Continuum Heat Model

Let $I\subseteq \mathbb{R}$ be an open interval measured in seconds, let $L_1,L_2,L_3>0$ be lengths in metres, and set $\mathrm{\Omega }={\prod }_{i=1}^3(0,L_i)\subseteq {\mathbb{R}}^3$ with closure $\bar{\mathrm{\Omega }}$. A rectangular local-equilibrium field record consists of an open temperature interval $\mathcal{T}\subset (0,\mathrm{\infty })$, a temperature field $T:\bar{\mathrm{\Omega }}\times I\to \mathcal{T}$ whose time derivative through order one and spatial derivatives through order two extend continuously to $\bar{\mathrm{\Omega }}\times I$, a function $u\in C^1(\mathcal{T},\mathbb{R})$ of internal-energy density in $\mathrm{J}{\mathrm{m}}^{-3}$, and its volumetric heat capacity $c(T):=u^{\mathrm{\prime }}(T)>0$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$. The coordinates $x_i$ are lengths in metres, and $T$ is measured in kelvins, using SI quantities and unit conventions.

The record may also include a thermal-energy flux-density field $j_Q:\bar{\mathrm{\Omega }}\times I\to {\mathbb{R}}^3$ with components in $\mathrm{W}{\mathrm{m}}^{-2}$ and a conductivity field $K:\bar{\mathrm{\Omega }}\times \mathcal{T}\to {\mathbb{R}}^{3\times 3}$ with entries in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. For any unit vector $n$, $j_Q\cdot n$ is the signed thermal-energy transfer rate per unit area in direction $n$, positive when energy flows along $n$. The flux and tensor are field data here; no relation between them is imposed by this Definition. In Euclidean coordinates, $\mathrm{\nabla }T=({\mathrm{\partial }}_{x_1}T,{\mathrm{\partial }}_{x_2}T,{\mathrm{\partial }}_{x_3}T)$. The classical divergence $\mathrm{div}{j}_Q(x,t):={\sum }_{i=1}^3{\mathrm{\partial }}_{x_i}(j_Q{)}_i(x,t)$ is defined at interior points where these three partial derivatives exist. No spatial differentiability of an optional flux is assumed here; any balance or result using its divergence throughout a domain must impose the needed regularity. On a flat face, $n$ denotes its outward unit normal.

This is a specified regular field class on a fixed rigid body. It does not assert that every physical medium has local-equilibrium fields, nor does it adopt a heat equation, a boundary condition, Fourier's law, or an entropy-production law.

The label “local-equilibrium field record” names this model's field class; it is not a claim about microscopic equilibration.

Fix a particle mass $m_p>0$ in kilograms and velocity space $V={\mathbb{R}}^3$ with components in $\mathrm{m}{\mathrm{s}}^{-1}$. The elastic collision set is $Q_{\mathrm{e}\mathrm{l}}=\{(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in V^4:v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }},\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2\}\mathrm{.}$ Thus each allowed event conserves momentum $m_p(v+w)$ and kinetic energy $m_p(\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2)\mathrm{/}2$. Equip $Q_{\mathrm{e}\mathrm{l}}$ with the Borel sets generated by its relative topology.

Collision-rate data consist of a nonnegative, countably additive, $\sigma$-finite Borel measure $\nu$ on these Borel sets. It is invariant under exchanging $v$ with $w$, exchanging $v^{\mathrm{\prime }}$ with $w^{\mathrm{\prime }}$, and exchanging the incoming pair $(v,w)$ with the outgoing pair $(v^{\mathrm{\prime }},w^{\mathrm{\prime }})$: for each of these maps $S$ and Borel set $E$, $\nu (S^{-1}E)=\nu (E)$. This is the three-dimensional physical restriction of the collision set and exchange conventions in Elastic collision model and weak collision identity, with $\sigma$-finiteness stated explicitly for the product integrals used by the entropy theorem. For physical applications of that weak identity, at a fixed position and time let $f:V\to (0,\mathrm{\infty })$ be the one-particle number density per spatial volume and velocity volume: $f(v)d^{3}v$ is number per spatial volume in the velocity element. Particle number is dimensionless, $d^{3}v$ has units ${\mathrm{m}}^3{\mathrm{s}}^{-3}$, and $f$ has units ${\mathrm{s}}^3{\mathrm{m}}^{-6}$. A collision term $Q_{\nu }(f)$, when a representative satisfying the absolutely convergent weak identity exists, has units ${\mathrm{s}}^2{\mathrm{m}}^{-6}$, namely $f$ per second. Test functions $\phi :V\to \mathbb{R}$ are measurable and have one common unit $U_{\phi }$ for their values (dimensionless for the tests $1$ and $\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$, where $f_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ has the units of $f$). Assign the measure $\nu$ itself units ${\mathrm{m}}^9{\mathrm{s}}^{-7}$: its numerical values are taken in these SI units, not as a density relative to an unstated measure on $Q_{\mathrm{e}\mathrm{l}}$. Thus ${\int }_V\phi Q_{\nu }(f)d^{3}v$ and the collision integral of $f^2$ times the test bracket both have units $U_{\phi }{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$. The magnitude and distribution of $\nu$ remain model inputs; dimensional consistency fixes neither a numerical collision rate nor existence of $Q_{\nu }(f)$. The measure is a declared model input; no differential cross section or microscopic limit is inferred from this Definition.

