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The smooth forced Maxwell Cauchy system on a slab: existence, uniqueness, Gauss constraints and the cone of dependence

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) through The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $c, ε, μ > 0$ with $ε μ c^{2} = 1$ , let $t_{0} < T$ , and let $ρ : R^{3} \times [t_{0}, T] \to R$ and $J : R^{3} \times [t_{0}, T] \to R^{3}$ be of class $C^{\infty}$ in the extension sense (restrictions of $C^{\infty}$ functions on $R^{3} \times I_{e x t}$ for some open interval $I_{e x t} \supset [t_{0}, T]$ , $C^{k}$ Euclidean maps and diffeomorphisms) satisfying the continuity equation

$\partial_{t} ρ + div J = 0 on R^{3} \times (t_{0}, T) .$

Let $E_{0}, B_{0} \in C^{\infty} (R^{3}; R^{3})$ satisfy $div E_{0} = ρ (\cdot, t_{0}) / ε$ and $div B_{0} = 0$ on $R^{3}$ . Then:

(a) the first-order system

$\partial_{t} E = c^{2} curl B - \frac{J}{ε}, \partial_{t} B = - curl E on R^{3} \times (t_{0}, T),$

with $E (\cdot, t_{0}) = E_{0}$ , $B (\cdot, t_{0}) = B_{0}$ , has exactly one solution $(E, B)$ with $E, B$ of class $C^{\infty}$ on $R^{3} \times [t_{0}, T]$ in the extension sense;

(b) this solution obeys both Gauss laws on $R^{3} \times (t_{0}, T)$ : $div E = ρ / ε$ and $div B = 0$ ;

(c) (domain of dependence) if $(\tilde{ρ}, \tilde{J}, {\tilde{E}}_{0}, {\tilde{B}}_{0})$ is second data of the same kind and, for some $x_{*} \in R^{3}$ and $t_{*} \in (t_{0}, T]$ , one has ${\tilde{E}}_{0} = E_{0}$ , ${\tilde{B}}_{0} = B_{0}$ on the closed base ball ${\overline{B}}_{c (t_{*} - t_{0})} (x_{*})$ and $\tilde{ρ} = ρ$ , $\tilde{J} = J$ on the closed backward cone $K (x_{*}, t_{*}) = {(y, v) : ∣ y - x_{*} ∣ \leq c (t_{*} - v), t_{0} \leq v \leq t_{*}}$ (Forward and backward wave cones, domain of dependence and influence), then $\tilde{E} (x_{*}, t_{*}) = E (x_{*}, t_{*})$ and $\tilde{B} (x_{*}, t_{*}) = B (x_{*}, t_{*})$ .

This is a statement about the stipulated smooth system with coefficients and sources given; no physical identification of $(E, B)$ with an electromagnetic field, no finite-domain boundary problem and no decay at spatial infinity is asserted.

Facts & Assumptions

Given: The constants $c, ε, μ > 0$ with $ε μ c^{2} = 1$ , the times $t_{0} < T$ , the extension-sense smooth sources $ρ, J$ with $\partial_{t} ρ + div J = 0$ , and the data $E_{0}, B_{0}$ with the two constraint identities.

[F1]

Put $W = c^{- 2} \partial_{t}^{2} - Δ$ . For every $a, b \in C^{\infty} (R^{3})$ and every extension-sense smooth source $h$ on the slab there is a unique extension-sense $C^{2}$ function $u$ with $W u = h$ on $R^{3} \times (t_{0}, T)$ , $u (\cdot, t_{0}) = a$ , $\partial_{t} u (\cdot, t_{0}) = b$ ; it is $C^{\infty}$ in the extension sense. Moreover, if $\tilde{a}, \tilde{b}, \tilde{h}$ are second data with $\tilde{a} = a$ , $\tilde{b} = b$ on a closed ball ${\overset{ˉ}{B}}_{c (t_{*} - t_{0})} (x_{*})$ and $\tilde{h} = h$ on the closed backward cone $K (x_{*}, t_{*})$ , then the corresponding solutions agree at $(x_{*}, t_{*})$ . The theorem assumes ${A C}_{ω}$ (The three-dimensional wave Cauchy problem: Kirchhoff representation, uniqueness, domain of dependence and compact support).

