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Uniform far-zone asymptotics and retarded-time flux limit for compact conserved smooth sources

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), as used by the declared polar/surface and change-of-variables suppliers. Let $c > 0$ , $μ_{0} > 0$ , $a > 0$ and $t_{0} < T$ be constants, let $ρ \in C^{\infty} (R^{3} \times R)$ and $J \in C^{\infty} (R^{3} \times R; R^{3})$ satisfy (H1) $ρ (y, v) = J (y, v) = 0$ whenever $∣ y ∣ > a$ or $v \notin [t_{0}, T]$ , and (H2) $\partial_{v} ρ + {div}_{y} J = 0$ on $R^{3} \times R$ . Here and below $B_{a} : = {y \in R^{3} : ∣ y ∣ < a}$ and ${\overline{B}}_{a} : = {y \in R^{3} : ∣ y ∣ \leq a}$ are the open and closed Euclidean balls of radius $a$ about the origin, and $S^{2} = {n \in R^{3} : ∣ n ∣ = 1}$ is the unit sphere. On the open exterior $U : = {x \in R^{3} : ∣ x ∣ > a}$ define the retarded scalar and vector potentials $φ (x, t) : = \frac{μ_{0} c^{2}}{4 π} \int_{R^{3}} \frac{ρ (y, t - ∣ x - y ∣ / c)}{∣ x - y ∣} d y, A (x, t) : = \frac{μ_{0}}{4 π} \int_{R^{3}} \frac{J (y, t - ∣ x - y ∣ / c)}{∣ x - y ∣} d y,$ and the fields $E : = - \nabla φ - \partial_{t} A$ , $B : = \nabla \times A$ , where $\nabla f = (\partial_{1} f, \partial_{2} f, \partial_{3} f)$ , $div F = \sum_{j = 1}^{3} \partial_{j} F_{j}$ and $\nabla \times F = (\partial_{2} F_{3} - \partial_{3} F_{2}, \partial_{3} F_{1} - \partial_{1} F_{3}, \partial_{1} F_{2} - \partial_{2} F_{1})$ are the spatial operators of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and Divergence and curl of a $C^{1}$ vector field. The fields are defined on $U \times R$ using these ordinary derivatives; the conclusions below use $r : = ∣ x ∣ \geq 2 a$ . Every displayed spatial integrand is evaluated on ${\overline{B}}_{a}$ and extended by zero outside it, including at $y = x$ , so no quotient $0 / 0$ is used. All vector-valued integrals are componentwise scalar Lebesgue integrals.

(a) (Fields as retarded integrals.) On the open set ${∣ x ∣ > 2 a}$ the potentials $φ, A$ are of class $C^{\infty}$ in $(x, t)$ , and there $E (x, t) = \frac{μ_{0}}{4 π} \int_{R^{3}} [\frac{N}{R^{2}} c^{2} ρ (y, t - \frac{R}{c}) + \frac{c N}{R} ρ_{t} (y, t - \frac{R}{c}) - \frac{J_{t} (y, t - \frac{R}{c})}{R}] d y, B (x, t) = - \frac{μ_{0}}{4 π} \int_{R^{3}} N \times [\frac{J (y, t - \frac{R}{c})}{R^{2}} + \frac{J_{t} (y, t - \frac{R}{c})}{c R}] d y,$ where $R : = ∣ x - y ∣$ , $N : = (x - y) / R$ , and $ρ_{t}, J_{t}$ are the partial derivatives in the second variable. Moreover, on ${∣ x ∣ \geq 2 a} \times [t_{0}, T]$ , $φ$ and each component of $A$ coincide with the corresponding solution constructed in the wave Cauchy problem of The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support for zero Cauchy data at $t_{0}$ and scalar source $h = μ_{0} c^{2} ρ$ , respectively $h = μ_{0} J_{k}$ .