For the field record of Rectangular local-equilibrium field data, adopt $j_Q(x,t)=-K(x,T(x,t))\mathrm{\nabla }T(x,t)\mathrm{.}$ Here $K$ has entries in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ and $j_Q$ in $\mathrm{W}{\mathrm{m}}^{-2}$, so the equation is dimensionally consistent. Write $K_s=(K+K^{\mathsf{T}})\mathrm{/}2$ for the symmetric part and assume $z^{\mathsf{T}}{K}_s(x,T)z\ge 0$ for every $z\in {\mathbb{R}}^3$. Define the thermal quadratic density by ${\sigma }_Q(x,t):=-\frac{j_Q(x,t)\cdot \mathrm{\nabla }T(x,t)}{T(x,t{)}^2}=\frac{(\mathrm{\nabla }T(x,t){)}^{\mathsf{T}}{K}_s(x,T(x,t))\mathrm{\nabla }T(x,t)}{T(x,t{)}^2},$ with units $\mathrm{W}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$. For the antisymmetric part $K_a=(K-K^{\mathsf{T}})\mathrm{/}2$, the scalar $z^{\mathsf{T}}{K}_{a}z$ equals its own negative by transposition, so it is zero. Thus $K_a$ may contribute to $j_Q$ but not to ${\sigma }_Q$, and ${\sigma }_Q\ge 0$. Interpreting this density as entropy production additionally requires a local entropy map $s\in C^1(\mathcal{T},\mathbb{R})$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$ with $s^{\mathrm{\prime }}(T)=u^{\mathrm{\prime }}(T)\mathrm{/}T$, the energy balance ${\mathrm{\partial }}_{t}u+\mathrm{div}{j}_Q=0$, and regularity permitting the product and chain rules; under those additional assumptions the entropy balance is ${\mathrm{\partial }}_{t}s(T)+\mathrm{div}(j_Q\mathrm{/}T)={\sigma }_Q$. This postulate alone adopts neither that caloric entropy relation nor the energy balance.

This is an adopted constitutive model in the stated continuum regime. It is neither a consequence of equilibrium entropy alone nor a universal law for all media. In particular, it does not assert a finite propagation speed or an empirical value for $K$.

For a field record on $\mathrm{\Omega }\times I$ from Rectangular local-equilibrium field data, require the supplied heat-flux field $j_Q$ and each of its first spatial partial derivatives to extend continuously to $\bar{\mathrm{\Omega }}\times [t_0,t_1]$ for every compact interval $[t_0,t_1]\subset I$. Thus $\mathrm{div}{j}_Q$ is the classical coordinate divergence in $\mathrm{\Omega }\times I$, and $j_Q\cdot n$ denotes its continuous trace on each closed face. For these records, adopt the local balance ${\mathrm{\partial }}_{t}u(T(x,t))+\mathrm{div}{j}_Q(x,t)=0$ and the insulated-boundary condition $j_Q(x,t)\cdot n(x)=0$ on every face of $\mathrm{\partial }\mathrm{\Omega }$. Both terms in the local equation have units $\mathrm{W}{\mathrm{m}}^{-3}$. The boundary condition says that no heat energy crosses the body's outer faces in this model.

This local continuum balance is an additional adopted postulate, with the sign and energy convention of First-law energy balance for closed process records. It is not deduced from that global process-record balance. Matter transfer, volume work, internal sources, moving boundaries, and external-field energy require separately specified balance models and are outside this item.

