The Millenium problem solution and a quantitiative turbulence model enabled by an ideal fluid H(-1/2) Hilbert space frmaework
The Navier-Stokes equations describe the motion of fluids. 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) 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).
Here we are:
The proposed solution concept is about rotation-invariant fluids (circulation) as elements of a Hilbert space with negative (distributional) Hilbert scale defined by the eigenpairs of the Stokes operator.
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) ).
With the notations of (TeR) the building of the alternative Stokes operator (which is about the definition of an alternative domain of the Stokes operator) is basically a replacement of
V := D(A(exp(1/2)) --> V := D(A(exp(1/4)) = V + V(ortho).
The fundamental property of the form b:
b(u,v,w)=-b(u,w,v) for all u,v,w ex V
leads to a vanishing "energy" term
b(u,v,v)=0 for all u,v ex V
which is not the case for all u,v ex V(ortho) .
With respect to the positive selfadjoint property of the fractional Stokes operator A(exp(a)) for -1<=a<=1 and the completion of the space D(A(exp(2a)) we refer to (SoH) III 2.2, 2.5.
The alternative H(1/2) energy (Hilbert) space framework enables a finite NSE energy inequality for all t>0 in case the solution is ex H(1). This then guarantees an unique global solution of the NSE.
The building of the corresponding Leray-Hopf (Helmholtz-Weyl) projection operator P with respect to the domain V is straightforward.
In three dimensions the curl operator and the Leray-Hopf operator are linked by the Laplacian equation (LeN). The action of the operator P on the Gaussian function is provided in (LeN), as well. The corresponding action of the operator P on the Dawson function (which is the Hilbert transform of the Gaussian function) seems to be worth for investigation.
The Hilbert transformed Gaussian in combination with the revisted one-dimensional CLM vorticity model with viscosity term in a H(-1/2) weak (variation) Hilbert space framework enables a space-scale turbulence model, which provides coherent (H(0) and incoherent (H(-1/2-H(0)) turbulent flows.
The Gaussian function f is the baseline function defining the Hermite polynomials, which build an orthogonal basis of L(2). Due to the properties of the Hilbert transform the same is valid for the Hilbert transformed Hermite polynomials. The Gaussian function is not a wavelet, but e.g. its second derivative, the Mexican hat, fulfills the admissibilty property. Let A denote the one-dimensional Symm operator and let g denote the first derivative of the Gaussian function f: then H(f)=-A(g). As the Hilbert transform of a wavelet is a wavelet, this provides the relationship of Fourier waves and Calderon wavelets within a Hilbert scale framework H(a). The Hilbert operator applied twice gives the identity operator with altered sign, i.e. -I . The counterpart of this property with respect to wavelets is given by Calderon's reproducing formula, providing the baseline for the development of the wavelet theory, e.g. Goupillaud P. Grossmann A., Morlet J., "cycle-octave and related transforms in seismic signal analysis", 1984,1985.
The concept can also be applied to the Maxwell equations for an alternative QED model whereby the closed subspace domain provides a model for the incoherent "mass element" flows.
The solution concept also addresses the D'Alembert "paradox" which is about unrealistic fluids (interaction) dynamics by which no aircraft would be able to fly, anyway.
In order to provide some (just purely technical) rationals supporting the proposed alternative fractional Hilbert space framework above we note the following (see also (SoH) 3.2, "Basic facts on Hilbert spaces"):
let "grad", "div", "S" and "R" denote the Gradient, the Divergence, the Symm and the Riesz operators, then for u ex H(1) it holds
- the non-linear term u*u is an element of the Hilbert space H(1/2)
- grad and -div are dual operators
- S(grad) = -R, i.e. S(-div) and R are dual operators .
(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
(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