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A unique physical 3D non-linear, non-stationary NSE solution

... enabled by rotating H(-1/2)-based "fluid elements" with H(1/2)-based angular momentum.

The classical Navier-Stokes Equations (NSE) describe a flow of incompressible, viscous fluid. It has been questioned whether the NSE really describe general flows: The difficulty with ideal fluids, and the source of the d'Alembert paradox, is that in such fluids there are no frictional forces. Two neighboring portions of an ideal fluid can move at different velocities without rubbing on each other, provided they are separated by a streamline. It is clear that such a phenomenon can never occur in a real fluid, and the question is how frictional forces can be introduced into a model of a fluid.

The three key foundational questions of every PDE is existence, and uniqueness of solutions, as well as whether solutions corresponding to smooth initial data can develop singularities in finite time, and what these might mean. The Navier–Stokes existence and smoothness problem for the three-dimensional NSE, given some initial conditions, is to prove that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass.

We provide a global unique (weak, generalized Hopf) H(1/2)-solution of the generalized 3D Navier-Stokes initial value problem. The global boundedness of a generalized energy inequality with respect to the energy Hilbert space  is a consequence of the Sobolevskii estimate of the non-linear term (1959). The proposed solution also overcomes the "Serin gap" issue, as a consequence of the bounded non-linear term with respect to the appropriate energy norm.

The proposed solution concept is about rotation-invariant fluids (circulation) modelled as elements of a Hilbert space with negative (distributional) Hilbert scale defined by the eigenpairs of the Stokes operator.

The solution concept also addresses the D'Alembert "paradox" which is not a paradox; it is about unrealistic fluid solutions (interaction) dynamics by which for example no aircraft would be able to fly.

The mathematical concept is about adequately defined fractional scaled (energy) Hilbert space. This corresponds to J. Plemelj's alternative normal derivative definition (with reduced regularity assumptions to its domain) and to the generalized Green identities valid for same domain. The resulting regularity requirement reduction is in the same size as a reduction from C(1) to C(0) regularity, which leads to "scale reduction" of weak (variation) partial differential equation representations governed by the inner product of H(1/2).


Braun K., Global existence and uniqueness of 3D Navier-Stokes equations

                                                   June 2016


Suporting data:


Federbush P., Navier and Stokes meet the Wavelet


Giga Y., Weak and Strong Solutions of the Navier-Stokes Initial Value Problem


Lerner N., A note on the Oseen kernels


Lions P. L., Compactness in Boltzmann's equation


Nitsche J. A., 1988, Direct Proofs of Some Unusual Shift-Theorems

        Analyse Mathematique et Applications, Gauthier-Villars, Paris, 1988


Nitsche J. A., On Korn s second inequality
            RAIRO - Analyse numerique, tome 15, no 3 (1981) p. 237-248

Nitsche J. A., Free boundary problems for Stokes´s flows and finite element methods


Peralta-Fabi R., An integral representation of the Navier-Stokes equation-1


Phillips, R. S., Dissipative operators and hyperbolic systems of partial differential equations

The Navier-Stokes equations describe the motion of fluids. The classical Navier-Stokes partial differential equations in a Sobolev space framework is about "fluid elements" modelled as elements of the Lebesgue L(2) Hilbert space and an related "fluid motion/velocity" modelled as elements of the  Sobolev (sub-) space H(1)=W(1,2).

The modelling framework of quantum mechanics are also Hilbert spaces.

The simple proposal of this homepage is to replace the standard Hilbert space L(2)=H(0) by the weaker distributional Hilbert space H(-1/2). The later one is proposed to replace Dirac's concept of the H(-n/2-e) Hilbert space (n denotes the space dimension, and e>0), which contains the "Dirac/Delta "function"", as his proposed model of a charged electron. We note that this kind of EP is twofold attributed, it is a "particle" with a given finite charge, or, in other words, this EP model is characterized by two specifications,

1. a kind of location in the considered space framework; mathematically speaking, this is basically a real number, which is with 100% probability "only" defined as the limit of a sequence of an infinite numbers of rational numbers, or, physically speaking, it is an whole universe by itself

2. "equipped" with an electric finite physical "potential difference" between the somewhere in the space located particle and its surrounding space framework.

The Navier–Stokes existence and smoothness problem for the three-dimensional NSE, (given some initial conditions) is to prove that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass. Continuity resp. differentiability of its solutions are ensured by the Sobolev embedding theorem, where the (space dimension n depending) inequality > n/2 occurs.

