Separate entropy-only radiation state maps

Fix a real parameter $a_R>0$ and let the temperature coordinate range over the open interval $T\in (0,\mathrm{\infty })$. Define the entropy-only radiation state maps

${\epsilon }_T(T):=a_R{T}^4,p_T(T):=\frac{1}{3}{a}_R{T}^4,s_T(T):=\frac{4}{3}{a}_R{T}^3\mathrm{.}$

Here ${\epsilon }_T$ and $p_T$ are scalar-valued maps on $(0,\mathrm{\infty })$, and $s_T$ is the entropy coordinate as a scalar-valued map on the same interval. Put $h_T(T):={\epsilon }_T(T)+p_T(T)$. No particle-number coordinate, particle-number current, or chemical-potential coordinate is part of this state space.

Also define maps on the positive entropy-coordinate interval by

$\tau (s):={\left(\frac{3s}{4a_R}\right)}^{1\mathrm{/}3},{\epsilon }_s(s):=a_R\tau (s{)}^4,p_s(s):=\frac{1}{3}{\epsilon }_s(s),s\in (0,\mathrm{\infty })\mathrm{.}$

The displayed formulas define functions on the stated domains. In the conventional SI physical reading, $T$ is measured in K, $a_R$ in J m${}^{-3}$ K${}^{-4}$, ${\epsilon }_T,p_T$ in J m${}^{-3}$, and $s_T$ in J m${}^{-3}$ K${}^{-1}$. These units do not assign an empirical value to $a_R$. In particular, this definition does not derive a quantum blackbody coefficient or assert that any physical system realizes these maps.

Fix $a_R>0$ and the maps of Separate entropy-only radiation state maps. First, their temperature and entropy coordinates are inverse, $d{\epsilon }_T=Td{s}_T$, $h_T=T{s}_T$, and $d{p}_T\mathrm{/}d{\epsilon }_T=1\mathrm{/}3$ on $T>0$.

Next, assume countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M$ be a smooth four-manifold without boundary and let $g$ be a smooth symmetric nondegenerate covariant $2$-tensor field of signature $(-,+,+,+)$. Write $g^{ab}$ for the inverse-metric components and $u_b:=g_{bc}{u}^c$. Use the Levi-Civita connection supplied by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Let $c>0$, let $u$ be a smooth vector field with $g_{ab}{u}^a{u}^b=-c^2$, and let $T:M\to (0,\mathrm{\infty })$ be smooth. Put $\epsilon ={\epsilon }_T(T)$, $p=p_T(T)$, $s=s_T(T)$, and $h=\epsilon +p$. Define the symmetric tensor field

${\mathsf{T}}^{ab}:=\frac{h}{c^2}{u}^a{u}^b+p{g}^{ab}\mathrm{.}$

If ${\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=0$, then ${\mathrm{\nabla }}_a(s{u}^a)=0$. This is a conditional mathematical identity for the specified tensor and state maps; it assumes no particle-number current.

Finally, on ${\mathbb{R}}^4$ with Minkowski coordinates $(x^0,x^1,x^2,x^3)=(ct,x)$ and metric $\eta =\mathrm{diag}(-1,1,1,1)$, specify the linearization as follows. For some $ϵ>0$, let $\lambda \mapsto (T_{\lambda },u_{\lambda })$ be jointly smooth on $(-ϵ,ϵ)\times {\mathbb{R}}^4$, with $T_{\lambda }(x)>0$, $T_0(x)=T_{\ast }$, $u_0=(c,0,0,0)$, and ${\eta }_{ab}{u}_{\lambda }^a{u}_{\lambda }^b=-c^2$. Form ${\mathsf{T}}_{\lambda }^{ab}$ by the same stress formula above with $(T,u,g)$ replaced by $(T_{\lambda },u_{\lambda },\eta )$, using ${\epsilon }_{\lambda }:={\epsilon }_T(T_{\lambda })$ and $p_{\lambda }:=p_T(T_{\lambda })$. Write $\delta f:={\frac{d}{d\lambda }{f}_{\lambda }\mid }_{\lambda =0}$; the linearized equation means $\delta ({\mathrm{\partial }}_a{\mathsf{T}}_{\lambda }^{ab})=0$. Put $h_{\ast }:={\epsilon }_T(T_{\ast })+p_T(T_{\ast })$, $r:=\delta \epsilon$, and $w^i:=\delta u^i\mathrm{/}c$. Its resulting constant-coefficient linear system is

