we trace back on the trajectory of the xtn the parts of the (eq. 4.13)

At initial time t=0. We use the advantage that as the incompressible flow is volume preserving, the |B(R, x0,)|= B R, x tn . We also utilize theorems 4.2, 4.3, and (eq. 4.5), (eq. 4.8) , which prove that at the initial conditions t=0 , this average vorticity is the same or higher than that at tn . From the (eq. 4.14) and (eq. 4.

3) of Lemma 4.2 we conclude that Fω > 00 + -ε (eq. 4.15)

But M00 was chosen in an independent way from ε>0 to be arbitrary large, while ε>0 can be chosen to be arbitrary small. Therefore, a contradiction. Thus there cannot be any blow-up either in finite or infinite time T*. QED.

Remark 4.3. Infinite initial energy. We must remark that we did not utilize anywhere that the initial energy was finite, only that the vorticity initially has finite supremum. Thus this result of no-blow-up can be with infinite initial energy too. But when applying it to the millennium problem we do have there also that the initial energy is finite.

Remark 4.4. Inhomogeneous case. It is interesting to try to extend this result of no blowup, for the inhomogeneous case too of the Euler and Navier-Stokes equations and investigate where it would fail, if at all, provided of course we have 1) the existence and uniqueness of a smooth solution local in time and 2) the criterion of the accumulation of the vorticity as in Theorem 2.2 still holds with external forcing as in the formulation of the Millennium problem . We would utilize the last inequality of remark 4.2

and we would anticipate for the choice of the constant M0 in (eq 4.11) , M0=M00 +M1+Fω . We would reason similarly as in the proof of the Theorem 4.4 and we end to a same contradiction Fω > 00 + -ε But since at least in the book [START_REF] Majda | Vorticity and incompressible flow Cambridge University texts in applied mathematics[END_REF] Majda, A.J-Bertozzi, A. L. 2002, that I took as reference on the subject, it does not claim the existence and uniqueness of a smooth solution locally in time, in the case of external forces, as we wrote in remark 2.4, I will avoid using it, and I remain only in the homogeneous case. Therefore, for the moment I will not spend space in this paper on the inhomogeneous case.

Remark 4.5. A strange blow up for any time t>0 of initially smooth data. We might be curious to ask the question if it is possible, starting with zero initial velocities and pressures, to create an artificial blow-up only with external forcing. A good candidate is the perfect circular vortex, where all the trajectory paths are perfect circles, which is known that it is an instance of the solution of the Euler and Navier-Stokes equations. We can formulate the circular vortex in 3D with cylindrical or spherical coordinates. But for simplicity we will formulate it in 2 dimensions, in spite the fact that we do know that in 2D dimensions there cannot be a blow up under the hypotheses of the millennium problem. So with an external forcing also as perfect circular vortex that in polar coordinates are as follows Fr=0 , Fθ=2ρ/(1+exp(r)) (eq 4. [START_REF] Majda | Vorticity and incompressible flow Cambridge University texts in applied mathematics[END_REF])

we raise the absolute initial rest within finite time t the flow to a circular vortex which has velocities in polar coordinates ur=0 , uθ=2t/(1+exp(r)) (eq 4.17)

Now it is elementary to show that 1) this flow follows the Euler and Navier-Stokes equations 2) Because curlω=0 , the viscosity has no effect it is as if an inviscid flow.

If ω is the vorticity then it is calculated in polar coordinates at the verical z-axis by the formula = + -(eq. 4.18)

4) Although the velocity has smooth polar coordinates, the vorticity is in steady blow-up (singularity) at r=0 for any t>0. That is although at t=0 the initial data are smooth, for any t>0, there is a blow-up.

5) The 4) is so because the external forcing although it has smooth polar coordinates, in the Cartesian coordinates, it has curl(F)=+∞, at r=0, thus it does not satisfy the smooth Schwartz condition external forcing of the millennium problem.

The solution of the Millennium problem for the Navier-Stokes equations but also for the Euler equations.

We are now in a position to prove the Conjectures (A) and (B) , non-periodic and periodic setting , homogeneous case of the Millennium problem.

(Millennium Homogeneous Case A) Existence and smoothness of Navier-Stokes solution on R 3 .

