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The rectangular Fourier model has instantaneous spatial support

Physical scope

A mathematical support property of the constant-coefficient Fourier heat model on an insulated rectangle; it is not an observation or a claim about finite-speed physical signals.

Statement

Assume Countable Choice as in Constant-conductivity heat evolution on an insulated rectangle. Let $L_{i} > 0$ be lengths in metres and $Ω = \prod_{i = 1}^{3} (0, L_{i})$ . Let $c > 0$ be a constant volumetric heat capacity in $J m^{- 3} K^{- 1}$ , $k > 0$ a constant scalar conductivity in $W m^{- 1} K^{- 1}$ , and $u_{⋆}$ an arbitrary reference energy density in $J m^{- 3}$ , with $u (θ) = u_{⋆} + c θ$ as in Constant-conductivity heat evolution on an insulated rectangle. Choose $x_{c} = (L_{1} / 2, L_{2} / 2, L_{3} / 2)$ and a radius $r$ in metres with $0 < r < \frac{1}{2} \min_{i} L_{i}$ , and let $T_{b}, ϵ > 0$ be temperatures in kelvins. Define on $\overline{Ω}$ $ϕ (x) = {\begin{cases} (1 - ∣ x - x_{c} ∣^{2} / r^{2})^{3}, & ∣ x - x_{c} ∣ < r, \\ 0, & ∣ x - x_{c} ∣ \geq r . \end{cases}$ For $L > 0$ , define the even periodic fold $ρ_{L} : R \to [0, L]$ by $ρ_{L} (s) = s - 2 n L$ for $2 n L \leq s \leq (2 n + 1) L$ and $ρ_{L} (s) = (2 n + 2) L - s$ for $(2 n + 1) L \leq s \leq (2 n + 2) L$ , $n \in Z$ . Put $ϕ_{p e r} (x) = ϕ (ρ_{L_{1}} (x_{1}), ρ_{L_{2}} (x_{2}), ρ_{L_{3}} (x_{3}))$ and take $g = T_{b} + ϵ ϕ_{p e r}$ . In the constant- $c, k$ insulated Fourier model, the heat-kernel solution satisfies $T (x, t) - T_{b} = ϵ \int_{R^{3}} Γ (x - y, (k / c) t) ϕ_{p e r} (y) d y > 0$ for every interior $x \in Ω$ and every $t > 0$ , although the initial perturbation is supported in the compact ball $\overline{B} (x_{c}, r) \subset Ω$ . Thus the adopted diffusion model has no finite propagation speed for this temperature perturbation.

Facts & Assumptions

Given: Countable Choice; lengths $L_{i}$ and radius $r$ in metres; the constants $c, k, u_{⋆}, T_{b}, ϵ$ with the units and hypotheses in the Statement; and the centre $x_{c}$ and profile $ϕ$ defined there.

[A1]

Countable Choice is the explicit hypothesis carried by the heat-evolution supplier The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

Constant-conductivity heat evolution on an insulated rectangle: under Countable Choice and the stated constant positive $c, k$ , a $C^{2}$ initial field $g : R^{3} \to (0, \infty)$ that is even in each coordinate and $2 L_{i}$ -periodic in coordinate $i$ , with every coordinate derivative through order two bounded and uniformly continuous and bounds $0 < T_{\min} \leq g \leq T_{\max} < \infty$ , has the stated insulated Gaussian-convolution solution. The Gaussian is strictly positive and has unit mass.

Example

The cutoff $ϕ$ is dimensionless, so $ϵ$ and $T_{b}$ are in kelvins. The perturbation is a constructed model profile, not measured data. It is zero outside a ball at $t = 0$ and is nonzero throughout the body for every positive time under the adopted Fourier equation.

Verification

1.1givenalgebra

Inside the ball write $h (x) = 1 - ∣ x - x_{c} ∣^{2} / r^{2}$ , so $ϕ = h^{3}$ . On the sphere $h = 0$ , the first derivatives $3 h^{2} \partial_{i} h$ and second derivatives $6 h (\partial_{i} h) (\partial_{j} h) + 3 h^{2} \partial_{i} \partial_{j} h$ tend to zero; outside the ball they are zero. Hence $ϕ$ is $C^{2}$ and nonnegative, and it is identically zero in a neighborhood of every face because $r < \frac{1}{2} \min_{i} L_{i}$ . The reflected periodic extension is therefore $C^{2}$ , bounded with bounded uniformly continuous derivatives through order two (by continuity and periodicity), and positive on the original open ball.

2.1A1F1step 1.1algebra

The initial field $g = T_{b} + ϵ ϕ_{p e r}$ satisfies $0 < T_{b} \leq g \leq T_{b} + ϵ$ and all the derivative and even-periodic hypotheses of [F1]. Unit mass of the Gaussian gives $T (x, t) - T_{b} = ϵ \int Γ (x - y, (k / c) t) ϕ_{p e r} (y) d y$ . For $t > 0$ and any $x \in Ω$ , the Gaussian $Γ (x - y, (k / c) t)$ is strictly positive on the ball. Choose a smaller closed ball on which the continuous $ϕ_{p e r}$ has a positive minimum; the Gaussian also has a positive minimum there. Its integral over that ball is positive, so the convolution in [F1] gives $T (x, t) - T_{b} > 0$ .

3.1step 1.1step 2.1given∎

At $t = 0$ , $ϕ_{p e r}$ restricted to $Ω$ vanishes outside $\overline{B} (x_{c}, r)$ , a proper compact subset of the rectangle, while step 2.1 shows the positive-time perturbation is nonzero at every interior point. This is instantaneous support growth in the specified diffusion model only.

The rectangular Fourier model has instantaneous spatial support

Statement

Facts & Assumptions

Example

Verification

Depends on

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