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Fixed-coefficient Cattaneo entropy and propagation

Statement

Let $C, κ, τ, T_{0}, L > 0$ . Let $Ω$ be either $R^{3}$ or the flat torus $T_{L}^{3} : = (R / L Z)^{3}$ , and let $I \subseteq R$ be an open interval containing $0$ . Consider real smooth fields $θ : I \times Ω \to R$ and $q : I \times Ω \to R^{3}$ satisfying

$C θ_{t} + div q = 0, τ q_{t} + q = - κ \nabla θ .$

The derivatives are ordinary derivatives of the Euclidean spatial coordinates; on the torus they are defined using periodic lifts. Write $U = (θ, q)$ and let $e_{j}$ be the $j$ th standard basis vector of $R^{3}$ . Define

$A_{j} : = (\begin{array}{cc} 0 & e_{j}^{T} / C \\ (κ / τ) e_{j} & 0_{3 \times 3} \end{array}), B : = diag (0, (1 / τ) I_{3}), M : = diag (C, (τ / κ) I_{3}) .$

Then the displayed system is $U_{t} + \sum_{j = 1}^{3} A_{j} \partial_{j} U + B U = 0$ . The matrix $M$ is a positive symmetrizer, meaning $M = M^{T}$ is positive definite and $M A_{j}$ is symmetric for each $j$ . For a unit direction $ν \in R^{3}$ , define the principal matrix $A (ν) : = \sum_{j = 1}^{3} ν_{j} A_{j}$ ; its eigenvalues are $0$ (multiplicity two) and $\pm v_{h}$ , where $v_{h} : = \sqrt{κ / (C τ)} > 0$ . For a fixed comparison speed $c_{*} > 0$ , $v_{h} \leq c_{*}$ if and only if $κ \leq C τ c_{*}^{2}$ .

For every such solution and every $s_{0} \in R$ , define the specified quadratic effective entropy density and flux by

$s_{e f f} : = s_{0} + \frac{C θ}{T_{0}} - \frac{C θ^{2}}{2 T_{0}^{2}} - \frac{τ ∣ q ∣^{2}}{2 κ T_{0}^{2}}, j_{s} : = \frac{q}{T_{0}} - \frac{θ q}{T_{0}^{2}} .$

Every smooth solution satisfies the exact identities

$\partial_{t} s_{e f f} + div j_{s} = \frac{∣ q ∣^{2}}{κ T_{0}^{2}} \geq 0, \partial_{t} e_{L} + div (θ q) = - \frac{∣ q ∣^{2}}{κ}, e_{L} : = \frac{1}{2} (C θ^{2} + \frac{τ}{κ} ∣ q ∣^{2}) .$

For every unit direction $ν$ , its quadratic flux satisfies $∣ θ q \cdot ν ∣ \leq v_{h} e_{L}$ . The density $e_{L}$ is a positive quadratic Lyapunov density, not a physical energy density. The entropy identity asserts nonnegative production for this specified quadratic effective current; it does not assert positivity of $s_{e f f}$ for arbitrary data or a nonlinear material entropy law. These equations define the mathematical subsystem considered here; the sources give model context and do not establish a material-specific realization.

Use the exact global-solution, Fourier-coefficient, and weighted-norm conventions in [F1]. For initial data $U_{0} = (θ_{0}, q_{0}) \in S (R^{3}; R^{4})$ or $C^{\infty} (T_{L}^{3}; R^{4})$ , the unique global smooth evolution, its weighted Sobolev contraction, and finite propagation at speed $v_{h}$ follow conditionally from [F1] after checking the matrices below. On $T_{L}^{3}$ , for every integer $r > 3 / 2$ , set

$K_{r, L, C} : = C^{- 1 / 2} {(\sum_{k \in Z^{3}} (1 + ∣ k / L ∣^{2})^{- r})}^{1 / 2} < \infty,$

where $C > 0$ is the heat-capacity coefficient and $∥ \cdot ∥_{H_{M}^{r} (T_{L}^{3})}$ is the norm in [F1]. The periodic global solution supplied conditionally by [F1] obeys $T_{0} + θ (t, x) > 0$ for all $t \geq 0$ whenever $K_{r, L, C} ∥ U_{0} ∥_{H_{M}^{r} (T_{L}^{3})} < T_{0}$ . The constant uses only the temperature component $C ∣ {\hat{θ}}_{k} ∣^{2}$ of the supplied quadratic norm, so it does not compare heat-flux and temperature coordinates by a least eigenvalue. Although [F1] states a generic periodic supremum bound with unspecified constant $C_{s, L, M}$ , the explicit $K_{r, L, C}$ bound here is derived directly from Fourier reconstruction and the stated norm contraction.