Take ν>0 and n=3. Let u0(x) be any smooth, divergent-free vector field satisfying (2.4). Take f(x,t) to be identically zero. Then there exist smooth functions p(x,t) , u(x,t) on R 3 x[0,+∞) that satisfy (2.1), (2.2), (2.3) , (2.6) , (2.7).

Proof: All the hypotheses of the no-blow-up theorem 4.4 are satisfied. After remark 2.4, with the current case of the millennium problem there exist indeed a unique smooth solution locally in time [0,t] (after A.J. Majda


Anastasiia Sheveleva

Ugo Andral

Bertrand Kibler

Pierre Colman

John M Dudley

Christophe Finot
email: christophe.finot@u-bourgogne.fr

Ideal Four Wave Mixing Dynamics in a Nonlinear Schrödinger Equation Fibre System

Near-ideal four wave mixing dynamics are observed in a nonlinear Schrödinger equation system using a new experimental technique associated with iterated sequential initial conditions in optical fiber. This novel approach mitigates against unwanted sideband generation and optical loss, extending the effective propagation distance by two orders of magnitude, allowing Kerr-driven coupling dynamics to be seen over 50 km of optical fiber using only one short fiber segment of 500 m. Our experiments reveal the full dynamical phase space topology in amplitude and phase, showing characteristic features of multiple Fermi-Pasta-Ulam recurrence cycles, stationary wave existence, and the system separatrix boundary. Experiments are shown to be in excellent quantitative agreement with numerical solutions of the canonical differential equation system describing the wave evolution.

INTRODUCTION

The nonlinear Schrödinger equation (NLSE) is one of the seminal equations of science, describing wave evolution in a dispersive medium subject to an intensity-dependent nonlinear phase shift. It applies to many different domains including plasma physics, hydrodynamics, Bose-Einstein condensates, analog gravity, optical self-focussing and filamentation, and wave propagation in optical fiber [START_REF] Sulem | The nonlinear Schrödinger equation: selffocusing and wave collapse[END_REF].

The key physical process in the NLSE is nonlinear four wave mixing (FWM), which arises from the dispersion-mediated energy exchange between discrete evolving frequency components [START_REF] Boyd | Nonlinear optics[END_REF]. From a fundamental perspective, the essential features of FWM are most clearly seen in the degenerate case when a single frequency pump generates only two sidebands of upshifted and downshifted frequency. In this case the system is fully described by a reduced system of three coupled differential equations which fully captures the rich dynamical landscape [START_REF]Nonlinear Fiber Optics[END_REF]. This includes effects such as the initial phase of modulation instability, Fermi-Pasta-Ulam recurrence, stationary waves (fixed points), and a separatrix boundary between different dynamical regimes. However although this canonical FWM system has been the subject of a number of previous theoretical and numerical studies, it is notoriously difficult to implement in practice. Consequently the expected ideal FWM dynamics have only been partially or imperfectly seen in experiments to date.

Here, we address this shortcoming directly through a new experimental technique which has allowed us to characterize nearideal FWM dynamics in a nonlinear Schrödinger equation system based on optical fibre propagation. In particular, we have developed a system where iterated initial conditions are sequentially injected into an optical fiber, extending the effective propagation distance by two orders of magnitude and mitigating against unwanted sideband generation and optical loss. As a result we are able to clearly follow the dynamical interactions between only four evolving frequency components over a distance of 50 km using only one fiber segment of only 500 m length. Our experiments reveal the full dynamical phase space topology, revealing characteristic features of multiple Fermi-Pasta-Ulam recurrence cycles, stationary wave existence, and the system separatrix boundary. We compare our experimental results with solutions of the canonical differential equation system describing the wave evolution, obtaining excellent agreement. This approach represents a significant improvement in both implementation and accuracy on previous approaches and moreover, it can be readily generalized to the study of any arbitrary number of interacting wave components. This represents a major advance in the development of experimental techniques in nonlinear fiber optics.