[F2]

If $ϕ$ is $C^{2}$ then $curl \nabla ϕ = 0$ wherever defined (The curl of the gradient of a $C^{2}$ function vanishes).

[F3]

If $F$ is a $C^{2}$ field then $div (curl F) = 0$ (The divergence of the curl of a $C^{2}$ field vanishes).

[F4]

If $F$ is a $C^{2}$ field on an open subset of $R^{3}$ , then $curl curl F = \nabla div F - Δ F$ (The curl of a curl is the gradient of the divergence minus the Laplacian).

[F5]

Divergence and curl are linear on $C^{1}$ fields, and $div (f F) = ⟨ \nabla f, F ⟩ + f div F$ (Divergence and curl are linear and satisfy the scalar product rules).

[F6]

If $ϕ$ is $C^{2}$ on an open subset of $R^{m}$ then $\partial_{i} \partial_{j} ϕ = \partial_{j} \partial_{i} ϕ$ (Clairaut--Schwarz theorem for continuous second partial derivatives).

[F7]

One-variable derivatives are linear and obey the sum rule, and total derivatives are additive and homogeneous (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 $g (c) \neq 0$ , Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives).

[F8]

$div F = \sum_{i} \partial_{i} F_{i}$ for $C^{1}$ $F$ , $curl F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x})$ for $C^{1}$ $F$ on open subsets of $R^{3}$ , $Δ f = div \nabla f$ for $C^{2}$ scalars, partial derivatives are one-variable derivatives along coordinate lines, and $\nabla f = (\partial_{0} f, \partial_{1} f, \partial_{2} f)$ (Divergence and curl of a $C^{1}$ vector field, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

[F9]

A map is of class $C^{k}$ when each component is, in the extension sense on a closed time slab when it is the restriction of a $C^{k}$ map on an open time neighbourhood ( $C^{k}$ Euclidean maps and diffeomorphisms).

[F10]

The closed backward cone with vertex $(x_{*}, t_{*})$ is $K (x_{*}, t_{*}) = {(y, v) : ∣ y - x_{*} ∣ \leq c (t_{*} - v), t_{0} \leq v \leq t_{*}}$ , with base ball ${\overline{B}}_{c (t_{*} - t_{0})} (x_{*})$ (Forward and backward wave cones, domain of dependence and influence).

[F11]

A continuous function on an interval with zero derivative at every interior point is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

[A1]

The Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenF1F8F9A1

Existence. Define $E$ componentwise by [F1] on the slab with data $E (\cdot, t_{0}) = E_{0}$ , $\partial_{t} E (\cdot, t_{0}) = c^{2} curl B_{0} - J (\cdot, t_{0}) / ε$ and source $- \nabla ρ / ε - μ \partial_{t} J$ , and $B$ componentwise by [F1] with data $B (\cdot, t_{0}) = B_{0}$ , $\partial_{t} B (\cdot, t_{0}) = - curl E_{0}$ and source $μ curl J$ . All data and sources are smooth in the extension sense because $ρ, J, E_{0}, B_{0}$ are, and $curl$ and $\nabla$ are finite combinations of first partial derivatives; hence $E$ and $B$ are of class $C^{\infty}$ on the slab in the extension sense.

2.1step 1.1F1F2F4F5F6F7

The first-order equations. Put $R : = E_{t} - c^{2} curl B + J / ε$ and $Q : = B_{t} + curl E$ . Since $\partial_{t}$ commutes with $W$ on $C^{\infty}$ functions and $curl$ commutes with $W$ on $C^{\infty}$ fields (both are finite combinations of constant-coefficient first and second partial derivatives, and mixed partials commute), the wave equations of step 1.1 give $W (E_{t}) = (W E)_{t} = - \nabla ρ_{t} / ε - μ J_{t t}$ , $W (curl B) = curl (W B) = μ curl curl J$ and $W (J / ε) = c^{- 2} J_{t t} / ε - Δ J / ε = μ J_{t t} - c^{2} μ Δ J$ . Adding, with $ε μ c^{2} = 1$ and so $c^{2} μ = 1 / ε$ , the $J_{t t}$ and $Δ J$ terms cancel and $W R = - \nabla ρ_{t} / ε - c^{2} μ \nabla div J = - \nabla (ρ_{t} + div J) / ε = 0$ by the continuity equation. Likewise $W Q = (W B)_{t} + curl (W E) = μ curl J_{t} + curl (- \nabla ρ / ε - μ J_{t}) = 0$ , using $curl \nabla ρ = 0$ and the interchange of $curl$ with $\partial_{t}$ . At $t_{0}$ : $R (t_{0}) = E_{t} (t_{0}) - c^{2} curl B_{0} + J (\cdot, t_{0}) / ε = 0$ and $Q (t_{0}) = B_{t} (t_{0}) + curl E_{0} = 0$ by the prescribed data. For the time derivatives at $t_{0}$ : $E_{t t} (t_{0}) = c^{2} Δ E_{0} + c^{2} (W E) (t_{0}) = c^{2} Δ E_{0} - c^{2} \nabla ρ (\cdot, t_{0}) / ε - c^{2} μ J_{t} (\cdot, t_{0})$ and $c^{2} curl curl E_{0} = c^{2} \nabla div E_{0} - c^{2} Δ E_{0} = c^{2} \nabla (ρ (\cdot, t_{0}) / ε) - c^{2} Δ E_{0}$ , so $R_{t} (t_{0}) = E_{t t} (t_{0}) + c^{2} curl curl E_{0} + J_{t} (\cdot, t_{0}) / ε = (- c^{2} μ + 1 / ε) J_{t} (\cdot, t_{0}) = 0$ ; similarly $B_{t t} (t_{0}) = c^{2} Δ B_{0} + c^{2} (W B) (t_{0}) = c^{2} Δ B_{0} + c^{2} μ curl J (\cdot, t_{0})$ and $Q_{t} (t_{0}) = B_{t t} (t_{0}) + curl E_{t} (t_{0}) = c^{2} Δ B_{0} + c^{2} μ curl J (\cdot, t_{0}) + c^{2} curl curl B_{0} - curl J (\cdot, t_{0}) / ε = c^{2} \nabla div B_{0} = 0$ . Thus $R$ and $Q$ are extension-sense smooth fields with $W R = W Q = 0$ on $R^{3} \times (t_{0}, T)$ and zero initial values and time derivatives; the uniqueness clause of [F1], applied componentwise, gives $R = Q = 0$ , that is, $E_{t} = c^{2} curl B - J / ε$ and $B_{t} = - curl E$ on the slab.

3.1step 1.1step 2.1F1F3F5F7

The Gauss laws. Put $D : = div E - ρ / ε$ . Since $W$ commutes with $div$ on extension-sense $C^{3}$ fields, $W D = div (W E) - (c^{- 2} ρ_{t t} - Δ ρ) / ε = - Δ ρ / ε - μ div J_{t} - (c^{- 2} ρ_{t t} - Δ ρ) / ε = - μ (div J_{t} + ρ_{t t}) = 0$ , the last equality being the time derivative of the continuity equation. Moreover $D (t_{0}) = div E_{0} - ρ (\cdot, t_{0}) / ε = 0$ and, using the first-order equation from step 2.1 and $div curl = 0$ , $D_{t} (t_{0}) = div (c^{2} curl B_{0} - J (\cdot, t_{0}) / ε) - ρ_{t} (\cdot, t_{0}) / ε = - (div J + ρ_{t}) (\cdot, t_{0}) / ε = 0$ . By the uniqueness clause of [F1] componentwise, $D = 0$ . Likewise $W (div B) = div (W B) = μ div curl J = 0$ , $div B (t_{0}) = div B_{0} = 0$ and $\partial_{t} div B (t_{0}) = - div curl E_{0} = 0$ , so $div B = 0$ on the slab.