(b) (Uniform far-zone expansion.) For $n \in S^{2}$ and $u \in [t_{0}, T]$ put $I (n, u) : = \int_{B_{a}} J (y, u + n \cdot y / c) d y$ and $V (n, u) : = \partial_{u} I (n, u) = \int_{B_{a}} J_{t} (y, u + n \cdot y / c) d y$ . Then $\int_{B_{a}} ρ_{t} (y, u + n \cdot y / c) d y = \frac{1}{c} n \cdot V (n, u)$ , and there are constants $Γ_{E}, Γ_{B}$ , depending only on $a, c, μ_{0}$ and on the suprema of $∣ ρ ∣, ∣ ρ_{t} ∣, ∣ ρ_{t t} ∣, ∣ J ∣, ∣ J_{t} ∣, ∣ J_{t t} ∣$ on ${\overline{B}}_{a} \times [t_{0}, T]$ , such that for every $n \in S^{2}$ , $u \in [t_{0}, T]$ and $r \geq 2 a$ , with $E_{l e a d} (n, u, r) : = \frac{μ_{0}}{4 π r} n \times (n \times V (n, u))$ , $∣ E (r n, u + \frac{r}{c}) - E_{l e a d} (n, u, r) ∣ \leq \frac{Γ_{E}}{r^{2}}, ∣ B (r n, u + \frac{r}{c}) - \frac{n \times E_{l e a d} (n, u, r)}{c} ∣ \leq \frac{Γ_{B}}{r^{2}} .$

(c) (Retarded-time flux limit.) Let $S : = μ_{0}^{- 1} E \times B$ and let $d Ω$ be the surface measure on $S^{2}$ (Surface integration on compact C1 hypersurfaces). Then for every $u \in [t_{0}, T]$ , $\lim_{r \to \infty} \int_{S^{2}} r^{2} n \cdot S (r n, u + \frac{r}{c}) d Ω (n) = \frac{μ_{0}}{16 π^{2} c} \int_{S^{2}} ∣ n \times V (n, u) ∣^{2} d Ω (n) .$ This is a limit at fixed retarded time $u$ , not at fixed observation time $t$ ; no decay or boundary condition at spatial infinity is assumed beyond the compact spatial support in (H1).

Facts & Assumptions

Given: ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )); constants $c > 0$ , $μ_{0} > 0$ , $a > 0$ , $t_{0} < T$ ; sources $ρ \in C^{\infty} (R^{3} \times R)$ , $J \in C^{\infty} (R^{3} \times R; R^{3})$ satisfying (H1) and (H2) of the Statement; the retarded potentials and fields of the Statement, with all vector integrals read componentwise. For $n \in S^{2}$ , $u \in [t_{0}, T]$ , $r \geq 2 a$ we write $x : = r n$ , $t : = u + r / c$ , $R : = ∣ x - y ∣ = ((r - n \cdot y)^{2} + ∣ y ∣^{2} - (n \cdot y)^{2})^{1 / 2}$ , $N : = (x - y) / R$ , $σ : = n \cdot y$ , $v_{y} : = u + σ / c$ and $δ_{y} : = (r - σ - R) / c$ , so that $t - R / c = v_{y} + δ_{y}$ ; $Γ_{ρ}, Γ_{ρ t}, Γ_{ρ t t}, Γ_{J}, Γ_{J t}, Γ_{J t t}$ denote the suprema of $∣ ρ ∣, ∣ ρ_{t} ∣, ∣ ρ_{t t} ∣, ∣ J ∣, ∣ J_{t} ∣, ∣ J_{t t} ∣$ on ${\overline{B}}_{a} \times [t_{0}, T]$ , and $λ (B_{a})$ the Lebesgue measure of the ball.

[F1]

A measurable $h$ is integrable exactly when $\int ∣ h ∣ < + \infty$ , and then the integral is computed from the positive and negative parts or, for complex $h = u + i v$ , as $\int u + i \int v$ ; $L^{1}$ denotes the integrable class and $\int_{E} f = \int f χ_{E}$ is the integral over a measurable set. (Integrable real and complex functions, and their integrals, The class $L^{1} (μ)$ of integrable functions, Integral over a measurable subset)

[F2]

The integral is linear, $∣ \int f ∣ \leq \int ∣ f ∣$ , $0 \leq f \leq g$ measurable implies $\int f \leq \int g$ , $\int c f = c \int f$ for $c \geq 0$ , and integrable functions that agree almost everywhere have the same integral. (The Lebesgue integral is linear on $L^{1} (μ)$ , The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree)

[F3]