For the local-equilibrium medium of Rectangular local-equilibrium field data, adopt a local entropy-density map $s:\mathcal{T}\to \mathbb{R}$ measured in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$ with $\frac{ds}{dT}(T)=\frac{c(T)}{T},$ where $c(T)=u^{\mathrm{\prime }}(T)>0$ is the volumetric heat capacity and $T>0$ is measured in kelvins. Equivalently, for a fixed reference $T_{\mathrm{r}\mathrm{e}\mathrm{f}}\in \mathcal{T}$ and reference density $s_{\mathrm{r}\mathrm{e}\mathrm{f}}\in \mathbb{R}$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$, set $s(T)=s_{\mathrm{r}\mathrm{e}\mathrm{f}}+{\int }_{T_{\mathrm{r}\mathrm{e}\mathrm{f}}}^T\frac{c(\theta )}{\theta }d\theta \mathrm{.}$ The volume integral of this field is the entropy assigned by this continuum model. This adopts a local thermodynamic relation extending the macroscopic entropy framework of Concave macroscopic fundamental entropy; it does not prove that every evolving medium has such a local entropy density or identify it with an arbitrary finite-system entropy function.

For constant $c(T)=c_0>0$, the adopted relation gives $s(T)=s_{\mathrm{r}\mathrm{e}\mathrm{f}}+c_0\log (T\mathrm{/}{T}_{\mathrm{r}\mathrm{e}\mathrm{f}})$. This is a restricted constitutive specialization, not a universal equation of state.

Let ${\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}={\prod }_{i=1}^3(0,L_i)$ with opposite faces identified, let $I\subseteq \mathbb{R}$ be an open time interval in seconds, and let $f(x,v,t)\ge 0$ be a one-particle number density for $x\in {\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}$, $v\in {\mathbb{R}}^3$, and $t\in I$. With $x$ in metres and $v$ in $\mathrm{m}{\mathrm{s}}^{-1}$, $f$ has units ${\mathrm{s}}^3{\mathrm{m}}^{-6}$ relative to $d^{3}x{d}^{3}v$; its moments used below are assumed finite. For this postulate, require $f$ to be jointly measurable and periodic in $x$, with $(x,t)\mapsto f(x,v,t)$ continuously differentiable for each fixed $v$ and its derivatives jointly measurable. Require $f$, ${\mathrm{\partial }}_{t}f$, and $v\cdot {\mathrm{\nabla }}_{x}f$ to be locally integrable with respect to product Lebesgue measure. Periodic boundary conditions are imposed in $x$.

Adopt the incoming molecular-chaos closure: at a collision location and time, the incoming two-particle velocity density used in the collision term is $f(x,v,t)f(x,w,t)$. This is a one-sided factorization assumption; no factorization of the outgoing pair and no derivation from reversible many-particle mechanics is asserted.

For the elastic data $Q_{\mathrm{e}\mathrm{l}},\nu$ of Regular single-species elastic collision data, adopt a collision operator $Q_{\nu }(f)$, with units of $f$ per second, as a jointly measurable locally integrable representative whose velocity-wise weak action satisfies, for almost every $(x,t)$ and every measurable test function $\phi$ for which both sides converge absolutely, ${\int }_{{\mathbb{R}}^3}\phi (v)Q_{\nu }(f)(x,v,t)d^{3}v=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(f(x,v^{\mathrm{\prime }},t)f(x,w^{\mathrm{\prime }},t)-f(x,v,t)f(x,w,t))(\phi (v)+\phi (w)-\phi (v^{\mathrm{\prime }})-\phi (w^{\mathrm{\prime }}))d\nu \mathrm{.}$ The adopted kinetic equation is ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f),$ as an equality of locally integrable functions almost everywhere on the product domain. These are admissibility conditions for solutions of the adopted model, not an existence assertion. The equation and weak collision identity are model postulates. This item does not assert a hard-sphere Boltzmann–Grad limit, propagation of chaos, or universal equilibration.

Let $d\ge 1$, let $L_1,\dots ,L_d>0$, and let $\mathrm{\Omega }={\prod }_{i=1}^d(0,L_i)$. If $w\in C^1(\bar{\mathrm{\Omega }},\mathbb{R})$ satisfies ${\int }_{\mathrm{\Omega }}w(x)dx=0$, write $\mathrm{\nabla }w=({\mathrm{\partial }}_{1}w,\dots ,{\mathrm{\partial }}_{d}w)$ and $\mathrm{\mid }\mathrm{\nabla }w{\mathrm{\mid }}^2={\sum }_{i=1}^d\mathrm{\mid }{\mathrm{\partial }}_{i}w{\mathrm{\mid }}^2$. Then ${\int }_{\mathrm{\Omega }}\mathrm{\mid }w(x){\mathrm{\mid }}^{2}dx\le \frac{d}{2}{\sum }_{i=1}^d{L}_i^2{\int }_{\mathrm{\Omega }}\mathrm{\mid }{\mathrm{\partial }}_{i}w(x){\mathrm{\mid }}^{2}dx\le C_{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }w(x){\mathrm{\mid }}^{2}dx,$ where $C_{\mathrm{\Omega }}=\frac{d}{2}{\max }_i{L}_i^2$. All integrals are the multidimensional Riemann integrals of continuous functions on the indicated closed rectangles, with the usual extension to their interiors.