THEORETICAL BACKGROUND AND PRINCIPLE

We first review the theoretical description of ideal FWM dynamics in the NLSE, and use numerical simulations to illustrate the expected dynamical behavior. In an ideal single mode and loss-free fiber, the evolution of a slowly-varying electric field envelope z,t is governed by the nonlinear Schrödinger equation:

2 2 2 2 0 2 i z t            , (1)
with z being the propagation distance and t the time in a reference frame traveling at the group velocity. The group-velocity dispersion is 2 and the nonlinear Kerr coefficient is . We can write the NLSE in normalized form:

2 2 2 1 0 2 AA i A A         , (2)
Here, normalized propagation and co-moving time variables  and  are linked to the dimensional quantities in fibre optics by

/ NL zL   and 2 / NL tL  
. The characteristic length scale is defined as: LNL = ( P0) -1 with P0 a power variable which in our case corresponds to the average power of the injected signal.

The normalized field is related to its dimensional equivalent z,T by

0 ( , ) ( , ) / A z T P    
. Note that this form of the NLSE that describes "focusing" dynamics is associated with a fibre dispersion parameter 2 < 0.

We discuss the fundamental wave mixing processes in the NLSE by considering the injection of a modulated pump wave A0 with two sidebands at frequencies ±:

    01 1 ( , ) ( ) ( ) exp ( ) exp A A A i Ai              , (3)
Note that we omit the carrier frequency here, and the normalized frequency  is related to dimensional frequency fm in Hz by:

20 2/ m fP     .
In general, the injection of such a modulated signal in an optical fibre leads to the generation of multiple additional sidebands, but the ideal truncated FWM system which describes only pump and first sideband energy exchange with distance is described by only three coupled equations:

      2 2 2 0 0 1 1 0 * 1 1 0 2 2 2 2 1 1 1 0 1 1 *2 10 2 2 2 2 1 1 1 0 1 1 *2 10 22 2 1 22 2 1 22 2 dA i A A A A d A A A dA i A A A A A d AA dA i A A A A A d AA                                              , (4) When 10 AA  
, amplification of the lateral sidebands can occur for  < 2, with maximum gain at a frequency

0 2    
. This is of course the same condition for maximum gain that is derived in the linear stability analysis of modulation instability [START_REF]Nonlinear Fiber Optics[END_REF]. Note also that even though this system describes degenerate FWM, the fact there are only three frequency components involved has led to it being described (somewhat confusingly) as a "three wave" system.

From a dynamical perspective, this system can be associated with the one-dimensional conservative Hamiltonian:

    22 3 2 1 cos 1 2 H            . ( 5
)
with canonical conjugate variables and  satisfying:

and

HH                 , (6)
and where  and are related to the amplitudes Ak() and phases k() of the evolving sidebands (k = 0, ±1) by:

2 0 2 2 2 0 1 1 1 1 0 2 A A A A                  , (7)
Here,  and  have physical interpretations as the fraction of the total power in the central frequency component and the sidebandpump frequency component phase difference respectively. Tracing the dynamics in the ( cossin ) plane fully captures all the physics of this ideal system.

To illustrate the physics of this system [START_REF] Cappellini | Third-order three-wave mixing in singlemode fibers: exact solutions and spatial instability effects[END_REF], Fig. 1 shows modelling results for different initial conditions. The parameters chosen correspond to maximum gain with  = 0, and we assume initially equal sideband intensities A1(0)=A-1(0). In panels a and b we show results of numerical integration of the ideal FWM system in Eq. 4 (blue), results from simulation of the segmented approach (described below) which we use in our experiments (red), and results from numerical solution of the full multiwave interactions from the NLSE (yellow). In panels c we show the temporal evolution computed from Eq. 3 for the ideal FWM case.

We now discuss these results in detail. We first consider an initial value  = 0.9 and in-phase initial components (shown in panels a1, b1, c1. The periodic evolution of  in Panel 1a clearly shows reversible energy transfer from the central mode to the sidebands, associated with the expected Fermi-Pasta-Ulam recurrence [START_REF] Van Simaeys | Experimental study of the reversible behavior of modulational instability in optical fibers[END_REF][START_REF] Mussot | Fibre multi-wave mixing combs reveal the broken symmetry of Fermi-Pasta-Ulam recurrence[END_REF]. This recurrence is also seen in the corresponding temporal intensity profile (panel c1), and is reflected in the closed trajectories in the phase space portrait in panel b1. These orbits are localized on the right-hand side of the ideal FWM separatrix orbit (dashed black line, also computed from Eqs 4) which divides the phase space into two distinct regimes.

