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Supported static heat model — Examples

1 · Prerequisites

2 · Summary

The cosine example gives a closed-form decaying mode in the supported parabolic heat model. Its redshifted temperature tends to a constant, while its proper-temperature limit varies with a nonconstant lapse. The formula and insulated-face conditions are verified directly; this constructed solution is not an experimental observation.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Redshifted cosine medium

Physical scope

One explicit compatible cosine profile in the fixed-lapse, externally supported parabolic heat model; the proper equilibrium is spatially nonuniform when the selected lapse is nonconstant.

Example

Use the box, positive lapse $N$ , constants, material and support of Conditional redshift heat relaxation. Assume $N$ is nonconstant on $Ω$ . Choose $\overline{ϑ} > 0$ and $δ \in R$ with $∣ δ ∣ < \overline{ϑ}$ . Set

$ϑ_{0} (x) = \overline{ϑ} + δ \cos ⁣ (\frac{π x^{1}}{L_{1}}), x \in \overline{Ω} .$

The unique redshifted-temperature solution and proper-temperature field are

$ϑ (t_{K}, x) = \overline{ϑ} + δ \exp ⁣ (- \frac{A_{0} π^{2} t_{K}}{C_{V} L_{1}^{2}}) \cos ⁣ (\frac{π x^{1}}{L_{1}}), T (t_{K}, x) = \frac{ϑ (t_{K}, x)}{N (x)} .$

The redshifted temperature converges to $\overline{ϑ}$ , while the proper temperature converges to the nonconstant profile $\overline{ϑ} / N (x)$ . This is a calculated solution of the postulated model, not an observation about an actual material.

Facts & Assumptions

Given: $L_{i}, A_{0}, C_{V} > 0$ , a smooth nonconstant lapse bounded above and strictly away from zero on $\overline{Ω}$ , and $\overline{ϑ} > 0$ with $∣ δ ∣ < \overline{ϑ}$ , all under the physical-model conditions of Conditional redshift heat relaxation.

[F1]

Conditional redshift heat relaxation: the fixed model solves $C_{V} \partial_{t_{K}} ϑ = A_{0} Δ_{x} ϑ$ with zero normal derivative on each face, and its solution is unique for the stated positive even-reflected periodic initial data. It also gives $T = ϑ / N$ and the limit $T_{e q} = \overline{ϑ} / N$ .

Verification

1.1givenalgebra

Extend the displayed initial profile to all $R^{3}$ by the same cosine formula. It is $C^{\infty}$ , even and $2 L_{i}$ -periodic in every coordinate. Its values lie between $\overline{ϑ} - ∣ δ ∣$ and $\overline{ϑ} + ∣ δ ∣$ , so the strict inequality on $δ$ gives positive lower and finite upper bounds. Its average on $Ω$ is $\overline{ϑ}$ , because the integral of $\cos (π x^{1} / L_{1})$ from $0$ to $L_{1}$ is $0$ .

2.1givenF1step 1.1algebra

Put $r = A_{0} π^{2} / (C_{V} L_{1}^{2}) > 0$ . Differentiating the proposed solution gives $\partial_{t_{K}} ϑ = - r δ e^{- r t_{K}} \cos (π x^{1} / L_{1})$ and $Δ_{x} ϑ = - (π^{2} / L_{1}^{2}) δ e^{- r t_{K}} \cos (π x^{1} / L_{1})$ , hence $C_{V} \partial_{t_{K}} ϑ = A_{0} Δ_{x} ϑ$ . Its initial trace is $ϑ_{0}$ . Also $\partial_{x^{1}} ϑ = - (π δ / L_{1}) e^{- r t_{K}} \sin (π x^{1} / L_{1})$ vanishes at $x^{1} = 0, L_{1}$ , and $\partial_{x^{2}} ϑ = \partial_{x^{3}} ϑ = 0$ , so every face has zero normal derivative. Uniqueness in [F1] identifies this function with the model solution.

3.1F1step 2.1algebra∎

The exponential factor tends to zero, so $ϑ (t_{K}, x) \to \overline{ϑ}$ . Because $N$ is time independent and positive, $T = ϑ / N \to \overline{ϑ} / N$ pointwise. The latter is nonconstant by the stipulated nonconstancy of $N$ . Direct integration using $\int_{0}^{L_{1}} \cos^{2} (π x^{1} / L_{1}) d x^{1} = L_{1} / 2$ gives the exact norm ${∥ ϑ (t_{K}, \cdot) - \overline{ϑ} ∥}_{L^{2} (Ω)} = ∣ δ ∣ \sqrt{∣ Ω ∣ / 2} e^{- r t_{K}}$ . Thus this example realizes a decaying redshifted mode with a spatially varying proper-temperature equilibrium in the selected model.