This homepage is dedicated to my mom, who died at April 9, 2020 in the age of 93 years. It considers multiple research areas. In retrospect, the proposed
solution concepts originate in some few simple ideas / basic conceptual
changes to current insufficient "solutions":
The Einstein field equations are classical non-linear,
hyperbolic PDEs defined on differentable manifolds coming along with the concepts
of „affine connexion“ and „external product“.
The Standard Model of Elementary Particles (SMEP) is basically about a sum of three Langragian equations, one equation, each for the considered three „Nature forces“. Quantum mechanics is basically about matter fields described in a L(2) Hilbert space framework modelling quantum states (position and momentum). Our proposed quantum gravity model is based on a properly extended distributional Hilbert space framework (avoiding the Dirac „function“ concept to model a „point“ charge), requiring some goodbyes from current postulates of the quantum and gravitation theories. The central changes are : - as the L(2) Hilbert space is reflexive, the current considered matter equations can be equivalently represented as variational equations with respect to the L(2) inner product; this representation is extended to a newly proposed quantum element
Hilbert space H(-1/2); we note that the Dirac function is only (at most,
depending from the space dimension) an element of H(-1/2-e), and that the main
gap of Dirac‘s related quantum theory of radiation is the small term
representing the coupling energy of the atom and the radiation field. - classical PDE equations are represented as variational equations in the H(-1/2) Hilbert space framework coming along with reduced regularity requirements to the correspondingly defined solutions; we note that the Einstein field equations and the wave equation are hyperbolic PDEs and that PDEs are only well defined in combination with approproiate initial and boundary value functions; we further note, that the main gap of the Einstein field equations is, that it does not fulfill Leibniz's requirement, that " there is no space, where
no matter exists"; the GRT field equations provide also solutions for
a vaccuum, i.e. the concept of "space-time" does not vanishes
in a matter-free universe.
The Friedrichs extension of the classical Laplace operator in a L(2) Hilbert space framework defines the inner product of a related „energy“ Hilbert space H(1). The extended Laplace operator in the newly proposed H(-1/2) framework leads to an extended energy Hilbert space H(1/2). The new energy Hilbert space H(1/2) is decomposed into the current " kinematical"
energy Hilbert space H(1) (with its corresponding underlying (fermion
elements) Hilbert space H(0)) and its complementary "ground state"
energy Hilbert space H(1,ortho) (with its corresponding underlying (boson
elements) Hilbert space H(0, ortho)). The "kinematical" energy
Hilbert space H(1) can be further decomposed into repulsive and attractive
kinematical energy spaces, in alignment with a corresponding underlying decomposition
of the (fermion elements) Hilbert
space H(0) into repulsive and attractive fermion element spaces.
Mathematically speaking, the decomposition H(1/2) = H(1) + H(1,ortho) is about a "coarse grained" Hilbert space H(1) (i.e. it is compactly and densely (with respect to the H(1/2) norm) embedded into H(1/2)) and its complementary closed (in the sense of Cantor’s cardinality measure, very much larger) sub-space H(1,ortho) of H(1/2). In the sense of Cantor, the decomposition corresponds to the "decomposition" of the field of real numbers R into rational (countable) numbers Q and irrational (non countable) numbers. We also mention that "distributions" are also called "ideal functions", (CoR) p. 766: the name "distributions" indicates that ideal functions, such that the Dirac delta function and its derivatives, may be interprested by mass distributions, dipole distributions, etc., concentrated in points, or along lines or on surfaces, etc.The considered Hilbert scale (defined by the eigenpair of the Laplace operator) resp. its underlying norm is governed by the sum of the standard (quantum mechanics / statistics) L(2)-Hilbert space norm and an " exponentical
decay" (entropy measurements, (BrK1) note 2, (BrK6)) norm, which is
weaker than any distributional "polynomial decay" norm (NiJ1).
