Entropy jump residual

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

Let $I\subseteq \mathbb{R}$ and $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be open, let $\mathcal{O}\subseteq {\mathbb{R}}^m$ be open and convex, and let $F_j\in C^1(\mathcal{O};{\mathbb{R}}^m)$, $1\le j\le d$. Let $n\in {\mathbb{R}}^d$ have $\mathrm{\mid }n\mathrm{\mid }=1$, let $s\in \mathbb{R}$, and set $\mathrm{\Sigma }=\{(t,x)\in I\times \mathrm{\Omega }:n\cdot x=st\}$. Suppose $U$ is piecewise $C^1$ across $\mathrm{\Sigma }$, satisfies $U_t+{\sum }_{j}D{F}_j(U){\mathrm{\partial }}_{x_j}U=0$ classically on each open side, and the state, flux, and entropy compositions are locally integrable. Let $\eta \in C^2(\mathcal{O};\mathbb{R})$ be convex and let $q_j\in C^1(\mathcal{O};\mathbb{R})$ satisfy $D{q}_j(V)=D\eta (V)D{F}_j(V)$ for every $V\in \mathcal{O}$. With $[G]=G_R-G_L$, the distributional entropy residual is ${\mathrm{\partial }}_t\eta (U)+{\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}{q}_j(U)=({\sum }_{j=1}^d{n}_j[q_j(U)]-s[\eta (U)]){\delta }_{\mathrm{\Sigma }}\mathrm{.}$ Here ${\delta }_{\mathrm{\Sigma }}$ uses the plane measure defined in the vector jump-residual theorem: integrate in $t$ and in $z\in {\mathbb{R}}^{d-1}$ with $x=Ez+stn\in \mathrm{\Omega }$, where $E:{\mathbb{R}}^{d-1}\to n^{\perp }$ is a linear isometry onto $n^{\perp }$, using Lebesgue measure ${\lambda }_{d-1}$ and singleton mass one when $d=1$. The entropy inequality ${\mathrm{\partial }}_t\eta (U)+{\sum }_j{\mathrm{\partial }}_{x_j}{q}_j(U)\le 0$ holds near the interface if and only if ${\sum }_j{n}_j[q_j(U)]-s[\eta (U)]\le 0$ at every interface point. Entropy admissibility is an additional condition; the conservation jump equality alone does not imply it.

vector jump-residual calculation

entropy-flux compatibility and inequality convention

Axiom of Countable Choice