Let ${\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}={\prod }_{i=1}^3(0,L_i)$ with opposite faces identified, and let $I\subseteq \mathbb{R}$ be an open time interval. Assume the current conditional mathematical result Conditional Boltzmann entropy inequality and equality form at its exact draft interface [L1] below. Let $f:{\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}\times {\mathbb{R}}^3\times I\to (0,\mathrm{\infty })$ satisfy the periodic Boltzmann postulate Periodic Boltzmann model with incoming molecular chaos and the regularity, integrability, and domination hypotheses in [A2]–[A3]. For the all-time conclusion below, require the kinetic equation ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f)$ to hold at every $(x,v,t)$ for the chosen collision representative, in addition to the postulate's product-almost-everywhere equation. Fix a positive constant reference density $f_{\mathrm{r}\mathrm{e}\mathrm{f}}$ with the same units as $f$, and define the kinetic entropy $S_{\mathrm{k}\mathrm{i}\mathrm{n}}(t)=-k_B{\int }_{{\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}}{\int }_{{\mathbb{R}}^3}f(x,v,t)\log \text{\hspace{0.17em}⁣}\left(\frac{f(x,v,t)}{f_{\mathrm{r}\mathrm{e}\mathrm{f}}}\right)d^{3}v{d}^{3}x\mathrm{.}$ Then $\frac{d{S}_{\mathrm{k}\mathrm{i}\mathrm{n}}}{dt}=\frac{k_B}{4}{\int }_{{\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(A-B)\log \text{\hspace{0.17em}⁣}\left(\frac{A}{B}\right)d\nu d^{3}x\ge 0,$ where $A=f(x,v,t)f(x,w,t)$ and $B=f(x,v^{\mathrm{\prime }},t)f(x,w^{\mathrm{\prime }},t)$; $\log (A\mathrm{/}B)$ is dimensionless because $A$ and $B$ have the same units. Equality holds exactly when $A=B$ for $\nu$-almost every collision at almost every spatial point. This is a conditional deduction inside the adopted incoming-molecular-chaos kinetic model. The cited mathematical carrier is currently draft and has not received independent proof acceptance; no microscopic derivation, experimental claim, or Maxwellian characterization is included.

Let $I\subseteq \mathbb{R}$ be an open time interval, $\mathrm{\Omega }={\prod }_{i=1}^3(0,L_i)$ a fixed rectangle, and $(T,u,j_Q,K)$ a field record from Rectangular local-equilibrium field data satisfying the adopted local-energy, Fourier, and local-entropy postulates Local energy balance for an insulated continuum body, Fourier heat-flux constitutive model, and Local-equilibrium caloric entropy density. Assume additionally that $j_Q$ and its first spatial derivatives are continuous on every compact subset of $\bar{\mathrm{\Omega }}\times I$. Then, throughout $\mathrm{\Omega }\times I$, ${\mathrm{\partial }}_{t}s(T)+\mathrm{div}\text{\hspace{0.17em}⁣}\left(\frac{j_Q}{T}\right)=\frac{(\mathrm{\nabla }T{)}^{\mathsf{T}}{K}_s(x,T)\mathrm{\nabla }T}{T^2}\ge 0,$ where $K_s=(K+K^{\mathsf{T}})\mathrm{/}2$. For every compact time interval $[t_0,t_1]\subset I$, the model's total entropy $S(t)={\int }_{\mathrm{\Omega }}s(T(x,t))dx$ is differentiable and nondecreasing, while ${\int }_{\mathrm{\Omega }}u(T(x,t))dx$ is constant. These are conditional deductions of the declared continuum model, not observations or a claim that every medium obeys Fourier's law.