$r_t+c{h}_{\ast }\mathrm{div}w=0,w_t+\frac{c}{3h_{\ast }}\mathrm{\nabla }r=0,$

where $\mathrm{\nabla }$ and $\mathrm{div}$ are the Euclidean spatial gradient and divergence. With $U=(r,w)$, write this system as $U_t+{\sum }_{j=1}^3{A}_j{\mathrm{\partial }}_{j}U=0$, where

$A_j:=c\left(\begin{array}{cc}0& h_{\ast }{e}_j^{\mathsf{T}}\\ (1\mathrm{/}(3h_{\ast }))e_j& 0_{3\times 3}\end{array}\right),e_j\text{ is the }j\text{th standard basis vector of }{\mathbb{R}}^3\mathrm{.}$

The matrix $M_{\ast }:=\mathrm{diag}(1\mathrm{/}(3h_{\ast }),h_{\ast }{I}_3)$ is a positive symmetrizer in the explicit sense that it is symmetric positive definite and $M_{\ast }{A}_j$ is symmetric for each $j$. For each unit spatial vector $\nu \in {\mathbb{R}}^3$, define the directional coefficient matrix $A(\nu ):={\sum }_{j=1}^3{\nu }_j{A}_j$; its eigenvalues are $0$ (multiplicity two) and $\pm c\mathrm{/}\sqrt{3}$. For Schwartz initial perturbations on ${\mathbb{R}}^3$ or smooth periodic initial perturbations on ${\mathbb{T}}_L^3$ with $L>0$, where Schwartz means each component belongs to $\mathcal{S}({\mathbb{R}}^3)$ as defined in Schwartz space and its seminorms, the global smooth evolution, weighted Sobolev contraction, and finite-propagation conclusions are conditional on the exact symmetrized-system interface of Complete symmetrized constant-coefficient evolution stated in [A1]. That current carrier is draft, so this item does not record those conclusions as independently accepted.

Let $C,\kappa ,\tau ,T_0,L>0$. Let $\mathrm{\Omega }$ be either ${\mathbb{R}}^3$ or the flat torus ${\mathbb{T}}_L^3:=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^3$, and let $I\subseteq \mathbb{R}$ be an open interval containing $0$. Consider real smooth fields $\theta :I\times \mathrm{\Omega }\to \mathbb{R}$ and $q:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ satisfying

$C{\theta }_t+\mathrm{div}q=0,\tau q_t+q=-\kappa \mathrm{\nabla }\theta \mathrm{.}$

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

$A_j:=\left(\begin{array}{cc}0& e_j^{\mathsf{T}}\mathrm{/}C\\ (\kappa \mathrm{/}\tau )e_j& 0_{3\times 3}\end{array}\right),B:=\mathrm{diag}(0,(1\mathrm{/}\tau )I_3),M:=\mathrm{diag}(C,(\tau \mathrm{/}\kappa )I_3)\mathrm{.}$

Then the displayed system is $U_t+{\sum }_{j=1}^3{A}_j{\mathrm{\partial }}_{j}U+BU=0$. The matrix $M$ is a positive symmetrizer, meaning $M=M^{\mathsf{T}}$ is positive definite and $M{A}_j$ is symmetric for each $j$. For a unit direction $\nu \in {\mathbb{R}}^3$, define the principal matrix $A(\nu ):={\sum }_{j=1}^3{\nu }_j{A}_j$; its eigenvalues are $0$ (multiplicity two) and $\pm v_h$, where $v_h:=\sqrt{\kappa \mathrm{/}(C\tau )}>0$. For a fixed comparison speed $c_{\ast }>0$, $v_h\le c_{\ast }$ if and only if $\kappa \le C\tau c_{\ast }^2$.