The good bye to current physical classical PDE model solutions is that those PDE are considered as approximation solutions to the underlying weak (H(-1/2)-based) variational representations and not the other way around. The current Lagrange equations are only valid in the classical sense resp. ist equivalent L(2)-based variational representations, whereby the weak variational models are governed by a common Hamiltonian (H(1/2)-based) formalism. Physically speaking, the currently modeled "forces" phenomena become part
of the specific corresponding classical PDE model, only, governed by the same (kinematical
and ground state) energy field. In other words, there is one common kinetical
and dynamical energy behind the several classical PDE physical (Lagrange
formalism based) models describing the current 3 Nature forces phenomena; all those
models can be derived from a common underlying (energy based) Hamiltonian
formalism, where the physical model specific (force based) Lagrange formalism
is only valid with additional regularity requirements to ensure the existence of
the classical PDE solutions. In other words, the (force based) Lagrange formalism
provides only an approximation model of the considered special physical
situation to the underlying quantum „world“.
The key ingredients of the proposed quantum gravity theory to integrate the Einstein field equations is about differential forms equipped with the inner product of the correspondingly defined distributional Hilbert space, with direct relationship to the Hilbert space H(1/2). An immediate consequence of the extended energy Hilbert space concept is the solution of the 3D-NSE and Yang-Mills mass problems. The correspondingly extended Cauchy problems of the NSE and Maxwell equations become long term stable and well-posed, while the extended Maxwell equations also allows standing (stationary) waves, i.e. the Yang-Mills equations (coming along with the physical mass gap problem) are no longer required: - regarding the 3D NSE problem the newly proposed " fluid element" Hilbert
space H(-1/2) with corresponding extended energy („momentum“,
"velocity") space H(1/2) leads to Ricci ODE estimates of order 1/2
enabling a corresponding bounded Sobolevskii (energy inequality) estimate. Regarding
the second unknown term of the NSE, the pressure, we note that
"pressure" corresponds to "energy density" (Nm/volume ~
N/area)
- the variational representation of the Maxwell equations in the proposed quantum
element/energy Hilbert space framwork (H(-1/2),H(1/2) conserves the
two H(1)-based progressive (1-parameter (space or time variable))
electric and magnetic waves concept while also allowing additional standing
(stationary) H(1,ortho)based (2-parameter) wavelets. The vaccuum solution of
the first ones conserves the linkage to the classical wave equations for the
electric and magnetic field (while this transformation still requires
additional, physical not relevant regularity requirements to the underlying
solution), while the second ones provides additional information regarding the
elementary particle dynamics.
With respect to „ The large scale structure of space-time“
and the role of gravity (Hawking S. W., Ellis G. F. R., Cambridge University
Press, 1973) and the positive answer regarding „the global nonlinear stability
of the Minkowski space“ (ChD1) we note that the notions „matter,
space-time, action, ..“ etc. are only defined in the H(1) energy Hilbert
space with its underlying Minkowski space governed by hyperbolic PDEs, while
the orthogonal Hilbert space H(1,ortho) is governed by elliptic PDE, only. The
(nonlinear) stability of the Minkowski space framework requires initial data
sets with finite energy and linear and angular momentum (ChD1). From (CoR) p. 763, we recall the following conjecture for the wave equation, which would show that the four-dimensional physical space-time world of classical physics enjoys an essential distinction: "families of spherical waves for arbitrary time-like lines exist only in case of two and four variables, and then only if the differential equation is equivalent to the wave equation (which includes also the radiation problem)."
