D'Alembert "paradox" which is about unrealistic fluids (interaction) dynamics by which no aircraft would be able to fly.The
mathematical concept is about adequately defined fractional scaled
(energy) Hilbert space. This corresponds to J. Plemelj's alternative
normal derivative definition (with reduced regularity assumptions to its
domain) and to the generalized Green identities valid for same domain.
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 governed by the inner product of H(1/2).
June 2016
Analyse Mathematique et Applications, Gauthier-Villars, Paris, 1988
RAIRO - Analyse numerique, tome 15, no 3 (1981) p. 237-248
As a shortcut reference to the underlying mathematical principles of classical fluid mechanics we refer to (SeJ). A central concept of the proposed solution Hilbert space frame is the alternative normal derivative concept of Plemelj. It is built for the logarithmic potential case based on the Cauchy-Riemann differential equations with its underlying concept of conjugate harmonic functions. Its generalization to several variables is provided in the paper below. It is based on the equivalence to the statement that a vector u is the gradient of a harmonic function H, that is u=gradH. Studying other systems thant his, which are also in a natural sense generalizations of the Cauchy-Riemann differential equations, leads to representations of the rotation group (StE). 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). Regarding the theory of turbulence we recall from (BrP): Turbulence is a three-dimensional time-dependent motion in which vortex stretching causes velocity fluctuations to spread to all wave lengths between a minimum determined by viscous forces and a maximum determined by the boundary conditions of the flow. It is the usual state of fluid motion except at low Reynolds numbers. ... Unother simplification in the study of turbulence is that itsgeneral behavious is apparently unaffected by compressibility if the pressure fluctuations within the turbulence are small compared with the absolute pressure, that is, if the fluctuating Mach number, u/(speed of sound) say, is small."
The Leray-Hopf operator plays a key role in existence and uniqueness proofs of weak solutions of the Navier-Stokes equations, obtaining weak and strong energy inequalities. For a related integral representation of the NSE solution we refer to (PeR).
(BrP) Bradshaw P., An Introduction to Turbulence and its Measurement, Pergamon Press, Oxford, New York, Toronto, Sydney, Braunschweig, 1971 | ||||||||||||||||||