Under ${A C}_{ω}$ the ball $B_{a}$ is Lebesgue measurable with $0 < λ (B_{a}) < + \infty$ ; every continuous map $R^{n} \to R^{m}$ is Borel measurable; and sums, scalar multiples and pointwise products of measurable functions are measurable. (Euclidean balls have positive finite Lebesgue measure, Continuous functions on Euclidean spaces are Borel measurable, Arithmetic and lattice operations preserve measurability whenever they are defined)

[F4]

Mean value theorem: if $ϕ$ is continuous on $[p, q]$ and differentiable on $(p, q)$ then $ϕ (q) - ϕ (p) = ϕ^{'} (ξ) (q - p)$ for some $ξ \in (p, q)$ . (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ )

[F5]

Parameter integrals over a fixed compact measurable set of finite measure: if $K \subseteq R^{n}$ is compact, measurable and of finite measure, if $a < b$ , and if the real-valued $f$ and $\partial_{t} f$ are continuous on a neighbourhood of $K \times [a, b]$ , then $t \mapsto \int_{K} f (x, t) d x$ is differentiable at every $t \in (a, b)$ with derivative $\int_{K} \partial_{t} f (x, t) d x$ and has the corresponding one-sided relative derivatives at the endpoints of $[a, b]$ ; the lemma is stated pointwise in its one real parameter and explicitly asserts nothing for moving parameter sets or for singularities crossing $K$ . Applying it once per parameter direction with its parameter renamed to $x_{1}, x_{2}, x_{3}$ or $t$ (all other variables held fixed), and iterating on the resulting integrals, yields the corresponding partial derivatives of a parameter integral whose integrand is smooth in all variables on a neighbourhood of the fixed $K$ . (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface)

[F6]

Chain rule and algebra of derivatives for the composite maps appearing below, such as $y \mapsto (y, u + n \cdot y / c)$ and $(x, t) \mapsto (y, t - ∣ x - y ∣ / c)$ . (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ) The spatial gradient, divergence and curl are the coordinate operators specified in the Statement (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Divergence and curl of a $C^{1}$ vector field).

[F7]

A $C^{k}$ map has continuous word partial derivatives through order $k$ , and $C^{\infty}$ means this for every $k$ ( $C^{k}$ Euclidean maps and diffeomorphisms). Finite sums, products and composites preserve these classes ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition). On $(0, \infty)$ , $(q^{α})^{'} = α q^{α - 1}$ and real powers agree with rational powers (Continuity and derivatives of positive-base real powers, The exponential definition of real powers agrees with the existing rational powers); iterating this rule makes $\sqrt{q} = q^{1 / 2}$ and $1 / q$ smooth there. Closed bounded Euclidean sets are compact, continuous functions on compact sets are uniformly continuous and attain finite extrema (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[F8]

Zero-data wave Cauchy problem of speed $c$ from time $t_{0}$ with source $h \in C^{\infty} (R^{3} \times [t_{0}, T])$ in the extension sense: its solution is $u (x, t) = c^{2} \int_{t_{0}}^{t} (t - v) M_{c (t - v)} h (\cdot, v) (x) d v$ on $R^{3} \times [t_{0}, T]$ , it is $C^{\infty}$ there, and it satisfies $c^{- 2} \partial_{t}^{2} u - Δ u = h$ ; here $M_{r} h (x, v) = \frac{1}{4 π} \int_{S^{2}} h (x + r n, v) d Ω (n)$ . (The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support)

[F9]

Polar-coordinates formula: for every nonnegative Borel $f : R^{3} \to [0, \infty]$ , $\int_{R^{3}} f (y) d y = \int_{0}^{\infty} \int_{S^{2}} f (r ω) r^{2} d σ (ω) d r$ ; here $σ$ is the polar surface measure of the cited formula, which equals the chart surface measure $d Ω$ of [F13] (Agreement with the existing polar sphere measure), so the two symbols are used interchangeably; a signed or complex $f$ is handled by applying the formula to positive and negative parts and subtracting. (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Agreement with the existing polar sphere measure)

[F10]

One-dimensional $L^{1}$ change of variables: for a $C^{1}$ diffeomorphism $T : U \to V$ of open subsets of $R$ and $f \in L^{1} (V)$ , $\int_{V} f = \int_{U} f \circ T ∣ T^{'} ∣$ . (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions)