For  = 0.9 but with  , very different features are observed. This highlights the crucial role of the initial phase in this system. Although the power redistribution between modes remains periodic (panel a2), the amplitude variation is increased compared to the previous case. Moreover, the closed phase space trajectory (panel b2) is found on the opposite side of the separatrix. Physically, this is associated with modified temporal evolution as shown in panel c2, with a temporal phase-shift taking place in each recurrence cycle associated with longitudinal period doubling [START_REF] Mussot | Fibre multi-wave mixing combs reveal the broken symmetry of Fermi-Pasta-Ulam recurrence[END_REF][START_REF] Trillo | Dynamics of the nonlinear modulational instability in optical fibers[END_REF].

. As a final example, panels a3-c3 show the dynamics observed for  = 0.715 and with  . This leads to a near-stationary solution with very low amplitude variation in the side band ratio  (panel a3) and in the temporal intensity profile (panel c3). The phase space trajectory in this case is a closed orbit of very small effective radius, close to a fixed point of the system (panel b3). Panels a1-a3 in Figure 1 also show the evolution of  computed from the numerical simulation of the full NLSE including multiple sideband generation (yellow). These results are clearly very different from the truncated ideal FWM system, with significantly more depletion of the central frequency component as additional sidebands are generated [START_REF] Hammani | Spectral dynamics of modulation instability described using Akhmediev breather theory[END_REF]. These differences are also very apparent when comparing the ideal FWM (blue) and NLSE (yellow) orbits in the phase space portraits in panels b1-b3. In fact, we see that the NLSE orbits actually cross the separatrix associated with the ideal FWM system (this is in fact expected here given that the NLSE separatrix for this case is associated with the Akhmediev breather which possesses an infinite number of sidebands.) Moreover, for  = 0.715, the stationary solution is clearly not recovered and the orbit is much more complex than a fixed point.

These examples clearly show the difficulties in observing the canonical dynamics of ideal FWM in a NLSE system -the generation of additional sidebands in the full NLSE leads to major quantitative disagreement. However, a segmented approach to propagation with reinjection of power-adjusted initial conditions allows us to overcome this limitation, and to develop a practical system that yields close to ideal FWM dynamics. The principle here is to replace a single long segment of fiber by a concatenation of segments that are sufficiently short such that additional sidebands cannot reach a significant level. Moreover, between sequential segments we cancel spectral components outside the four principle modes, and we use amplification to restore the same average power. Results illustrating this segmented approach are shown as the red lines in Fig. 1, and clearly show how this approach yields excellent agreement with the ideal FWM model: all the main features previously discussed are now quantitatively reproduced. Note that for these results, we use fiber segment lengths of L = 0.12, a choice which is motivated by our experiments described in Section 4.

EXPERIMENTAL SETUP

Attempts to measure the complex longitudinal NLSE wave mixing dynamics in optical fibre have been previously reported using various methods such as destructive cut-back measurements [START_REF] Hammani | Spectral dynamics of modulation instability described using Akhmediev breather theory[END_REF], distributed optical time domain reflectometry [START_REF] Naveau | Heterodyne Optical Time Domain Reflectometer Combined With Active Loss Compensation: A Practical Tool for Investigating Fermi Pasta Ulam Recurrence Process and Breathers Dynamics in Optical Fibers[END_REF][START_REF] Hu | Distributed Measurement of Fermi-Pasta-Ulam Recurrence in Optical Fibers[END_REF] or evolution in a recirculating loop [START_REF] Goossens | Experimental realization of Fermi-Pasta-Ulam-Tsingou recurrence in a long-haul optical fiber transmission system[END_REF][START_REF] Kraych | Nonlinear Evolution of the Locally Induced Modulational Instability in Fiber Optics[END_REF]. However, deviation between experiments and ideal FWM dynamics becomes significant very quickly in these cases, and is dramatically impacted by even small amounts of distributed loss or gain [START_REF] Van Simaeys | Experimental study of the reversible behavior of modulational instability in optical fibers[END_REF][START_REF] Naveau | Heterodyne Optical Time Domain Reflectometer Combined With Active Loss Compensation: A Practical Tool for Investigating Fermi Pasta Ulam Recurrence Process and Breathers Dynamics in Optical Fibers[END_REF][START_REF] Kimmoun | Modulation Instability and Phase-Shifted Fermi-Pasta-Ulam Recurrence[END_REF].