If the symbols receive their conventional SI thermal interpretation, take $θ, T_{0}$ in K, $C$ in J m $^{- 3}$ K $^{- 1}$ , $κ$ in W m $^{- 1}$ K $^{- 1}$ , $τ$ in s, $L$ in m, $q$ in W m $^{- 2}$ , and the comparison speed $c_{*}$ in m s $^{- 1}$ . Use the spatial coordinates in the inherited Fourier norms as their numerical values in metres, so the unit-frequency term in $1 + ∣ k / L ∣^{2}$ uses the reference length $1$ m. Then $s_{0}$ and $s_{e f f}$ have entropy-density units J m $^{- 3}$ K $^{- 1}$ , $j_{s}$ has entropy-flux units J m $^{- 2}$ s $^{- 1}$ K $^{- 1}$ , and $e_{L}$ has units J K m $^{- 3}$ rather than energy-density units. The norm $∥ U ∥_{H_{M}^{r}}$ has units $\sqrt{J K m^{- 3}}$ and $K_{r, L, C}$ has units $\sqrt{m^{3} K / J}$ , so their product has temperature units. These unit labels do not assert a physical realization of the linear subsystem.

Facts & Assumptions

Given: Positive constants $C, κ, τ, T_{0}, L$ , an open time interval $I$ containing $0$ , a smooth solution of the displayed system on $I$ , and the stated spatial setting. For the conditional global-evolution clause, take $U_{0}$ in the stated Schwartz or smooth-periodic data class.

[F1]

Conditional evolution supplier Complete symmetrized constant-coefficient evolution: assume ${A C}_{ω}$ , an integer $d \geq 1$ , $U : R \times R^{3} \to R^{d}$ , and constant real matrices $A_{1}, A_{2}, A_{3}, B, M \in R^{d \times d}$ with $M = M^{T}$ positive definite, each $M A_{j}$ symmetric, and $S : = (M B + B^{T} M) / 2$ positive semidefinite. For $U_{t} + \sum_{j} A_{j} \partial_{j} U + B U = 0$ , the supplier gives a unique global smooth solution for Schwartz data on $R^{3}$ and smooth periodic data on $T_{L}^{3}$ . Its negative-sign, $2 π$ -normalized Fourier transform is $\hat{U} (ξ) : = \int_{R^{3}} e^{- 2 π i x \cdot ξ} U (x) d x$ , and $∥ U ∥_{H_{M}^{s}}^{2} : = \int_{R^{3}} (1 + ∣ ξ ∣^{2})^{s} \hat{U} (ξ)^{*} M \hat{U} (ξ) d ξ$ contracts for every $s \geq 0$ and $t \geq 0$ . On the torus, with ${\hat{U}}_{k} : = L^{- 3} \int_{[0, L]^{3}} U (x) e^{- 2 π i k \cdot x / L} d x$ , $∥ U ∥_{H_{M}^{s} (T_{L}^{3})}^{2} : = \sum_{k \in Z^{3}} (1 + ∣ k / L ∣^{2})^{s} {\hat{U}}_{k}^{*} M {\hat{U}}_{k}$ also contracts. Its finite propagation speed is $v_{*} : = \sup_{∣ ν ∣ = 1} \max {∣ λ ∣ : λ \in spec (M^{1 / 2} A (ν) M^{- 1 / 2})}$ : if $v_{*} > 0$ , $R > v_{*} t$ , and $U_{0}$ vanishes on $B_{R} (x_{0})$ , then $U (t)$ vanishes on $B_{R - v_{*} t} (x_{0})$ for $0 \leq t < R / v_{*}$ , with the same result on $T_{L}^{3}$ using geodesic distance; if $v_{*} = 0$ , every $A_{j} = 0$ . The current carrier and proof are draft; this exact interface is used conditionally.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) defines ${A C}_{ω}$ as a choice function for each countable family of nonempty sets. This is the only choice strength used through [F1].