We note that the Fourier analysis based applied spectral analysis methods (e.g. cosmoligical distance measurement or the Doppler effect in combination with the Hubble diagram leading to the interpretations of moving apart galaxies from each other galaxies with superluminal velocity in an expanding universe) is only defined in the „granular“ kinematical Hilbert space framework H(1), i.e. the proposed quantum gravity model allows an re-interpretation of the observed cosmological background radiation phenomenon. At the same point in time H. Weyl's requirement concerning a truly
infinitesimal geometry are fulfilled as well, because ...(WeH0): "… a truly infinitesimal geometry (wahrhafte
Nahegeometrie) … should know a transfer principle for length measurements
between infinitely close points only ..."(WeH0) Weyl H., Gravitation und Elektrizität, Sitzungsberichte Akademie der Wissenschaften Berlin, 1918, 465-48. https://arxiv.org/ The proposed model is only about truly bosons w/o mass, modelled as elements of the H(1)-complementary sub-space of the overall energy Hilbert space H(1/2). Therefore, the main gap of Dirac‘s quantum theory of radiation, i.e. the small term representing the coupling energy of the atom and the radiation field, becomes part of the H(1)-complementary (truly bosons) sub-space of the overall energy Hilbert space H(1/2) . It allows to revisit Einstein's
thoughts onETHER AND THE THEORY OF RELATIVITY An Address delivered on May 5th, 1920, in the University of Leyden in the context of the space-time theory and the kinematics of the
special theory of relativity modelled on the Maxwell-Lorentz theory of the
electromagnetic field.Einstein’s field equations are hyperbolic and allow so called „time bomb solutions“ which spreads along bi-characteristic or characteristic hyper surfaces. Actual quantum theories are talking about „inflations“, which blew up the germ of the universe in the very first state. The inflation field due to these concepts are not smooth, but containing fluctuation quanta. The action of those fluctuations create traces into a large area of space. The existence of quantum fluctuations (in a „world“ without a time arrow and without entropy) has been verified by the Casimir and the Lamb shift effects. The standard „big bang“ theory assumes that the creation of the first mass particle (fermion) was the „birthday“ of the universe. This event was caused by an „inflation“ energy field triggered by a „disturbance“, called fluctuations. In the proposed quantum gravity model the „birthday“ of the „granular“, compactly embedded fermion-energy Hilbert (sub-) space H(1) of H(1/2) (coming along with the (kinematical) notions "space", "time", "action", etc.) is interpreted as first disturbance of the purely (pre-universe) boson energy field H(1,ortho) with not existing entropy. The latter one can be interpreted as the (in sync with the Casimir effect) not empty quantum vaccuum; its oscillation is the cosmic background radiation, which contains all features of dynamic energies. With the „birthday“ of the fermions the correspondingly adapted variational representation of the wave equation is then governed by the purely kinematical (fermions) energy Hilbert space H(1), while its underlying initial values are purely (undistorbed) vacuum (CBR, bosons) energy data from H(1,ortho). As a consequence, the wave equation becomes time-asymmetric and the second law of (kinematical) thermodynamics (the entropy phenomenon coming along with the notions „mass“, „time“, „space“ etc.) can be interpreted (and derived from this wave equation) as „action“ principle of the ground state energy to damp and finally eliminate (remedy the deficiency) of any kinematical energy „disturbance“.
The proposed quantum gravity model is based on a Hilbert space framework. Wavelet analysis can be used as a mathematical microscope, looking at the details that are added if one goes from a scale "a" to a scale " a-da", where "da" is infinitesimally small. We mention that an alternative model for an "a" to a scale " a-da" model is the concept of the ordered field of ideal points, an extension to the ordered field of real numbers with same cardinality, but having additionally infinitesimal elements (also called non-Archimedean numbers). The mathematical microscope wavelet tool 'unfolds' a function over the one-dimensional space R into a function over the two-dimensional half-plane of "positions" and "details". This two-dimensional parameter space may also be called the position-scale half-plane. The wavelet duality relationship provides an additional degree of freedom to apply wavelet analysis with appropriately (problem specific) defined wavelets in a (distributional) Hilbert scale framework where the " microscope observations"
of two wavelet (optics)
functions f and g can be compared with each other by the "reproducing"
("duality") formula. Plasma is an inonized gas consisting of approximately equal numbers of positively charged inos and negatively charged electrons. One of the key differentiator to neutral gas is the fact that its electrically charged particles ares trongly influeneced by electric and magnetic fields, while neutral gas in not. There are two nonlinear equations that have been treated extensively in connection with nonlinear plasma waves: The Korteweg-de Vries equation and the nonlinear Schrödinger equation. .... " When
an electron plasma wave goes nonlinear, the dominant new effect is that
the ponderomotive force of the plasma waves causes the background
plasma to move away, causing a local depression in density called
caviton. Plasma waves trapped in this cavity then form an isolated structure called envelope soliton or envelope solitary wave. Considering
the difference in both the physical model and the mathematical form of
the governing equations, it is surprising that solitons and evelopes
solitons have almost the same shape", (ChF) 8.8.Plasma physics modelling is basically about statistical evolution of a large number of particles interacting through „collisions“. The mathematical models are the Boltzmann and Landau equations, where the unknown function f corresponds at each time t to the density of particles at the point x with velocity v, (LiP): „ If the related non-local, quadratic operator
Q(f,f) were zero, the kinetic Boltzmann and Landau equations would simply mean
that the particles do not interact an the density f would be constant along
particle paths“. The operator Q(f,f) was introduced by Maxwell and
Boltzmann for the case, that collisions occur.