By formally operating with "div" operator on the NSE the pressure field must satisfy the Neumann problem. It follows that the prescription of the pressure at the bounding walls or at the initial time independently of the velocity u, could be incompatible with the initial boundary values of the NSE, and therefore, could render the problem ill-posed (GaG). Plemelj's alternative normal derivative concept enables initial boundary value "functions" to define a pressure operator with domain H(1/2) overcoming this issue. We further note that both physical concepts, "pressure" and "energy density", do have the same unit of measure (  N / (m*m) = Nm / (m*m*m) ).

The Serrin gap in case of a space dimension n=3 isa consequence of the Sobolev embedding theorem applied to the energy Hilbert space H(1), which is equipped with the (Dirichlet integral based inner product. The proposed alternatively model deals with an extended energy Hilbert space H(1/2). It enables an appropriate, currently missing, energy norm estimate for the 3-D non-stationary, non-linear NSE taking into account energy values of the non-linear terms.

As a shortcut reference to the underlying mathematical principles of classical fluid mechanics we refer to (SeJ).

A central concept of the proposed solution Hilbert space frame is the alternative normal derivative concept of Plemelj. It is built for the logarithmic potential case based on the Cauchy-Riemann differential equations with its underlying concept of conjugate harmonic functions. Its generalization to several variables is provided in the paper below. It is based on the equivalence to the statement that a vector u is the gradient of a harmonic function H, that is u=gradH. Studying other systems thant his, which are also in a natural sense generalizations of the Cauchy-Riemann differential equations, leads to representations of the rotation group (StE).

We provide a global unique (weak, generalized Hopf) NSE solution of the variational H(-1/2)-representation of the generalized 3D Navier-Stokes initial value problem. The global boundedness of a generalized energy inequality with respect to the energy Hilbert space H(1/2) is a consequence of the Sobolevskii estimate of the non-linear term (1959).

Regarding the theory of turbulence we recall from (BrP):

"Indeed, turbulence studies may be defined as the art of understanding the Navier-Stokes equations without actually solving them …

We can now define turbulence:

Turbulence is a three-dimensional time-dependent motion in which vortex stretching causes velocity fluctuations to spread to all wave lengths between a minimum determined by viscous forces and a maximum determined by the boundary conditions of the flow. It is the usual state of fluid motion except at low Reynolds numbers. ...

Unother simplification in the study of turbulence is that itsgeneral behavious is apparently unaffected by compressibility if the pressure fluctuations within the turbulence are small compared with the absolute pressure, that is, if the fluctuating Mach number, u/(speed of sound) say, is small."

The Leray-Hopf operator and the linearized Landau collision operator

The Leray-Hopf operator plays a key role in existence and uniqueness proofs of weak solutions of the Navier-Stokes equations, obtaining weak and strong energy inequalities. For a related integral representation of the NSE solution we refer to (PeR).

In a weak H(-1/2) Hilbert space framework in the context of the Landau damping phenomenon the linerarized Landau collision operator can be interpreted as a compactly disturbed Leray-Hopf operator. 

Both operators, the "Leray-Hopf (or Helmholtz-Weyl) operator and the linearized Landau collision operator are not classical pseudo-differential operators, but Fourier multipliers with same continuity properties as those of the Riesz operators(LiP1).

For the related Oseen operators Fourier multiplier we refer to (LeN).

The related hypersingular integral equation theory, including the Prandtl operator, is provided in (LiI).


(BrP) Bradshaw P., An Introduction to Turbulence and its Measurement, Pergamon Press, Oxford, New York, Toronto, Sydney, Braunschweig, 1971

(FeP) Federbush P., Navier and Stokes Meet the Wavelet, Commun. Math. Phys. 155, 219-248 (1993)

(FoC) Foias C., Temam R., Some Analytic and Geometric Properties of the Solutions of the Evolution Navier-Stokes Equations, J. Math. Pures et Appl. 58, 339 (1979)

(GaG) Galdi G. P., The Navier-Stokes Equations: A Mathematical Analysis, Encyclopedia of Complexity and System Science, Springer Verlag, 2009

(LeN) Lerner, N., A note on the Oseen kernels, Advances in Phase Space Analysis of Partial Differential Equations, pp. 161-170, 2007

(LiP) Lions P.L., Boltzmann and Landau equations

(LiP1) Lions P.L., Compactness in Boltzmann’s Fourier integral operators and applications

(LiI) LifanovI. K., Poltavskii L. N., Vainikko G. M., Hypersingular integral equations and their applications, Chapman & Hall, CRC Press Company, Boca Raton, London,New York, Washington, 2004

(PeR)Peralta-Fabi R., A integral representation of the Navier-Stokes equations

(SoH) Sohr H., The Navier-Stokes Equations, An Elementary Functional Analytical Approach, Birkhäuser Verlag, Basel, Boston, Berlin, 2000

(TeR) Teman R., Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1983

(WeH) Weyl H., Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, 1949, 2009