4.1step 1.1step 3.1F1F2F3F4F5F6F11

Uniqueness and the forced wave problems. Let $(\tilde{E}, \tilde{B})$ be any second $C^{\infty}$ solution of the first-order system on the slab in the extension sense with the same data. Taking divergence directly in its first-order equations gives $\partial_{t} (div \tilde{E} - ρ / ε) = - (div J + ρ_{t}) / ε = 0$ and $\partial_{t} div \tilde{B} = - div curl \tilde{E} = 0$ by [F3] and the continuity equation. For each fixed spatial point these defects have zero derivative, hence are constant in time by [F11], and their initial values vanish; thus $div \tilde{E} = ρ / ε$ and $div \tilde{B} = 0$ . Differentiating ${\tilde{E}}_{t} = c^{2} curl \tilde{B} - J / ε$ in time and using $curl curl = \nabla div - Δ$ gives $c^{- 2} {\tilde{E}}_{t t} = - curl curl \tilde{E} - μ J_{t}$ , hence $W \tilde{E} = - \nabla div \tilde{E} - μ J_{t} = - \nabla ρ / ε - μ J_{t}$ , while $W \tilde{B} = μ curl J$ . Its initial data and initial time derivatives coincide with those used in step 1.1 (the derivative values are determined by the system and the data: ${\tilde{E}}_{t} (\cdot, t_{0}) = c^{2} curl B_{0} - J (\cdot, t_{0}) / ε$ and ${\tilde{B}}_{t} (\cdot, t_{0}) = - curl E_{0}$ ). By the uniqueness clause of [F1] applied componentwise, $\tilde{E} = E$ and $\tilde{B} = B$ .

5.1step 1.1step 4.1F1F10∎

Domain of dependence. Let $(\tilde{ρ}, \tilde{J}, {\tilde{E}}_{0}, {\tilde{B}}_{0})$ be second data as in the Statement, with corresponding solution $(\tilde{E}, \tilde{B})$ furnished by step 1.1, and fix $x_{*} \in R^{3}$ , $t_{*} \in (t_{0}, T]$ . By step 4.1's reduction, $E$ solves the wave problem with source $- \nabla ρ / ε - μ \partial_{t} J$ and initial data $E_{0}$ , $c^{2} curl B_{0} - J (\cdot, t_{0}) / ε$ , and $\tilde{E}$ solves the corresponding wave problem with hatted data. The data agree on the closed base ball ${\overline{B}}_{c (t_{*} - t_{0})} (x_{*})$ : the displacement data agree by hypothesis, and the time-derivative data $c^{2} curl B_{0} - J (\cdot, t_{0}) / ε$ agree because $B_{0} = {\tilde{B}}_{0}$ and $J (\cdot, t_{0}) = \tilde{J} (\cdot, t_{0})$ there (the latter is the base of the cone in [F10]). The sources agree on the closed backward cone: $ρ, J$ and their first partial derivatives agree on the cone, since the identity holds on the open cone (for $ρ, J$ by hypothesis, for the derivatives by the same local differential expressions) and all these functions are continuous on the closed cone, on which they agree by continuity. Applying the cone clause of [F1] to each component of the wave problems for $E$ and $\tilde{E}$ gives $\tilde{E} (x_{*}, t_{*}) = E (x_{*}, t_{*})$ ; the same argument, with source $μ curl J$ and initial data $B_{0}$ , $- curl E_{0}$ , gives $\tilde{B} (x_{*}, t_{*}) = B (x_{*}, t_{*})$ .

Remarks

Choice accounting. The countable-choice assumption enters only through the existence, uniqueness and cone clauses of the wave theorem [F1]; all differential-algebraic steps are choice-free. The statement records this by assuming ${A C}_{ω}$ globally.
Role of the constraint hypotheses. $div E_{0} = ρ (\cdot, t_{0}) / ε$ and $div B_{0} = 0$ are used in step 2.1 to propagate the Gauss laws; without them the first-order system would still have a unique solution but the first-order divergence identities would keep the corresponding initial Gauss defects constant in time at each spatial point. The continuity equation is used in steps 2.1, 3.1 and 4.1.
No physical claim. The constants $c, ε, μ$ are parameters of the mathematical system; no identification with an electromagnetic field or with SI quantities is made here.

The smooth forced Maxwell Cauchy system on a slab: existence, uniqueness, Gauss constraints and the cone of dependence

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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