Stoichiometric conservation and dissipation functionals

Let $F:U\to {\mathbb{R}}^m$ be the reaction vector field of Reaction vector fields and quasi-positivity, and let $R\subset U$ be a state set. A vector $c\in [0,\mathrm{\infty }{)}^m$ is a stoichiometric conservation vector on $R$ if $c^{T}F(z)=0$ for every $z\in R$. It is a stoichiometric dissipation vector on $R$ if $c^{T}F(z)\le 0$ for every $z\in R$. The corresponding pointwise mass is $c^{T}z$.

For a Lebesgue measurable domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ and a measurable field $u:\mathrm{\Omega }\to {\mathbb{R}}^m$ such that ${\int }_{\mathrm{\Omega }}\mathrm{\mid }{c}^{T}u(x)\mathrm{\mid }dx<\mathrm{\infty }$, the continuous total mass is ${\mathcal{M}}_c(u)={\int }_{\mathrm{\Omega }}{c}^{T}u(x)dx$, with Lebesgue integration. In particular, this is defined for bounded measurable fields on finite-volume domains. For any finite index set $I$ with assigned positive weights $m_i$, the discrete total mass of states $U_i\in {\mathbb{R}}^m$ is ${\mathcal{M}}_{c,h}(U)={\sum }_{i\in I}{m}_i{c}^T{U}_i$. The weights and index set are part of the discrete model. Conservation and dissipation always refer to these stated functionals and the stated state set; no claim is made for a different weight vector or for reactions outside $R$.