enabled by the energy Hilbert space H(1/2)

Planck proposed a distinction between dynamic and statistical regularities

The current dynamic laws in theoretical physics (e.g., in thermo-statistics) are based on statistical regularities accompanied by the statistical L(2) Hilbert space framework. 

Planck proposed a distinction between dynamic and statistical regularities:

“Wie kann man (..) aus der Betrachtung von Vorgängen, deren Verlauf im ganzen wie im einzelnen vorläufig noch vollständig dem blinden Zufall überlassen bleibt, wirkliche Gesetze ableiten? … Wer also nur wirklich bestimmte Zahlen, nicht zugleich auch einen Fehlerbereich zulassen wollte, müßte auf die Verwertung von Messungen und konsequenterweise auf induktive Erkenntnis überhaupt Verzicht leisten", PlM).

"Immerhin erhellt aus der geschilderten Sachlage wohl hinreichend deutlich, die überaus hohe Bedeutung, welche die Durchführung einer sorgfältigen und grundsätzlichen Trennung der beiden besprochenen Arten von Gesetzmäßigkeit: der dyynamischen, streng kausalen, und der lediglich statistischen, für das Verständnis des eigentlichen Wesens jedlicher naturwissenschaftlicher Erkenntnis besitzt”, (PlM).

For further details of the below we refer to

unified-field-theory.de

The proposed framework for two types of energy concepts

The proposed framework to enable a well posed 3D-NSE system is based on two types of energy concepts, the current mechanical (kinetical and potential) energy type, and a new dynamic (potential) energy type.Technically spoken, the Laplacian operator based statistical (momentum) energy space H(1) is extended to H(1/2) = H(1)xH(1,ortho) accompanied by H(0)  H(-1/2) extension. The construction of the underlying Hilbert scale is based on the Stokes operator. This is a self-adjoint operator with appropriately defined Hilbert space domain, (GiY). It is the Friedrichs extension of a non-negative symmetric operator build from an orthogonal projection operator and the Laplacian.

A dynamic fluid concept enabling weak NSE solutions with bounded energy inequality

Conceptually, the mechanical fluid particle concept is enhanced to a dynamic fluid particle. The dynamic fluid H(1/2) energy concept is in line with a well-defined Plemelj’s double layer potential function, (BrK11). The related Prandtl operator accompanied by a Hilbert scale domain H(r) (where  ½ smaller or equal than r smaller than 1) provides a unique solution of the underlying Neumann boundary value problem for the pressure p(x,t), (BrK7), (BrK11), (LiI) p. 95 ff., (PlJ).

The two physical NSE intrinsic modelling problems

The two physical modelling problems are the NSE intrinsic Neumann pressure problem and the the d’Alembert problem. The two problems are addressed by

- the Noetherian Prandtl operator defined on a closed connected surface in the three-dimension space providing a unique solution of the underlying Neumann boundary value problem

- J. Plemelj’s concept of a surface intrinsic mass element dm and a related well-defined flux concept through that surface.

The proposed extended energy Hilbert space framework in a nutshell

In potential theory the standard variational domain of the mechanical (self-adjoint) potential operator is the H(1) energy space. It enables the definition of (distributional Hilbert scales H(a), a real.

The Stokes operator is a projector from the L(2) Hilbert space onto its sub-space with vanishing divergence functions. The definition of a appropriately defined Hilbert scales H(a) for absolute values of a less or equal 1 is enabled by the corresponding positive selfadjoint a-fractional powers of A , (SoH), IV15. The related Leray-Hopf projector P = Id – RxR, where R denotes the Riesz operator, is an orthogonal projection, (BrK9) in unified-field-theory.de.

The pressure p of the solution pair (u,p) of the NSE are related by the Riesz transform operator in the form p = R(u x u), where u x u is a 3x3 matrix. It enables a representation of the sum of the non-linear NSE term and the negativ pressure by the Helmholtz-Weyl projection operator and the divergence operator, (CuS).

The proposed H(1/2) Hilbert space may be interpreted as an extension of the standard variational mechanical energy Hilbert space H(1), which may be characterized by its “discrete” eigenpair spectrum. By design the H(1) becomes a compactly embedded sub-Hilbert space of H(1/2) = H(1) x H(1,ortho). This complementary closed sub-space may be interpreted as dynamic fluid energy space.The related (not self-adjoint) dynamic potential operator may be interpreted as a compact disturbance of the self-adjoint Friedrichs extension of the Stokes operator accomapnied by an underlying coerciveness (Garding type) inequality, (BrK0) p. 26 in unified-field-theory.de.

The dynamics within the H(1/2) dynamic fluid system is governed by the intrinsic energy difference between the energy Hilbert space H(1) and its related complementary closed sub-space H(1,ortho). While the H(1) Hilbert space may be governed by Fourier waves, the complementary sub-space may be governed by Calderon wavelets.