For every such solution and every $s_0\in \mathbb{R}$, define the specified quadratic effective entropy density and flux by

$s_{\mathrm{e}\mathrm{f}\mathrm{f}}:=s_0+\frac{C\theta }{T_0}-\frac{C{\theta }^2}{2T_0^2}-\frac{\tau \mathrm{\mid }q{\mathrm{\mid }}^2}{2\kappa T_0^2},j_s:=\frac{q}{T_0}-\frac{\theta q}{T_0^2}\mathrm{.}$

Every smooth solution satisfies the exact identities

${\mathrm{\partial }}_t{s}_{\mathrm{e}\mathrm{f}\mathrm{f}}+\mathrm{div}{j}_s=\frac{\mathrm{\mid }q{\mathrm{\mid }}^2}{\kappa T_0^2}\ge 0,{\mathrm{\partial }}_t{e}_L+\mathrm{div}(\theta q)=-\frac{\mathrm{\mid }q{\mathrm{\mid }}^2}{\kappa },e_L:=\frac{1}{2}(C{\theta }^2+\frac{\tau }{\kappa }\mathrm{\mid }q{\mathrm{\mid }}^2)\mathrm{.}$

For every unit direction $\nu$, its quadratic flux satisfies $\mathrm{\mid }\theta q\cdot \nu \mathrm{\mid }\le v_h{e}_L$. 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 $s_{\mathrm{e}\mathrm{f}\mathrm{f}}$ 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 $U_0=({\theta }_0,q_0)\in \mathcal{S}({\mathbb{R}}^3;{\mathbb{R}}^4)$ or $C^{\mathrm{\infty }}({\mathbb{T}}_L^3;{\mathbb{R}}^4)$, 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 ${\mathbb{T}}_L^3$, for every integer $r>3\mathrm{/}2$, set

$K_{r,L,C}:=C^{-1\mathrm{/}2}{({\sum }_{k\in {\mathbb{Z}}^3}(1+\mathrm{\mid }k\mathrm{/}L{\mathrm{\mid }}^2{)}^{-r})}^{1\mathrm{/}2}<\mathrm{\infty },$

where $C>0$ is the heat-capacity coefficient and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H_M^r({\mathbb{T}}_L^3)}$ is the norm in [F1]. The periodic global solution supplied conditionally by [F1] obeys $T_0+\theta (t,x)>0$ for all $t\ge 0$ whenever $K_{r,L,C}\mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{H_M^r({\mathbb{T}}_L^3)}<T_0$. The constant uses only the temperature component $C\mathrm{\mid }{\widehat{\theta }}_k{\mathrm{\mid }}^2$ 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 $\theta ,T_0$ in K, $C$ in J m${}^{-3}$ K${}^{-1}$, $\kappa$ in W m${}^{-1}$ K${}^{-1}$, $\tau$ in s, $L$ in m, $q$ in W m${}^{-2}$, and the comparison speed $c_{\ast }$ 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 $1+\mathrm{\mid }k\mathrm{/}L{\mathrm{\mid }}^2$ uses the reference length $1$ m. Then $s_0$ and $s_{\mathrm{e}\mathrm{f}\mathrm{f}}$ 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 $\mathrm{\parallel }U{\mathrm{\parallel }}_{H_M^r}$ has units $\sqrt{\mathrm{J}\mathrm{K}{\mathrm{m}}^{-3}}$ and $K_{r,L,C}$ has units $\sqrt{{\mathrm{m}}^3\mathrm{K}\mathrm{/}\mathrm{J}}$, so their product has temperature units. These unit labels do not assert a physical realization of the linear subsystem.