Our experimental setup is shown in Fig. 2 and is made of commercially-available telecommunications components. First, a laser operating at 1550 nm emits a continuous wave (CW). A phase modulator driven by a 40 GHz RF sinusoidal modulation converts the monochromatic laser spectrum into a set of equally spaced spectral lines [START_REF] Hammani | Applications of sinusoidal phase modulation in temporal optics to highlight some properties of the Fourier transform[END_REF]. The resulting symmetrical comb is then processed using a programmable filter (waveshaper device based on liquid crystal on silicon [START_REF] Clarke | Reconfigurable Optical Pulse Generator Employing a Fourier-Domain Programmable Optical Processor[END_REF]) that simultaneously implements several operations: elimination of unwanted spectral components, and the precise adjustment of the ratio  between the central and lateral components as well as their relative phase . Special care has been devoted to ensure that no unwanted phase/intensity coupling occurs during this shaping. The tailored three-component signal with the target i and i is then amplified by an erbium-doped fiber amplifier that can deliver a tunable power. The amplifier runs in a power controlled mode so that the average power does not depend on the input spectral properties such that the system can be considered as quasi-conservative.

Nonlinear propagation takes place in single mode optical fiber with dispersion and nonlinear parameters being respectively -7.6 ps 2 .m -1 and 1.7 W -1 .m -1 . The fiber length is 500 m, with this length selected as a tradeoff between the sensitivity of the detection stage of our setup and the appearance of Brillouin scattering: with 500-m of this fiber, the changes experienced by the optical field are significant enough to be conveniently detected and we have checked the absence of Brillouin backscattering for the range of powers under investigation. We can therefore work with CW signals without having to involve additional strategies of temporal pulse carving and associated synchronization. In order to limit polarization mode dispersion, the input state of polarization is optimized using polarization controllers.

The output signal is then attenuated and split into two in order to record both its spectral phase and intensity. An optical spectrum analyzer (OSA, resolution 0.1 nm) provides directly the ratio i+1. The spectral phase offset i+1 is retrieved from the temporal delay between the central and lateral sidebands as measured with a highspeed sampling oscilloscope. The experimentally measured values can then be imprinted as new input values and the process can be iterated at will without any accumulation of deleterious amplified spontaneous emission and without any growth of unwanted spectral sidebands or noise. Potentially unlimited propagation can therefore be emulated, similarly to methods that have for example been implemented in the field of hydrodynamics [START_REF] Chabchoub | Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves[END_REF].

EXPERIMENTAL RESULTS

A. Phase space and longitudinal reconstruction of the dynamics

We first study the dynamics of the system at maximum gain with 0 2     , i.e. for Pin = 21.5 dBm. In terms of normalized units, LNL = 4.1 km and the 500m length of our fiber segment corresponds to a normalized length L = 0.12, similar to the one used in the discussion of section 2C. The experimental phase space portraits obtained for different initial values 0 and 0 are shown in Fig. 3(a) with the orbits shown as circles connected by dotted lines. Note that in these experiments, we checked that the energy contained in the unwanted sidebands located at ±2 fm always remained well below 3% of the total energy of the signal.

For each value of 0, we examined the dynamics at two values of initial phase: 0 = 0 and 0 =  which yielded trajectories on the right and the left of the separatrix as expected from the results in Section 2. The dynamics were measured over 25 km (i.e. 50 iterations) and the results yield immediate insight into the phase space topology. The experimental orbits are seen to be in very good agreement with the predictions from the ideal system described in section 2 which are shown as thick solid lines. Indeed, many fundamental features of the ideal FWM dynamics can be seen from these results. Specifically, we clearly confirm the importance of the separatrix dividing the phase plane into two well-defined regions, with the measurements for 0 = 0.95 in particular providing a very clear indication of its location. We also see that the different experimental trajectories are nearly closed orbits and do not intersect. The slight discrepancies between experiment and prediction here are attributed to the accumulation of small errors in the phase/intensity measurements and residual depolarization effects not included in our scalar model.

Significantly, with complete experimental knowledge of the spectral phase and intensity of the three interacting frequency components of the evolving field, it is straightforward to fully reconstruct the evolving intensity profiles in the temporal domain.

Over a propagation distance of 50 km (a normalized distance  = 12, 100 iterations), Fig. 3(b) shows these results for an initial value of 0 =0.9 and both initial values of phase 0 = 0 and 0 =  plotted beneath the corresponding orbits on the right and left side of the phase space plot. These results show the expected recurrence dynamics as seen in Section 2, and for the case of 0 =, also highlight the evolution phase shift of half temporal period, leading, as expected by theory to a period doubling [START_REF] Trillo | Dynamics of the nonlinear modulational instability in optical fibers[END_REF]. Note that these doubleperiodic solutions have been the subject of particular recent interest [START_REF] Naveau | Heterodyne Optical Time Domain Reflectometer Combined With Active Loss Compensation: A Practical Tool for Investigating Fermi Pasta Ulam Recurrence Process and Breathers Dynamics in Optical Fibers[END_REF][START_REF] Barthelemy | Unusual modulation instability in fibers with normal and anomalous dispersions[END_REF]. which appear respectively on the right and left side of the separatrix (dashed black line). The experimental results over 50 iterations (circles joined with dotted lines) are compared with the theoretical solution of system (4) (solid thick line). (b) Longitudinal evolution of the temporal intensity profiles reconstructed from the experimental spectral measurements for 0 of 0.9 and phases of 0 and  as indicated.

B. Influence of the gain

Tuning the input power, we can also explore the modulationally unstable dynamics for higher values of gain. Phase space portraits obtained for an average value of 23.7 dBm leading to  = 1.1 are plotted in Fig. 4. Once again, the experimental results are in good agreement with the theoretical predictions. And when comparing with Fig. 3(a), we note how the dynamics at higher gain are associated with the change of the shape of the trajectories and the displacement of the separatrix. which appear respectively on the right and left side of the separatrix (dashed black line). The experimental results accumulated over 50 iterations (circles joined with dotted lines) are compared with the theoretical solution of (4) (solid thick line).

A more exhaustive study of the influence of the average power for a fixed value of 0 = 0.65 and 0 = 0.9 and phase offset 0 of 0 and  is shown in Fig. 5. Average powers between 19.7 and 23.7 dBm were tested, leading in terms of normalized frequency  of a range between 1.74 and 1.1.

The measurements of the instability process achieved at 0 =0.9 (panels a) confirm that with increasing powers the separatrix progressively shifts: the intersection point between the separatrix and the horizontal axis  = 0 continuously decreases. Consequently, the phase space available for the evolution of initial conditions 0= 0 gets larger and larger whereas initial conditions 0 =  evolve in more and more restricted areas.

Further measurements at 0 =0.65 (panels b) are also of interest, especially for 0 = 0. Indeed, we note that for the lowest powers (19.7 and 20.2 dBm), the initial condition 0 =.65 and 0 = 0 leads to orbits that are on the left side of the separatrix. For these powers, the trajectory obtained for 0 =  is therefore surrounded within the trajectory for 0 = 0. When increasing the power, the separatrix is crossed and each initial condition evolves on a different side of the phase plane. For powers between 20.7 and 22.7 dBm, the orbits get smaller and smaller up to the stage where they reach a fixed point for 22.7 dBm. When further increasing the average power, the orbit becomes increasingly open.

C. Observation of a fixed point

Finally, we investigate in more detail the properties observed at one of the two fixed points of the phase plane. For 0 = 0.65 and 0 = 0 and Pin = 22.7 dBm (= 1.23), the longitudinal evolution of the temporal intensity reconstructed from the spectral measurements is plotted in Fig. 6(a) over 50 km. We clearly see in this case that the temporal profile is invariant with propagation. Panel (b) explicitly compares these results with temporal measurements made with a picosecond-resolution optical sampling oscilloscope and the agreement is such that they cannot be visually distinguished. Moreover, both experimental profiles agree with the expected temporal profile computed from Eq. 3 which in the case of a fixed point simply consists of a stationary temporal profile formed from three wave interference.