“ In case the described particles of the
Boltzmann equation interact with a two-body force (collisions case), this leads
to a Vlasov-like force (or self-consistent force, or mean field...) F”, (LiP1). Its underlying potential function V(x) is governed by the Laplace operator “div(grad)”
based potential equation, given by -div(grad(V))=grad(F). In (NiJ*) corresponding
unusual (Sobolev and Hölder) norm estimates are provided, enjoying
appreciated shift theorems for the Landau damping phenomenon critical Coloumb potential
case; the shift theorems also well fit to the proposed H(-1/2) Hilbert space framework. The provided proofs are all based on standard estimates for the Newtonian potential.In case the Boltzmann collision kernel B of the non-local, quadratic operator Q(f(t,x,*),f(t,x,*)) presents singularities of an arbitrarily high order, it is about so-called grazing collisions, (LiP): when almost all collisions are grazing this leads to Landau collision operator resp. the Landau equation (also called the Fokker-Landau equation). The central tool of a proposed distributional Hilbert space based plasma physics to govern grazing
collisions is the exponential decay distributional norm, (NiJ1), (BrK*) p.24. The
central tool of a proposed distributional Hilbert space based plasma physics to govern non-grazing
collisions are the polynomial decay distributional norms, (NiJ1). In case of grazing
collisions, the kernel function B presents singularities with
bounded orders, e.g. the Coloumb (Newtonian) potential related singularity, (BrK*) p.24. In case of non-grazing
collisions, the kernel function B presents singularities of an
arbitrarily high order. We mention that also the NSE deals with two unknown functions. Regarding
the second unknown term of the NSE, the pressure, we note that
"pressure" corresponds to "energy density" (N/area ~ Nm/volume).
Applying formally the div-operator to the classical NSE the
pressure field must satisfy a Neumann problem with normal derivative boundary
conditions, (BrK*) p. 40, 55, (GaG). As a consequence the prescription of the pressure at the
boundary walls or at the initial time independently of u, could be incompatible
with and, therefore, could retender the NSE problem ill-posed.
For related unusual (Sobolev and Schauder) shift theorems concerning the Stokes flow equation we also refer to (NIJ*).1. The proposed appropriate "Landau damping phenomenon" model The proposed appropriate Landau damping phenomenon model is about a Landau-Poisson-Maxwell PDO system, where "Landau" specifies the (well defined) Landau equation with a linearized Landau collision operator, where "Poisson" specifies the extended (well-posed, including appropriate boundary values compatible with the Maxwell equations, (WeP), covering also other than gravitation attractive elementary particles) potential equation of the Newton gravity model, and where "Maxwell" specifies the extended (well-posed, (CoM), including appropriate boundary values covering also other than electro-magnetic elementary particles) Maxwell equations. A correspondingly well defined H(-1/2)-based weak variational representation of the proposed Landau-Poisson-Maxwell PDO system enables corresponding (combined) standard (Ritz) and complementary (Noble) energy minimization approximation methods for the Poisson and the Maxwell equations, e.g. finite element, boundary element or wavelet approximation methods for pseudo-differential equations. The proposed „energy“ Hilbert space H(1/2) also allows to apply the method of Noble to analyze the (nonlinear) linearized Landau collision operator. In this case the Noble method can be applied to two self- adjoint operator equations defined by the linearized Landau collision operator, leading to a common “Hamiltonian” function , based on the “Gateaux derivative” concept, (ArA) 4.2, (BrK*) p. 25, (VeW) 6.2.4). Mathematically speaking, the proposed H(-1/2)-based (strong and weak) Landau-Poisson-Maxwell PDO systems cover all types of PDE, which are parabolic-elliptic-hyperbolic PDE, while the differentiated (!) standard Maxwell equations result into the (hyperbolic) wave equation, defining the principle of maximal electro-magnetic information exchange by the speed of light and all other related special and general relativity theory aspects. Conceptually speaking the parabolic Landau (evolution) equation connects the elliptic and hyperbolic (space-time) quantum world. The
elliptic vs. hyperbolic “worlds” are very much in line with D.
