Distributional Maxwell equations with point and stationary-sheet sources

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, the convention of the surface-measure interface inherited from Surface distributions and interfacial jumps (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $U\cong I\times \mathrm{\Omega }$ be the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the units and constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let ${\mathcal{D}}^{\mathrm{\prime }}(U)$ be the space of distributions on $U$ (Distribution, Test function space d of an open set); recall that the pairing $⟨T,\psi ⟩$ is bilinear with no conjugation and that the distributional partial derivatives are $⟨{\mathrm{\partial }}_{\mu }T,\psi ⟩=-⟨T,{\mathrm{\partial }}_{\mu }\psi ⟩$ (Distributional derivative).

(a) [Fields as distributions] Locally integrable fields and source densities on $U$ are identified with their regular distributions (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions).

(b) [Point sources] Let $I_0\subseteq \mathbb{R}$ be a nonempty open interval, let $z:I_0\to {\mathbb{R}}^3$ be a $C^1$ trajectory whose graph map $\gamma (t)=(z(t),t)$ is proper into the open set $U$, and let $q\in \mathbb{R}$ (coulombs). By Point-support spacetime densities: distributions, continuity equation and endpoint fluxes,

$⟨{\rho }_{\mathrm{p}\mathrm{t}},\psi ⟩:=q{\int }_{I_0}\psi (z(t),t)dt,⟨J_{\mathrm{p}\mathrm{t}},\psi ⟩:=q{\int }_{I_0}{z}^{\mathrm{\prime }}(t)\psi (z(t),t)dt$

define distributions on $U$ with ${\mathrm{\partial }}_t{\rho }_{\mathrm{p}\mathrm{t}}+\mathrm{div}{J}_{\mathrm{p}\mathrm{t}}=0$. If instead $I_0=(a,b)$ has finite endpoints, $z$ extends continuously to $[a,b]$, the absolute improper integral ${\int }_a^b\mathrm{\mid }{z}^{\mathrm{\prime }}(t)\mathrm{\mid }dt$ is finite and both endpoint events lie in $U$, the same formulas define distributions with

${\mathrm{\partial }}_t{\rho }_{\mathrm{p}\mathrm{t}}+\mathrm{div}{J}_{\mathrm{p}\mathrm{t}}=q{\delta }_{\gamma (a)}-q{\delta }_{\gamma (b)}$

(Dirac delta and its derivatives); a finite-history point source therefore contributes an endpoint term unless that term is included explicitly, and the pairings above exclude it only by the stated properness or endpoint hypotheses.

(c) [Stationary-sheet sources] Let $S\subset \mathrm{\Omega }$ be a fixed compact embedded $C^2$ hypersurface with a chosen unit normal $n$ pointing from the side ${\mathrm{\Omega }}_{-}$ to the side ${\mathrm{\Omega }}_{+}$ (Surface distributions and interfacial jumps, Surface integration on compact C1 hypersurfaces), and let $\sigma :S\times I\to \mathbb{R}$ and $K:S\times I\to {\mathbb{R}}^3$ be continuous with $K(x,t)\cdot n(x)=0$ for all $(x,t)$; $\sigma$ carries C m${}^{-2}$ and $K$ carries A m${}^{-1}$ (a tangential surface current density; the sheet is stationary). Then

$⟨{\rho }_S,\psi ⟩:={\int }_I{\int }_S\sigma (x,t)\psi (x,t)dS(x)dt,⟨J_S,\psi ⟩:={\int }_I{\int }_{S}K(x,t)\psi (x,t)dS(x)dt$

define distributions on $U$: for $\psi$ with compact support the inner integrals are continuous in $t$ on a compact parameter interval and the functionals are bounded by $(\sup \mathrm{\mid }\sigma \psi \mathrm{\mid })\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}(S)\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}$ of the time projection of the support, and likewise in the vector case. A normal sheet current (one with $K\cdot n\mathrm{\ne }0$), a line current and a moving sheet are not source types of this definition.

(d) [Unrestricted weak fields and optional interface branches] The general field class for the solution predicate (e) consists of all locally integrable pairs $(E,B)$ on $U$, also when the source has a stationary-sheet part. No classical one-sided trace or off-sheet $C^1$ regularity is required to define those four distributional pairings. Additional trace hypotheses are optional hypotheses of interface and jump results, not admission requirements of the general weak model.

For a pointwise classical jump branch, one may additionally assume that $E,B$ are $C^1$ off $S\times I$ and, around every $(x_0,t_0)\in S\times I$, each side is the restriction of a $C^1$ function on an open spacetime neighbourhood of the corresponding closed side of a smaller stationary graph cylinder. These joint one-sided extensions make the field traces and first spatial and temporal derivatives continuous up to that side, locally uniformly on compact cylinders.

For the broader weak-trace branch, retain spacetime $C^1$ regularity on $(\mathrm{\Omega }\setminus S)\times I$ and assume the original one-sided spatial $C^1$ extensions at each time, with the explicit additional conditions that the piecewise first spatial and temporal derivatives are in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(U)$ and the field traces are measurable and locally integrable on $S\times I$ for $dSdt$. This branch supports sheet-distribution and almost-everywhere jump identities; spatial continuity at each time gives pointwise spatial identities for almost every time, and joint continuity of the jump densities gives the further all-time upgrade. Separate spatial $C^1$ traces at each time alone do not imply that joint regularity or upgrade. Whenever such traces exist, $[F]:=F_{+}-F_{-}$ denotes their jump in the chosen normal convention.

(e) [Weak Maxwell equations with a distributional source pair] Let $E,B$ be a field pair of class (d), let $\rho ={\rho }_v+{\rho }_s$ and $J=J_v+J_s$ be source distributions on $U$ whose volume parts ${\rho }_v,J_v$ are locally integrable volume densities with the charge and current quantity types of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (the weak predicate requires this stated $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ regularity, not the classical continuous-source regularity) and whose singular parts $({\rho }_s,J_s)$ are point sources as in (b) or stationary-sheet sources as in (c), and let ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Say that $E,B$ satisfy the weak Maxwell equations with the source pair $(\rho ,J)$ when, for every $\psi \in C_c^{\mathrm{\infty }}(U)$, the four pairings

$⟨\mathrm{div}E-\rho \mathrm{/}{\epsilon }_0,\psi ⟩,⟨\mathrm{div}B,\psi ⟩,⟨\mathrm{curl}E+{\mathrm{\partial }}_{t}B,\psi ⟩,⟨\mathrm{curl}B-{\mu }_{0}J-c^{-2}{\mathrm{\partial }}_{t}E,\psi ⟩$

all vanish, all derivatives being distributional derivatives of the regular distributions of the fields and the source pairings being those of (b) and (c). This clause defines a solution predicate. When the singular parts vanish, its four identities are the same as the locally integrable weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches. That earlier postulate explicitly excludes point and sheet distributions and does not imply their physical adoption here; adoption for the declared singular-source models is made separately, after this definition. No physical field equation is deduced merely from defining this predicate.

(f) [Scope] No product of two distributions is formed anywhere: the fields occur only through their distributional derivatives paired with a test function, and the source densities $\sigma ,K$ occur only inside the ordinary surface and time integrals of (b) and (c); products such as ${\rho }_{s}E$ or $J_s\times B$ are not defined by this item. The definition asserts no existence theorem for fields with singular sources, no conservation law (continuity for these sources is proved separately where used) and no regularisation. Sources that are not locally integrable and are not point or stationary-sheet sources, moving sheets, line currents, and arbitrary products of distributions are outside the definition.

The distributional treatment of point charges and surface sources follows the standard classical presentations cited in the frontmatter; the pairings are defined with the library's own distribution and surface-measure conventions, and the continuity statements quoted in (b) are the conclusions of Point-support spacetime densities: distributions, continuity equation and endpoint fluxes, consumed here as the exact interface of the point-source pairings.