[F11]

Divergence theorem for a bounded $C^{1}$ Euclidean domain $Ω$ and $F \in C^{1} (\overline{Ω}; R^{n})$ : $\int_{Ω} div F d x = \int_{\partial Ω} F \cdot ν d S$ , with the outward normal. (Divergence on a bounded C1 Euclidean domain)

[F12]

The Euclidean norm satisfies the triangle and Cauchy-Schwarz inequalities (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation). Cross product and inner product on $R^{3}$ : $u \times v$ is bilinear and antisymmetric, $n \cdot (u \times v) = \det (n, u, v)$ , and for $∣ n ∣ = 1$ the identities $n \cdot (n \times w) = 0$ and $n \times (n \times w) = n (n \cdot w) - w$ hold, so $∣ n \times (n \times w) ∣ = ∣ n \times w ∣$ . (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ )

[F13]

Surface integral on the compact embedded $C^{1}$ hypersurface $S^{2}$ : for a nonnegative Borel $f$ the integral $\int_{S^{2}} f d Ω$ is defined by finitely many regular parametrizations and an ambient partition of unity, the empty surface integrates to $0$ , the definition is independent of the chosen presentation, the total surface measure $\int_{S^{2}} 1 d Ω$ is finite, and the integral is monotone in $f$ ; the definition carries the ${A C}_{ω}$ convention already declared. (Surface integration on compact C1 hypersurfaces)

[F14]

$C^{1}$ change of variables for nonnegative Borel functions: under ${A C}_{ω}$ , for a $C^{1}$ diffeomorphism $T : U \to V$ of open subsets of $R^{m}$ and every nonnegative Borel $h : V \to [0, \infty]$ , $\int_{V} h (y) d y = \int_{U} h (T (z)) ∣ \det D T (z) ∣ d z$ with equality in $[0, \infty]$ ; in particular the translation $T (z) = x + z$ of $R^{3}$ is a $C^{1}$ diffeomorphism with $\det D T = 1$ , and a signed integrand is handled through its positive and negative parts by linearity. (Borel change of variables from the compact-support formula and Radon uniqueness, The Lebesgue integral is linear on $L^{1} (μ)$ )

Proof

technique · direct; part (a) differentiates the retarded integrals and identifies them with the wave Cauchy solution, part (b) expands the field integrals uniformly in the far zone and derives the continuity identity, part (c) passes to the surface integral

1.1F5F6F7F3

Smoothness of the potentials: for $x \in U$ and $y \in {\overline{B}}_{a}$ one has $R = ∣ x - y ∣ \geq ∣ x ∣ - a > 0$ , so the maps $(x, t, y) \mapsto ρ (y, t - R / c) / R$ and $(x, t, y) \mapsto J (y, t - R / c) / R$ are $C^{\infty}$ on a neighbourhood of $U \times R \times {\overline{B}}_{a}$ by [F6] and [F7]. Apply [F5] to the fixed compact set $K = {\overline{B}}_{a}$ once for each parameter direction with its one real parameter renamed to $x_{1}$ , $x_{2}$ , $x_{3}$ and $t$ (the remaining variables held fixed) and iterate on the resulting integrals: every differentiated integrand is again continuous on a neighbourhood of $K$ , so each application is covered by the lemma and $φ$ and each component of $A$ have continuous partial derivatives of every order on $U \times R$ , these differentiated parameter integrals are jointly continuous: on each compact parameter neighbourhood the continuous differentiated integrands are uniformly continuous on its product with $K$ , so the integral of their difference is bounded by $λ (K)$ times their uniformly vanishing difference [F2]. Hence the potentials are $C^{\infty}$ on $U \times R$ by [F7], so the derivatives defining the fields are ordinary derivatives even at $∣ x ∣ = 2 a$ .