Assume Countable Choice. Let $L_i>0$ be lengths measured in metres, set $\mathrm{\Omega }={\prod }_{i=1}^3(0,L_i)$, and take the open temperature interval $\mathcal{T}=(0,\mathrm{\infty })$ with temperatures measured in kelvins. Fix a constant volumetric heat capacity $c>0$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$, a constant scalar conductivity $k>0$ in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, and an arbitrary reference energy density $u_{\star }$ in $\mathrm{J}{\mathrm{m}}^{-3}$. Define $u:\mathcal{T}\to \mathbb{R}$ by $u(\theta )=u_{\star }+c\theta$, so $u^{\mathrm{\prime }}(\theta )=c$, and define $K:\bar{\mathrm{\Omega }}\times \mathcal{T}\to {\mathbb{R}}^{3\times 3}$ by $K(x,\theta )=k{I}_3$ as in Fourier heat-flux constitutive model. Put $a=k\mathrm{/}c$ in ${\mathrm{m}}^2{\mathrm{s}}^{-1}$.

Let $g:{\mathbb{R}}^3\to \mathcal{T}$ be $C^2$ and satisfy, for every $i\in \{1,2,3\}$ and $x\in {\mathbb{R}}^3$, $g(x_1,\dots ,-x_i,\dots ,x_3)=g(x_1,\dots ,x_i,\dots ,x_3),g(x_1,\dots ,x_i+2L_i,\dots ,x_3)=g(x_1,\dots ,x_i,\dots ,x_3)\mathrm{.}$ Assume every coordinate derivative $D^{\alpha }g$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le 2$ is bounded and uniformly continuous, and that $0<T_{\min }\le g(y)\le T_{\max }<\mathrm{\infty }$ for all $y$, with these bounds measured in kelvins.

For $x\in \bar{\mathrm{\Omega }}$ and $t>0$ define $T(x,t)={\int }_{{\mathbb{R}}^3}\mathrm{\Gamma }(x-y,at)g(y)dy,\mathrm{\Gamma }(z,\theta )=(4\pi \theta {)}^{-3\mathrm{/}2}{e}^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}(4\theta )},$ and set $T(x,0)=g(x)$. Here $y$ is measured in metres, $\theta =at$ in the kernel has units ${\mathrm{m}}^2$, and the normalized kernel $\mathrm{\Gamma }(\cdot ,at)$ has units ${\mathrm{m}}^{-3}$, so the integral has units of kelvins. Also $\mathrm{\nabla }T=({\mathrm{\partial }}_{x_1}T,{\mathrm{\partial }}_{x_2}T,{\mathrm{\partial }}_{x_3}T)$, $\mathrm{\Delta }T:={\sum }_{i=1}^3{\mathrm{\partial }}_{x_i}^{2}T$, and on a face with outward unit normal $n$, ${\mathrm{\partial }}_{n}T:=n\cdot \mathrm{\nabla }T$. For every finite $\tau >0$, this is the unique function $T$ continuous on $\bar{\mathrm{\Omega }}\times [0,\tau ]$ whose first time derivative for $t>0$ and spatial derivatives through order two extend continuously to $\bar{\mathrm{\Omega }}\times [0,\tau ]$, solving $c{\mathrm{\partial }}_{t}T=k\mathrm{\Delta }T$ on $\mathrm{\Omega }\times (0,\tau ]$, with ${\mathrm{\partial }}_{n}T=0$ on each face and the given initial value. The time derivative at $t=0$ is interpreted from the right. The field satisfies $T_{\min }\le T(x,t)\le T_{\max }$ for all $t\ge 0$, its spatial derivatives through order two are continuous to $t=0$, and its right time derivative there is $a\mathrm{\Delta }g$.

On $I=(0,\mathrm{\infty })$, set $j_Q(x,t)=-k\mathrm{\nabla }T(x,t)$. Then $(T,u,j_Q,K)$ is a field record of Rectangular local-equilibrium field data satisfying Local energy balance for an insulated continuum body and Fourier heat-flux constitutive model, so the solution is global in time for this restricted physical continuum model.

Assume Countable Choice and the hypotheses and solution $T$ of Constant-conductivity heat evolution on an insulated rectangle, let $V=\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }=L_1{L}_2{L}_3$, define the spatial mean $\bar{T}(t)=V^{-1}{\int }_{\mathrm{\Omega }}T(x,t)dx$, and put $E(t)={\int }_{\mathrm{\Omega }}\mathrm{\mid }T(x,t)-\bar{T}(t){\mathrm{\mid }}^{2}dx$. With $a=k\mathrm{/}c$ and $C_{\mathrm{\Omega }}=\frac{3}{2}{\max }_i{L}_i^2$, the mean is constant and $E(t)\le e^{-2at\mathrm{/}{C}_{\mathrm{\Omega }}}E(0)(t\ge 0)\mathrm{.}$ Thus this model solution converges exponentially to its initial spatial mean in $L^2(\mathrm{\Omega })$; no pointwise-uniform convergence rate is asserted.