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In the proposed UFT the most granular extended H(1/2)-energy Hilbert space of the standard variational mechanical H(1)-energy Hilbert space enables a well-posed 3D non-linear, non-stationary Navier-Stokes problem (existence, uniqueness, and smoothness by a bounded H(1/2)-energy norm estimate); this solves the related problem of the Clay Mathematics Institute, Cambridge.
The concept of a H(1/2) energy field based dynamic fluid element is accompanied by a well-posed exterior Neumann problem, a well defined related Prandtl operator, (BrK7), and Plemelj's double layer potential, where the mass density (du/ds)(s) of the single layer potential is replaced by the differential du(s), (PlJ).
The inner product of H(1/2) is isometric to an inner product in the form (Qx,Px), where Q resp. P denote Schrödinger‘s position & momentum
operators. The combination with the Riesz transform operator provides the link to the considered operators in (BrK0), and is in line with the alternatively proposed Schrödinger operator in (BrK6) and related early thoughts on a new ground state energy model in (BrK8).
For early thoughts on NSE, YME, and GUT solutions based on an orthogonal decomposition of H(1/2) into the mechanical energy Hilbert space H(1) and its complementary closed sub-space of H(1/2) we refer to (BrK9), (BrK10).
By the way, an extended H(1/2) energy field based variational framework allows ‘optimal’ FEM approximation
error estimates
for non-linear parabolic
problems
with
non-regular initial value data using the Stefan (free boundary) model problem, (BrK12).
 Braun K., Global existence and uniqueness of 3D Navier-Stokes equations |
www.navier-stokes-equations.com