[F2]

Schwartz space and its seminorms: $S (R^{3})$ consists of smooth complex-valued functions for which every seminorm $\sup_{x \in R^{3}} ∣ x^{α} \partial^{β} f (x) ∣$ is finite; real components are viewed as complex-valued functions with zero imaginary part, and this condition is imposed componentwise on $R^{4}$ -valued data.

Proof

1.1givenalgebra

Differentiate the first system equation in time and take the divergence of the second. Substituting $div q = - C θ_{t}$ gives $C τ θ_{t t} + C θ_{t} - κ Δ θ = 0$ , or $τ θ_{t t} + θ_{t} - (κ / C) Δ θ = 0$ .

1.2givenalgebra

Expanding the derivative of $s_{e f f}$ and the divergence of $j_{s}$ gives $\partial_{t} s_{e f f} + div j_{s} = \frac{C θ_{t} + div q}{T_{0}} - \frac{θ (C θ_{t} + div q) + q \cdot (τ q_{t} + κ \nabla θ) / κ}{T_{0}^{2}}$ . The first equation makes both occurrences of $C θ_{t} + div q$ zero, and the second gives $τ q_{t} + κ \nabla θ = - q$ . Thus the expression equals $∣ q ∣^{2} / (κ T_{0}^{2})$ .

1.3givenalgebra

Differentiating $e_{L}$ and adding the flux divergence gives $\partial_{t} e_{L} + div (θ q) = θ (C θ_{t} + div q) + q \cdot (\frac{τ}{κ} q_{t} + \nabla θ) = - \frac{∣ q ∣^{2}}{κ}$ , where the first term vanishes by the energy equation and the second equality uses the Cattaneo equation. Since $C, τ, κ > 0$ , $e_{L}$ is a positive definite quadratic form in $(θ, q)$ .

1.4givenalgebra

Write $U = (θ, q)$ , $e_{j}$ for the $j$ th standard basis vector of $R^{3}$ , $A_{j} = (\begin{array}{cc} 0 & e_{j}^{T} / C \\ (κ / τ) e_{j} & 0_{3 \times 3} \end{array})$ , and $B = diag (0, (1 / τ) I_{3})$ . With $M = diag (C, (τ / κ) I_{3})$ , $M A_{j} = (\begin{array}{cc} 0 & e_{j}^{T} \\ e_{j} & 0_{3 \times 3} \end{array})$ is symmetric, $M$ is positive definite, and $S = (M B + B^{T} M) / 2 = diag (0, (1 / κ) I_{3})$ is positive semidefinite. For each unit $ν$ , the longitudinal block of $A (ν)$ has eigenvalues solving $λ^{2} = κ / (C τ)$ and the two transverse eigenvalues are zero. Since $M$ is positive definite, $M^{1 / 2}$ is invertible and the supplier's matrix $M^{1 / 2} A (ν) M^{- 1 / 2}$ is similar to $A (ν)$ , so it has the same eigenvalues and maximum absolute eigenvalue $v_{h} = \sqrt{κ / (C τ)}$ . Finally, $∣ θ q \cdot ν ∣ \leq ∣ θ ∣ ∣ q ∣ \leq \frac{1}{2} \sqrt{κ / (C τ)} (C θ^{2} + (τ / κ) ∣ q ∣^{2}) = v_{h} e_{L}$ , since $2 \sqrt{C τ / κ} ∣ θ ∣ ∣ q ∣ \leq C θ^{2} + (τ / κ) ∣ q ∣^{2}$ .