Bohm’s notions of implicate and explicate order, (BoD). With respect to the elliptic “world” we recall from (BoD) A.2: " Rather,
an entirely diļ¬erent sort of basic connection of elements is possible, from
which our ordinary notions of space and time, along with those of separately
existent material particles, are abstracted as forms derived from the deeper
order. These ordinary notions in fact appear in what is called the explicate or
unfolded order, which is a special and distinguished form contained within the
general totality of all the implicate orders… Explicate order arises primarily as a
certain aspect of sense perception and of experience with the content of such
sense perception. It may be added that, in physics, explicate order generally
reveals itself in the sensibly observable results of functioning of an
instrument. … „What is common to the functioning of instruments generally used
in physical research is that the sensibly perceptible content is ultimately
describable in terms of a Euclidean system of order and measure, i.e., one that
can adequately be understood in terms of ordinary Euclidean geometry. … The general transformations are considered to
be the essential determining features of a geometry in a Euclidean space of
three dimensions; those are displacement operators, rotation operators and the
dilatation operator."The hyperbolic world in the standard statistics (reflexiv) Hilbert space framework L(2) is about statistical thermodynamics and related Shannon (discrete) entropy, based on the countable spectrum of the considered differential operators with range = L(2). The norm of the quantum H(-1/2) elements is governed by the sum of the corresponding "observables" L(2) norm and the exponential decay norm, while both summands are interwoved by a parameter, which can be appropriately choosen to model the influencing & balancing contribution of underlying "ground state" energy effect ("time-independent "action""), (BrK*) p. 24. With regards to the proposed integrated SMEP and gravity model, the "between bodies interacting" force in the Boltzmann equation is decomposed into two "forces" defined by a corresponding
(Hamiltonian formalism based) integrated (kinematical & dynamical) energy concept. This is achived
by considering the Landau integral operator equation in a weak H(-1/2)
Hilbert space framework. The Coloumb force (Poisson equation based) force and the Lorentz (electro-magnetic) force (Maxwell equation
based) are replaced by the concept
of underlying related kinematical
and (complementary, not only electro-magnetic) dynamical energy, modelled as decomposition of the (energy) Hilbert space H(1/2)=H(1)+H(1,ortho). With regards to the Maxwell equations we recall that the components of the electric and magnetic field forces E, H build the 4-dimensional electromagnetic field force tensor F(i,k)=(E,H). The Maxwell stress tensor s(i,k) is built on the field force tensor in combination with the Dirac function. The standard Maxwell operator is not coercive. For the time-harmonic Maxwell equations, (KiA), there is a coercive bilinear form provided, containing tangential derivatives of the normal and tangential components of the field on the boundary, vanishing on the subspace H(1), (CoM) below. In the proposed H(-1/2) framework the Dirac function is replaced by H(-1/2 distributions to model point/surface densities. The Laplace operator of the Poisson equation also defines a coercive bilinear form (see also (WeP) below. Thus, in the proposed new framework standard and complementary variational methods can be applied, based on coercive bilinear forms. With regards to the changes coming along with the above proposed quantum element/quantum energy distributional Hilbert space framework we note : - normal and tangential derivatives, mass density, and „flow through a surface“ are replaced by Plemelj’s Stieltjes‘ integral based concept of the notions „mass“ and „flux“ at each point of a surface (PlJ), requiring less regularity assumptions to the underlying potential function; we mention that the Vlasov-Poisson-Boltzmann system is about the Poisson potential function defining the forces term F in the general Boltzmann equation, (LiP1) - the extended Maxwell equations (making the Yang-Mills equation superfluous & enabling an unique stabil 3D-NSE Cauchy problem solution with appropriately defined distributional initial value function) define a coercive bilinear form in the related variational equation representation; we mention that the Vlasov-Maxwell-Boltzmann system is about the (collision-free) Lorentz potential function defining the forces term F in the general Boltzmann equation, (LiP1) - the role of the Gaussian density function to measure the statistics in the observable space H(0) can be extended by the mathematical microscope wavelet tool 'unfolding' a function over the one-dimensional space R into a function over the two-dimensional half-plane of "positions" and "details" - in general, the usage of a H(-1/2) Hilbert space framework allows a variational calculus with differentials and related pseudo-differential equations, including Gateaux and Frechet differentials, (VaM). 2. The challenges of the current "Landau damping phenomenon" Vlasov model The Landau damping phenomenon is
about “wave damping w/o energy
dissipation by collisions in plasma”, because electrons are
faster or slower than the wave and a Maxwellian distribution has a higher
number of slower than faster electrons as the wave. As a consequence, there are
more particles taking energy from the wave than vice versa, while the wave is
damped ((BiJ)). The Landau damping property is complementary to the properties of electro-magnetic forces, which weaken themselves spontaneously over time w/o increase of entropy or friction. " It involves coupling between
single-particles and collective aspects of plasma behavior. ..this topic
is related to one of the main unsolved questions in physics. ....
Landau damping involves a flow of energy between single particles on the
one hand side, and collective excitations of plasma on the other side", (DeR) p. 94." In
its purest form, Landau damping represents a phase-space behavior
peculiar to collisionless systems. Analogs to Landau damping exist, for
example, in the interactions of stars in a galaxy at the Lindblad
resonances of a spiral dwnsity wave. Such resonances in an inhomogeneous
medium can produce wave absorption (in space rather than in time),
which does not usually happen in fluid systems in the absence of
dissipative forces. An exception in the behavior of corotation
resonances for density waves in a gaseous medium)", (ShF) p. 402. In other words, the
Landau damping phenomenon can be interpreted as the capability of stars to
organize themselves in a stable arrangement. The microscopic kinetic description of plasma fluids leads to a continuity equation of a system of (plasma) “particles” in a phase space (x,v). In case of a Lorentz force the equation reduces to the so-called collisions-less (kinetic) Vlasov equation (ShF) (28.1.2)), where the force F of the baseline Boltzmann equation, acting on the particles, is entirely electromagnetic (ChF) 7.2. Physically speaking, collisions are neglegted in case of sufficiently hot plasma, i.e. in case of sufficiently high plasma energy. The related Vlasov formula for the plasma dielectric for the longitudinal oscillators is only a kind of Hilbert transform representation; the integral is divergent in case of the important physical phenomenon of electrons travelling with exactly the same material speed w/k and the wave speed v, (ShF) p. 392. The correct definition for the Vlasov formula is a threefold integral definition using also principle value integrals, (ShF) p. 395. Vlasov’s mathematical argument against the Landau equation (leading to the Vlasov equation) was, that “ this model of pair
collisions is formally (!) not applicable to Coulomb interaction due to the
divergence of the kinetic terms”. This argument is being overcome by the proposed
distributions framework in part B. It also enables a truly Hilbert
transform integral based longitudinal plasma oscillation analysis and a corresponding plasma extended dielectric formula for longitudinal oscillations, (ShF) chapter 29.In (MoC) a proof is provided for the Landau damping phenomenon based on the Vlasov equation using analytical norm estimates. We note that the only considered force and wave-type in the Vlasov equation is electro-magnetic force/wave. Neither the Vlasov equation itself (a collisions-less equation to model wave damping w/o energy
dissipation by collisions in plasma) nor the application of
analytical norm estimates (a hammer being used for nuclear fission) are
appropriate to model or measure hot plasma
physical phenomena. Alternatively, we propose a weak variational PDE
representation of
the kinematical Landau equation in a distributional Hilbert space
framework coming along with distributional Hilbert norm estimates
(including convergent kinetic terms) accompanied by wavelet analysis
capabilities for a physically relevant proof of the Landau phenomenon. The alternative
approach also avoids the (physically not relevant) Penrose stability criterion
assumption.For "a truly proof of the physical Landau damping phenomenon" we propose the following changes, (BrK6) : - the Vlasov (Lorentz force connected) equation is replaced by the Landau-Poisson-Maxwell equations system with underlying attractive and repulsive elementary particles type related potential functions, where the collision term is represented by the linerized Lndau collision (integral) operator, (LiP)), (resp. in modern terms, as a system of three pseudo-differential operator equations, (PeB)). We note that in (IwC) the fundamental solution for the Fokker–Planck (Laundau) equation is constructed by calculus of pseudo-differential operators, from which an eigenfunction expansion of the Landau operator is obtained - the related weak (PDO) equation representation is given by a system of corresponding weak variational equations in a H(-1/2) Hilbert space framework resp. by a system of H(-1/2) based ((energy) Hamiltonian formalism based) minimization problems. The H(-1/2) framework allows an alternative modelling of cold, warm, hot electronic gases by 3 different wave types, by one 2-parameter wavelet concept. We emphasis that he proposed quantum element space H(-1/2) is also not restricted to electro-magnetic particles, only - the collision operator of the Landau equation can be split into a sum of a linearized collision operator of order zero (which is bounded in the considered H(-1/2) Hilbert space framework with respect to the H(-1/2) norm) and a remaining term with order less than zero, (BrK*) p. 6 - the H(-1/2) norm of the linearized collision operator is bounded by the sum of two (with a balancing parameter "delta" connected) terms, the corresponding H(0) and exponential decay norms of the linearized collision operator, (BrK*) p. 24, 34 - the weak variational presentation enables e.g. the methods of Ritz and Noble in the context of standard and complementary non-linear extremal problems, (ArA), (VaM), (VeW), (BrK*) pp. 25/26 - the analytical norms in (MoC) are replaced by the exponential decay inner product resp. norm, (BrK*) p. 33, additionally defined to the (standard) polynomial decay inner products resp. norms, coming along with the PDO resp. Sobolev space concept; we emphasis that the exponential decay norm is weaker than any polynomial distributional norms, (NiJ1). The exponential decay distributional norm is proposed to govern grazing collisions, (LiP1), whereby the polynomial decay distributional norm is proposed to govern non-grazing collisions, i.e. the kernel function B presents singularities with bounded order, (BrK*) p.24, (NiJ1). Additional References to (BrK*)(BrK*) Braun K., 3D-NSE, YME, GUT solutions, July 31, 2019(CoM) Costabel M., Coercive Bilinear Form for Maxwell's Equations (IwC) Iwasaki C., A Representation of the Fundamental Solution for the Fokker–Planck Equation and Its Application, Fourier Analysis Trends in Mathematics, 211–233, Springer International Publishing, Switzerland, 2014 (NiJ*) Nitsche J. A., Direct Proofs of Some Unusual Shift-Theorems, Anal. Math. Appl., Gauthier-Villars, Montrouge, pp.383-400, 1988, Dedicated to Prof. Dr. Jacques L. Lions on His 60th Birthday (WeP) Werner P., Self-Adjoint Extensions of the Laplace Operator with Respect to Electric and Magnetic Boundary Conditions (WeP1) Werner P., Spectral Properties of the Laplace Operator with respect to Electric and Magnetic Boundary Conditions
| ||||||||||||||||||||||||||||||