1.2F12F1

Elementary estimates in the far zone: for $y \in B_{a}$ , $r \geq 2 a$ and $n \in S^{2}$ , expanding $R^{2} = (r - σ)^{2} + ∣ y ∣^{2} - σ^{2}$ with $σ = n \cdot y$ gives $∣ R - (r - σ) ∣ = ∣ ∣ y ∣^{2} - σ^{2} ∣ / (R + r - σ) \leq a^{2} / r$ because $R + r - σ \geq r - σ + ∣ r - σ ∣ \geq r$ ; consequently $∣ r - R ∣ \leq a + a^{2} / r$ , $∣ N - n ∣ = ∣ (r - R) n - y ∣ / R \leq (a + a^{2} / r + a) / (r - a) \leq 5 a / r$ , $∣ 1 / R - 1 / r ∣ \leq (a + a^{2} / r) / (r (r - a)) \leq 3 a / r^{2}$ , and $∣ δ_{y} ∣ \leq a^{2} / (r c)$ , all uniformly in $y, n$ .

2.1step 1.1F5F6F12

Field formulas: by [F6], $\nabla_{x} R = N$ , $\nabla_{x} (t - R / c) = - N / c$ and $\nabla_{x} \times [J (y, t - R / c)] = - N \times J_{t} (y, t - R / c) / c$ , while $\nabla_{x} (1 / R) = - N / R^{2}$ ; hence $\nabla_{x} [ρ (y, t - R / c) / R] = - N [ρ / R^{2} + ρ_{t} / (c R)]$ and $\nabla_{x} \times [J (y, t - R / c) / R] = - N \times [J / R^{2} + J_{t} / (c R)]$ . Each of these first derivatives of the integrands is continuous on a neighbourhood of ${\overline{B}}_{a}$ , so by [F5] the derivatives may be taken under the integral: $- \nabla φ - \partial_{t} A$ and $\nabla \times A$ are given on $U$ , in particular for $∣ x ∣ \geq 2 a$ , by the two displayed zero-extended integrals of the statement, since $E = - \nabla φ - \partial_{t} A$ and $B = \nabla \times A$ .

2.2step 1.1F9F14F1F3

Polar-coordinates identity: fix $x \in U$ and $t \in R$ , and let $h$ be either $ρ$ or one component $J_{k}$ ; then $y \mapsto h (y, t - ∣ x - y ∣ / c) / ∣ x - y ∣$ vanishes off ${\overline{B}}_{a}$ and $∣ x - y ∣ \geq ∣ x ∣ - a > 0$ there, so the quotient on ${\overline{B}}_{a}$ , extended by zero elsewhere, defines a bounded compactly supported Borel function $f$ on all of $R^{3}$ . The translation $T (z) : = x + z$ is a $C^{1}$ diffeomorphism of $R^{3}$ with $\det D T = 1$ , so [F14] applied to the positive and negative parts of $f$ gives $\int_{R^{3}} f (y) d y = \int_{R^{3}} f (x + z) d z$ ; applying [F9] to the positive and negative parts of $z \mapsto f (x + z)$ , which at $z = ρ ω$ equals $h (x + ρ ω, t - ρ / c) / ρ$ and therefore has inner angular integral $\frac{1}{ρ} \int_{S^{2}} h (x + ρ n, t - ρ / c) d Ω (n) = \frac{4 π}{ρ} M_{ρ} h (x, t - ρ / c)$ , gives $\frac{1}{4 π} \int_{R^{3}} \frac{h (y, t - ∣ x - y ∣ / c)}{∣ x - y ∣} d y = \int_{0}^{\infty} r M_{r} h (x, t - r / c) d r$ with $M_{r} h (x, v) = \frac{1}{4 π} \int_{S^{2}} h (x + r n, v) d Ω (n)$ , because the inner angular integral is $4 π M_{r} h$ and the radial factor is $r^{2} / r = r$ .

2.3step 1.2F11F6F2

Continuity identity: fix $n, u$ and let $ρ_{0} > a$ ; the map $F (y) : = J (y, u + n \cdot y / c)$ is $C^{1}$ on ${\overline{B}}_{ρ_{0}}$ , and [F6] gives $div F (y) = {div}_{y} J (y, v_{y}) + \frac{n}{c} \cdot J_{t} (y, v_{y})$ with $v_{y} = u + n \cdot y / c$ . By [F11] on $B_{ρ_{0}}$ , whose boundary flux vanishes because $J = 0$ for $∣ y ∣ > a$ , $\int_{B_{ρ_{0}}} div F d y = 0$ ; substituting the chain-rule expression and using (H2), $0 = \int_{B_{ρ_{0}}} [- ρ_{t} (y, v_{y}) + \frac{1}{c} n \cdot J_{t} (y, v_{y})] d y$ . The integrands vanish for $∣ y ∣ > a$ by (H1), so [F2] lets the integrals be taken over $B_{a}$ ; the second is $\frac{1}{c} n \cdot V (n, u)$ by the definition of $V$ , whence $\int_{B_{a}} ρ_{t} (y, v_{y}) d y = \frac{1}{c} n \cdot V (n, u)$ .