2.1F1step 1.4algebra

Fix a periodic solution from [F1]. For $k \neq 0$ , choose a coordinate $j$ with $∣ k_{j} ∣ \geq ∣ k ∣ / \sqrt{3}$ ; periodic integration by parts $N$ times and $∣ {\hat{\partial_{j}^{N} U}}_{k} ∣ \leq \sup_{x} ∣ \partial_{j}^{N} U (t, x) ∣$ give $∣ {\hat{U}}_{k} (t) ∣ \leq C_{N, t, L} (1 + ∣ k ∣)^{- N}$ . For each integer $n \geq 1$ , the disjoint unit cubes centered at lattice points with $n \leq ∣ k ∣ < n + 1$ lie in an annulus of bounded extra width and volume $O ((n + 1)^{2})$ , so the shell has $O ((n + 1)^{2})$ points. Hence $\sum_{k} (1 + ∣ k ∣)^{- N} < \infty$ for $N > 3$ , and the Fourier series converges absolutely and uniformly. To identify its sum with $U$ , use the product Fejer kernel: in one dimension the normalized kernel is $F_{N} (y) = \frac{1}{L (N + 1)} ∣ \sum_{ℓ = 0}^{N} e^{2 π i ℓ y / L} ∣^{2}$ . It is nonnegative with integral one, and for torus distance $∣ y ∣ \geq δ$ with $0 < δ < L / 2$ , the geometric-sum formula bounds it by $[L (N + 1) \sin^{2} (π δ / L)]^{- 1}$ . For the three-dimensional product, the mass outside the cube of coordinate distances less than $δ$ is bounded by the sum of the three one-dimensional tail masses, hence tends to zero. Uniform continuity of $U (t, \cdot)$ then shows that convolution with these kernels converges uniformly to $U (t, \cdot)$ . The convolutions are finite Fourier sums with weights $\prod_{j = 1}^{3} (1 - ∣ k_{j} ∣ / (N + 1))_{+}$ ; absolute convergence makes the same sums converge uniformly to $\sum_{k} {\hat{U}}_{k} (t) e^{2 π i k \cdot x / L}$ . Thus this series equals $U (t, x)$ .

2.2F1A1F2step 1.4

The matrices in step 1.4 have four components, positive-definite $M$ , symmetric $M A_{j}$ , and $S \geq 0$ , so they satisfy [F1]'s hypotheses under [A1]. The data class in the Statement is exactly the Schwartz/periodic class in [F1], using the componentwise Schwartz definition in [F2]. The eigenvalue computation in step 1.4 gives $v_{*} = v_{h} > 0$ . Applying [F1] therefore supplies the unique global smooth evolution for each stated data class, contraction of every stated weighted Sobolev norm for $t \geq 0$ , and the shrinking-ball finite-propagation conclusion at speed $v_{h}$ on $R^{3}$ and on $T_{L}^{3}$ .

3.1F1step 2.2step 2.1algebra

The same shell count in step 2.1 shows the lattice sum in $K_{r, L, C}$ converges: its shell contribution is $O ((j + 1)^{2 - 2 r})$ , summable for $r > 3 / 2$ . Set $a_{k} : = (1 + ∣ k / L ∣^{2})^{- r / 2}$ and $b_{k} : = (1 + ∣ k / L ∣^{2})^{r / 2} \sqrt{C} ∣ {\hat{θ}}_{k} (t) ∣$ . For each finite Fourier box, expanding $\sum_{k} (a_{k} - λ b_{k})^{2} \geq 0$ and minimizing over $λ \in R$ gives $(\sum_{k} a_{k} b_{k})^{2} \leq (\sum_{k} a_{k}^{2}) (\sum_{k} b_{k}^{2})$ , with the zero- $b$ case immediate; increasing boxes and taking monotone limits gives the same inequality for the series. The explicit diagonal matrix $M = diag (C, (τ / κ) I_{3})$ gives $C ∣ {\hat{θ}}_{k} (t) ∣^{2} \leq {\hat{U}}_{k} (t)^{*} M {\hat{U}}_{k} (t)$ for every $k$ . Therefore $∣ θ (t, x) ∣ \leq \sum_{k} ∣ {\hat{θ}}_{k} (t) ∣ \leq K_{r, L, C} ∥ U (t) ∥_{H_{M}^{r} (T_{L}^{3})} \leq K_{r, L, C} ∥ U_{0} ∥_{H_{M}^{r} (T_{L}^{3})}$ , where the Fourier reconstruction is justified in step 2.1 and the last inequality is [F1] contraction. If the last quantity is less than $T_{0}$ , then $T_{0} + θ (t, x) > 0$ for every $x$ and $t \geq 0$ .

4.1givenstep 1.4algebra∎

Since $C, κ, τ, c_{*} > 0$ , squaring the positive quantities in $v_{h} \leq c_{*}$ gives the equivalent condition $κ \leq C τ c_{*}^{2}$ , and reversing the same steps proves the converse.

Fixed-coefficient Cattaneo entropy and propagation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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