axiom of observables (each
physical observable is represented as a linear Hermitian
operator of the state Hilbert space). The corresponding mathematical model and
its solutions are governed by the Heisenberg uncertainty inequality. As the
observable space needs to support statistical analysis the Hilbert space, this
Hilbert space needs to be at least a subspace of A. At the same
point in time, if plasma is considered as sufficiently collisional, then it can
be well-described by fluid-mechanical equations. There is a hierarchy of such
hydrodynamic models, where the magnetic field lines (or magneto-vortex lines)
at the limit of infinite conductivity is “frozen-in” to the plasma. The “mother
of all hydrodynamic models is the Hcontinuity equation treating
observations with macroscopic character, where fluids and gases are considered
as continua. The corresponding infinitesimal volume “element” is a volume,
which is small compared to the considered overall (volume) space, and large
compared to the distances of the molecules. The displacement of such a volume
(a fluid particle) then is a not a displacement of a molecule, but the whole
volume element containing multiple molecules, whereby in hydrodynamics this
fluid is interpreted as a mathematical point.In fluid description of
plasmas ( The The The
The common distributional Hilbert space
framework is also proposed for a
Earliest examples of complementary variational principles are provided
by the energy principle of Dirichlet in the theory of electrostatics, together
with the Thomson principles of complementary energy. As a short cut reference
in the context of the considered Maxwell equations we refer to (ShM1).
A central concept of the proposed solution
Hilbert space frame is the alternative normal derivative concept of Plemelj. It
is built for the logarithimc 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 than
this, which are also in a natural sense generalizations of the Cauchy-Riemann
differential equations, leads to representations of the rotation group (StE).
(LiP)
Lions P. L., On Boltzmann and Landau equations, Phil. Trans. R. Soc. Lond. A,
346, 191-204, 1994 (LiP1)
Lions P. L., Compactness in Boltzmann’s equation via Fourier integral operators
and applications. III, J. Math. Kyoto Univ., 34-3, 539-584, 1994 (ShM1) Shimoji M., Complementary variational formulation of Maxwell s equations
in power series form (StE)
Stein E. M., Conjugate harmonic functions in several variables | ||||||||||||||||||||||||