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Exponential relaxation of the constant-conductivity Neumann heat model

Physical scope

Conditional L2 relaxation for the compatible positive-temperature solution of the constant-capacity, constant-conductivity model on a three-dimensional insulated rectangle.

Statement

Assume Countable Choice and the hypotheses and solution $T$ of Constant-conductivity heat evolution on an insulated rectangle, let $V = ∣ Ω ∣ = L_{1} L_{2} L_{3}$ , define the spatial mean $\overline{T} (t) = V^{- 1} \int_{Ω} T (x, t) d x$ , and put $E (t) = \int_{Ω} ∣ T (x, t) - \overline{T} (t) ∣^{2} d x$ . With $a = k / c$ and $C_{Ω} = \frac{3}{2} \max_{i} L_{i}^{2}$ , the mean is constant and $E (t) \leq e^{- 2 a t / C_{Ω}} E (0) (t \geq 0) .$ Thus this model solution converges exponentially to its initial spatial mean in $L^{2} (Ω)$ ; no pointwise-uniform convergence rate is asserted.

Facts & Assumptions

Given: Countable Choice, the smooth positive solution and homogeneous Neumann boundary data from Constant-conductivity heat evolution on an insulated rectangle, the rectangular mean-zero inequality on $Ω$ , and $a = k / c > 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every sequence of nonempty sets has a choice function.

[F1]

Constant-conductivity heat evolution on an insulated rectangle: $T_{t} = a Δ T$ on $Ω$ , $\partial_{n} T = 0$ on every face, and $T$ is $C^{1}$ in time and $C^{2}$ in space up to the boundary on finite intervals.

[F2]

Mean-zero Poincare inequality on a rectangle: if $w$ has zero spatial mean, then $\int_{Ω} w^{2} \leq C_{Ω} \int_{Ω} ∣ \nabla w ∣^{2}$ with $\nabla w = (\partial_{x_{1}} w, \partial_{x_{2}} w, \partial_{x_{3}} w)$ and $C_{Ω} = \frac{3}{2} \max_{i} L_{i}^{2}$ .

[F3]

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative: a continuous function on a closed interval with an integrable interior derivative has endpoint increment equal to the derivative integral.

[F4]

Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections: iterated Riemann integrals of continuous functions on product rectangles equal the multidimensional integral.

Proof

1.1A1F1F3F4given

On each finite time interval the time derivative is continuous on the compact box, so its difference quotients converge uniformly under the spatial integral. By [F1] and [F4], coordinatewise application of [F3] gives ${\overline{T}}^{'} (t) = a V^{- 1} \int_{Ω} Δ T d x = 0$ : the integral of each second coordinate derivative is the difference of the corresponding first derivative on its two faces, and both normal derivatives vanish. Hence $\overline{T} (t) = \overline{T} (0)$ .

2.1F1F3F4step 1.1algebra

Set $w (x, t) = T (x, t) - \overline{T} (0)$ . Then [F4] and step 1.1 give $\int_{Ω} w = 0$ , while $w_{t} = a Δ w$ and $\partial_{n} w = 0$ . Differentiating $E (t) = \int_{Ω} w^{2}$ and using [F4], the product rule, and [F3] in each coordinate gives $E^{'} (t) = 2 a \int_{Ω} w Δ w = - 2 a \int_{Ω} ∣ \nabla w ∣^{2}$ ; each endpoint term is $w \partial_{n} w = 0$ .

3.1F2step 2.1

Apply [F2] to $w (\cdot, t)$ in step 2.1. It gives $E (t) \leq C_{Ω} \int_{Ω} ∣ \nabla w ∣^{2}$ , so $E^{'} (t) \leq - (2 a / C_{Ω}) E (t)$ .

4.1F3step 3.1algebra∎

The product rule gives $(e^{2 a t / C_{Ω}} E (t))^{'} = e^{2 a t / C_{Ω}} (E^{'} (t) + (2 a / C_{Ω}) E (t)) \leq 0$ by step 3.1. Integrating this derivative with [F3] from $0$ to $t$ yields $e^{2 a t / C_{Ω}} E (t) \leq E (0)$ , which is the asserted estimate.

Exponential relaxation of the constant-conductivity Neumann heat model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources