# A global unique H(1/2) based dynamical energy inner product based weak solution of the 3D-Navier-Stokes equations

**Note**: The proposed solution concept is concerned with an additional (additive) "dynamical energy" norm concept (complementary to the standard Dirichlet integral based H(1) energy norm with respect to the extended H(1/2) Hilbert space norm) as introduced in www.unified-field-theory.de

https://www.fuchs-braun.com/

Physically speaking, …

… in the current purely H(1) based mechanical energy based modelling framework the crucial 3D-NSE non-linear term governs a mechanical particle flow accompanied by an additional pressure variable governing the interactions between those mechanical particles and the interaction between mechanical flow and mechanical boundary particles enabling Kolmogorov’s statistical L(2)=H(0) based turbulence (disturbance) theory

… in the proposed extended H(1/2) mechanical + dynamical energy based modelling framework the mechanical particle flow may be interpreted as physical disturbance (mathematically modelled by a compact operator) of a corresponding overall dynamical quanta energy field system, where the classical 3D-NSE initial & boundary conditions become only relevant for the „mechanical world“ and Kolmogorov’s statistical L(2)=H(0) resp. energetical H(1)-based mechanical turbulence theory may still be applied.

**The 3D-NSE solution concept**

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 H(1/2) is a consequence of the Sobolevskii estimate of the non-linear term (1959). The extended (energy) Hilbert space is in line with the proposed Krein space based quanta potential energy Hilbert space concept in unified-field-theory.de. It enables an alternative mathematical model for Mie’s concept of an electric pressure enhancing the Maxwell equations. The second unknown function in the NSE is the pressure p; the pressure function p can be represented as Riesz operator transforms of (u x u), while the gradient (force) operator applied to the unknown pressure function p becomes the Calderón-Zygmund integrodifferential operator applied to the (velocity) NSE-solution function u (EsG) p. 44.

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