The Millenium problem solution and a quantitiative turbulence model enabled by a physical H(-1/2) Hilbert space
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.
The Serrin gap occurs in case of space dimension n=3 as a consequence of the Sobolev embedding theorem with respect to the energy Hilbert space H(1) with the Dirichlet integral as its inner product.
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).
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 concept can also be applied to the Maxwell equations for an alternative QED model whereby the Heisenberg inquality is localized to the closed subspace domain for the incoherent "mass element" flows.
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.
It enables a problem adequately defined fractional scaled (energy) Hilbert space. This Hilbert space framework is related to the RH solution concepts of P1 and P2. It is enabled by J. Plemelj's alternative normal derivative definition, which requires less regularity assumptions, and corresponding generalized Green identities with same reduced regularity assumptions. 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 by 1/2.
The solution concept also addresses also theD'Alembert "paradox" which is about unrealistic (fluid interaction) assumptions of today's (mathematical model) fluid dynamics by which no aircraft would be able to fly, anyway.