Corrected bulk relaxation and exact extended entropy

Use the smooth $(-,+,+,+)$ Lorentzian data, local state, and conventions of Perfect stress and particle balances with entropy identity and Frame-qualified dissipative entropy cancellation, with $n>0$, $T>0$, $c>0$, $x^0=ct$, and smooth future unit $u$. On the nonempty open spacetime domain $O$ of those suppliers, let $\zeta ,\tau \in C^1(O;(0,\mathrm{\infty }))$ be scalar coefficient fields, set $\beta :=\tau \mathrm{/}\zeta \in C^1(O;(0,\mathrm{\infty }))$, and let $\mathrm{\Pi }:O\to \mathbb{R}$ be smooth. In the bulk-only branch take

$T^{ab}=ϵu^a{u}^b+(p+\mathrm{\Pi }){\mathrm{\Delta }}^{ab},J^a=n{u}^a,{\mathrm{\nabla }}_a{T}^{ab}=0,{\mathrm{\nabla }}_a{J}^a=0,$

where ${\mathrm{\Delta }}^{ab}=g^{ab}+u^a{u}^b$, $D=u^a{\mathrm{\nabla }}_a$, $\theta ={\mathrm{\nabla }}_a{u}^a$, and $a_{\mathrm{a}\mathrm{c}\mathrm{c}}^b=D{u}^b$. Adopt the corrected relaxation law

$\beta D\mathrm{\Pi }+\frac{\mathrm{\Pi }}{\zeta c}=-\theta -\frac{T\mathrm{\Pi }}{2}{\mathrm{\nabla }}_a\text{\hspace{0.17em}⁣}\left(\frac{\beta u^a}{T}\right)$

and the extended entropy current $S^a=[s-\beta {\mathrm{\Pi }}^2\mathrm{/}(2T)]u^a$. Then the stress and particle equations imply

$Dϵ+(h+\mathrm{\Pi })\theta =0,(h+\mathrm{\Pi })a_{\mathrm{a}\mathrm{c}\mathrm{c}}^b+{\mathrm{\Delta }}^{ba}{\mathrm{\nabla }}_a(p+\mathrm{\Pi })=0,Dn+n\theta =0,$

and every actual smooth solution of the stated corrected law satisfies

${\mathrm{\nabla }}_a{S}^a=\frac{{\mathrm{\Pi }}^2}{\zeta cT}\ge 0.$

Here $\beta$ has units $\mathrm{P}{\mathrm{a}}^{-1}$, $\mathrm{\Pi },ϵ,p,h$ have units Pa, $D,\theta ,a_{\mathrm{a}\mathrm{c}\mathrm{c}}$ have inverse-length components, and the correction $\beta {\mathrm{\Pi }}^2\mathrm{/}(2T)$ has the units of entropy density. The law is an explicit model assumption; dropping its derivative term does not yield the stated sign identity in general.

There is an exact homogeneous solution on the explicitly specified Minkowski spacetime $M={\mathbb{R}}^4$ with global length coordinates $(x^0,x^1,x^2,x^3)=(ct,x^1,x^2,x^3)$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, and future direction ${\mathrm{\partial }}_{x^0}$; its Levi-Civita derivative is ordinary coordinate differentiation. Fix a state point $(v_0,{\sigma }_{p0})$ in the supplied local state domain with $v_0>0$ and $T_0=e_{{\sigma }_p}(v_0,{\sigma }_{p0})>0$, and take the constant state map $(v,{\sigma }_p)(x)=(v_0,{\sigma }_{p0})$, so $n_0=1\mathrm{/}{v}_0$, ${ϵ}_0=n_{0}e(v_0,{\sigma }_{p0})$, and $p_0=-e_v(v_0,{\sigma }_{p0})$ are constant. For constant positive coefficients ${\zeta }_0,{\tau }_0$, let $u={\mathrm{\partial }}_{x^0}=(1,0,0,0)$, $T=T_0$, and define the smooth scalar field $\mathrm{\Pi }(x)={\mathrm{\Pi }}_0{e}^{-t\mathrm{/}{\tau }_0}$ on all of $M$, where $t=x^0\mathrm{/}c$. The stress and particle balances and the corrected law hold, with strict entropy production wherever $\mathrm{\Pi }(x)\mathrm{\ne }0$. This mathematical witness is not a general nonlinear existence theorem or a claim of physical applicability in any separately selected near-equilibrium regime.