More generally, a fixed point of the ideal FWM system is predicted to exist at any value of gain for a particular choice of sideband ratio. Expressed in terms of normalized frequency , the dependence is given by [START_REF] Cappellini | Third-order three-wave mixing in singlemode fibers: exact solutions and spatial instability effects[END_REF]:

2 3 7 e    , (8)
and this can be readily tested experimentally. Specifically, for different input powers, the waveshaper is used to experimentally determine the value of e associated with the fixed point, and the results are shown in Fig. 3(c). The agreement between experimental results (blue) and the prediction of Eq. 8 is excellent.

CONCLUSION AND OUTLOOK

Optical systems are well known to provide flexible testbeds with which to study the physics of diverse nonlinear systems, and the results here show the success of a new experimental approach allowing the dynamics of ideal four wave mixing to be fully explored. The use of iterated initial conditions in a short fiber segment mitigates against effects of loss, inelastic scattering and high-order sideband generation, allowing clear observation of the predicted evolution dynamics of the FWM system. Amongst the dynamics that are seen experimentally are Fermi-Pasta-Ulam recurrence, qualitatively different evolution on either side of a separatrix, and the existence of system fixed points.

It is important to stress the adaptability of this experimental technique. Although we have focused on spectrally-symmetric initial conditions, experiments can be readily adapted to asymmetric initial sidebands [START_REF] Cappellini | Third-order three-wave mixing in singlemode fibers: exact solutions and spatial instability effects[END_REF] and to evolution in fibers with normal dispersion. This opens up the possibility to study even more complex nonlinear wave mixing [START_REF] Barthelemy | Unusual modulation instability in fibers with normal and anomalous dispersions[END_REF][START_REF] Trillo | Nonlinear dynamics of dualfrequency-pumped multiwave mixing in optical fibers[END_REF]. Vector instability processes in birefringent fibers offer additional perspectives [START_REF] Murdoch | Nonlinear dynamics of polarization modulation instability in optical fiber[END_REF], and the principle of the technique can be extended into the spatial domain using spatial light modulators and phase-intensity characterization [START_REF] Pierangeli | Observation of Fermi-Pasta-Ulam-Tsingou Recurrence and Its Exact Dynamics[END_REF]. We also anticipate extension to other branches of nonlinear physics where discrete wave mixing plays a central role in the system evolution [START_REF] Thompson | Statistical fluctuations in multiple fourwave mixing in a single-mode optical fiber[END_REF].

Fig. 1 .

 1 Fig. 1. Evolution properties of an strong pump and two lateral sidebands spaced by

.

Fig. 2 .

 2 Fig. 2. Experimental setup. The insets 0-4 in the bottom panels represent the optical spectrum generated at different stages of the experiment.

Fig. 3 .

 3 Fig. 3. Experimental results for

Fig. 4 .

 4 Fig. 4. Experimental phase portraits obtained at an average power Pin = 23.7 dBm and for initial values of 0 of 0.65, 0.85 and 0.95 (green, yellow and blue lines respectively). Results are plotted for an initial phase offset 0 of 0 or .

Fig. 5 .

 5 Fig. 5. Influence of the input average power on the phase space portraits obtained for initial values of 0 of 0.9, and 0.65 (panels a and b respectively) and initial phase offset of 0 or . The theoretical predictions (panels 1) are compared with the experimental results (panels 2).

Fig. 6 .

 6 Fig. 6. (a) Longitudinal evolution of the temporal intensity profile reconstructed from the spectral properties of the signal obtained for an average power of 22.7 dBm and initial conditions 0 = 0.65 and 0 = 0. (b) Comparison of the temporal intensity profile reconstructed from spectral measurements (blue line) and directly measured with an optical sampling oscilloscope (red dashed line). Results from theoretical predictions are also displayed with black circles. (c) Evolution of the value of 0 leading to a fixed point. The experimental data (blue) are compared with the analytical prediction (dashed black line, Eq. (8)).

Funding.

  OPTIMAL (ANR-20-CE30-0004); EIPHI-BFC (ANR-17-EURE-0002); I-SITE BFC (ANR-15-IDEX-0003); Région Bourgogne-Franche-Comté; Institut Universitaire de France.

Acknowledgments. The numerical simulations used the HPC resources of DNUM CCUB (Centre de Calcul de l'Université de Bourgogne). The authors also thank GDR Elios (GDR 2080).

Disclosures. The authors declare no conflicts of interest.

Data availability. The data that support the findings of this study are available from the corresponding author, CF, upon reasonable request.