3.1step 2.2F10F8F1

Identification with the Cauchy solution: fix $x \in U$ . For $t > t_{0}$ the radial integrand of step 2.2 vanishes when $r > c (t - t_{0})$ , because then its source time is less than $t_{0}$ . Apply [F10] to the affine diffeomorphism $v \mapsto c (t - v)$ from $(t_{0}, t)$ onto $(0, c (t - t_{0}))$ , with absolute derivative $c$ ; the integrand is bounded on this finite interval, hence integrable. Endpoint values do not affect the Lebesgue integrals, and this gives $\int_{0}^{\infty} r M_{r} h (x, t - r / c) d r = c^{2} \int_{t_{0}}^{t} (t - v) M_{c (t - v)} h (x, v) d v$ . At $t = t_{0}$ both sides are zero by (H1), so the identity holds throughout $[t_{0}, T]$ . Multiplying it by $μ_{0} c^{2}$ for $h = ρ$ , and by $μ_{0}$ for $h = J_{k}$ , identifies $φ$ and $A_{k}$ respectively with the zero-data Cauchy solutions of [F8] with sources $μ_{0} c^{2} ρ$ and $μ_{0} J_{k}$ . This holds on $U \times [t_{0}, T]$ and in particular at $∣ x ∣ = 2 a$ .

3.2step 1.2step 2.3F2F3F4F12

Uniform error estimates: fix $n, u, r$ , let $r e t$ denote the argument $v_{y} + δ_{y}$ and $0$ the argument $v_{y}$ , and write $F_{t o t} : = c^{2} N ρ_{r e t} / R^{2} + c N ρ_{t, r e t} / R - J_{t, r e t} / R$ and $F_{l e a d} : = \frac{1}{r} (c n ρ_{t, 0} - J_{t, 0})$ , so that the field formula of step 2.1 and the definition of $V$ give $E (r n, u + r / c) - E_{l e a d} (n, u, r) = \frac{μ_{0}}{4 π} \int_{B_{a}} (F_{t o t} - F_{l e a d}) d y$ by the linearity of the integral. Every source factor is bounded by its supremum on ${\overline{B}}_{a} \times [t_{0}, T]$ , and (H1) makes $ρ$ and $J$ vanish on the open set $R^{3} \times ((- \infty, t_{0}) \cup (T, \infty))$ , hence makes all their partial derivatives vanish there too, so these suprema also dominate the retarded arguments $v_{y} + δ_{y}$ ; the mean value theorem [F4] gives $∣ ρ_{t, r e t} - ρ_{t, 0} ∣ \leq Γ_{ρ t t} ∣ δ_{y} ∣$ and $∣ J_{t, r e t} - J_{t, 0} ∣ \leq Γ_{J t t} ∣ δ_{y} ∣$ : when the difference is nonzero, apply [F4] to its scalar projection onto the unit direction of that difference, and use Cauchy-Schwarz from [F12], and using $R^{- 1} \leq 2 / r$ , $R^{- 2} \leq 4 / r^{2}$ and $∣ N / R - n / r ∣ \leq ∣ N - n ∣ / R + ∣ 1 / R - 1 / r ∣ \leq 13 a / r^{2}$ yields $∣ F_{t o t} - F_{l e a d} ∣ \leq Γ_{E}^{'} / r^{2}$ , with $Γ_{E}^{'} : = 4 c^{2} Γ_{ρ} + 13 c a Γ_{ρ t} + a^{2} Γ_{ρ t t} + 3 a Γ_{J t} + a^{2} Γ_{J t t} / c$ for all $r \geq 2 a$ ; hence, by the triangle inequality for the integral and the finiteness of $λ (B_{a})$ , $∣ E (r n, u + r / c) - E_{l e a d} (n, u, r) ∣ \leq Γ_{E} / r^{2}$ with $Γ_{E} : = \frac{μ_{0}}{4 π} Γ_{E}^{'} λ (B_{a})$ . The same computation applied to $B (r n, u + r / c) = - \frac{μ_{0}}{4 π} \int_{B_{a}} N \times [J_{r e t} / R^{2} + J_{t, r e t} / (c R)] d y$ and to $n \times E_{l e a d} / c = - \frac{μ_{0}}{4 π} \int_{B_{a}} \frac{1}{c r} n \times J_{t, 0} d y$ gives $∣ B (r n, u + r / c) - n \times E_{l e a d} (n, u, r) / c ∣ \leq Γ_{B} / r^{2}$ , for $Γ_{B} : = \frac{μ_{0} λ (B_{a})}{4 π} (4 Γ_{J} + 13 a Γ_{J t} / c + a^{2} Γ_{J t t} / c^{2})$ .

4.1step 3.2F5F6F7F12

Smoothness of the far-zone moments: the map $(n, u) \mapsto I (n, u) = \int_{B_{a}} J (y, u + n \cdot y / c) d y$ and its partial derivative $\partial_{u} I = \int_{B_{a}} J_{t} (y, u + n \cdot y / c) d y = V (n, u)$ are $C^{\infty}$ in $(n, u) \in R^{3} \times R$ : apply [F5] repeatedly in the directions $n_{1}, n_{2}, n_{3}, u$ with the fixed compact set ${\overline{B}}_{a}$ , each differentiated integrand again being continuous on a neighbourhood of ${\overline{B}}_{a}$ ; the integrals over $B_{a}$ and ${\overline{B}}_{a}$ agree because the smooth sources and all their derivatives vanish on $∣ y ∣ = a$ by continuity from $∣ y ∣ > a$ . In particular $n \mapsto ∣ n \times V (n, u) ∣^{2}$ is continuous for each fixed $u$ .

4.2step 3.2F12F2

Pointwise flux asymptotics: by step 3.2 write $E = E_{l e a d} + ρ_{E}$ and $B = n \times E_{l e a d} / c + ρ_{B}$ at $(r n, u + r / c)$ with $∣ ρ_{E} ∣ \leq Γ_{E} / r^{2}$ , $∣ ρ_{B} ∣ \leq Γ_{B} / r^{2}$ ; then $n \cdot E_{l e a d} = 0$ and $E_{l e a d} \times (n \times E_{l e a d}) = ∣ E_{l e a d} ∣^{2} n$ by [F12], while $∣ E_{l e a d} ∣ = \frac{μ_{0}}{4 π r} ∣ n \times V (n, u) ∣$ because $∣ n \times (n \times V) ∣ = ∣ n \times V ∣$ . Expanding the cross product bilinearly, the error terms are $O (r^{- 3})$ , so $r^{2} n \cdot S (r n, u + r / c) = \frac{μ_{0}}{16 π^{2} c} ∣ n \times V (n, u) ∣^{2} + R (n, u, r)$ with $∣ R (n, u, r) ∣ \leq C / r$ for a constant $C$ depending only on $a, c, μ_{0}$ and the source suprema, uniformly in $n \in S^{2}$ and $u \in [t_{0}, T]$ .

5.1step 4.2step 4.1F13F2∎

Flux limit: by [F13] the surface integral over the compact sphere of a bounded Borel function is finite and monotone, so for every $u \in [t_{0}, T]$ and $r \geq 2 a$ the difference $\int_{S^{2}} r^{2} n \cdot S (r n, u + r / c) d Ω (n) - \frac{μ_{0}}{16 π^{2} c} \int_{S^{2}} ∣ n \times V (n, u) ∣^{2} d Ω (n) = \int_{S^{2}} R (n, u, r) d Ω (n)$ has absolute value at most $\frac{C}{r} \int_{S^{2}} 1 d Ω$ , which tends to $0$ ; the function $n \mapsto ∣ n \times V (n, u) ∣^{2}$ is continuous by step 4.1, so its surface integral is the displayed finite constant. Hence the stated limit at fixed retarded time $u$ follows; no step compares different values of $u$ , so the limit is asserted only at fixed retarded time.

Uniform far-zone asymptotics and retarded-time flux limit for compact conserved smooth sources

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources