Convex entropy--entropy flux pairs

Let $n\ge 1$, let $f\text{\hspace{0.17em}⁣}:\mathbb{R}\to {\mathbb{R}}^n$ be $C^1$, and let $\eta \text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$ be a finite convex locally Lipschitz function (Convex and strictly convex functions on Euclidean convex sets). An entropy flux is specified by the coordinatewise formula $q_i(s)=C_i+{\int }_0^s{\eta }^{\mathrm{\prime }}(r)f_i^{\mathrm{\prime }}(r)dr\mathrm{.}$ One may use the left derivative of $\eta$ in this formula. Convex secant inequalities show that it is bounded and nondecreasing on compact intervals, and that it equals the ordinary derivative except at at most countably many points: assign a distinct rational to each nonempty gap between left and right slopes. A bounded monotone function is Riemann integrable, since the difference of upper and lower sums on an equal mesh is at most the mesh size times its total increase. The same holds after multiplication by the continuous $f_i^{\mathrm{\prime }}$: uniform continuity controls the additional oscillation in the product sums. Thus the displayed integrals exist and give locally Lipschitz $q$. The convex secant bounds also squeeze the telescoping sum $\eta (b)-\eta (a)$ between the left and right derivative sums, proving $\eta (b)-\eta (a)={\int }_a^b{\eta }^{\mathrm{\prime }}$ without a choice principle. At every continuity point of ${\eta }^{\mathrm{\prime }}$, averaging the integrand over a shrinking interval gives $q^{\mathrm{\prime }}(s)={\eta }^{\mathrm{\prime }}(s)f^{\mathrm{\prime }}(s)$; the exceptional points are countable. This includes nonsmooth entropies such as $\mathrm{\mid }s-k\mathrm{\mid }$. The constants $C_i$ are the only normalization freedom in this construction. For $\eta \in C^1$, ordinary FTC makes $q\in C^1$ (Every continuous function on an interval has a primitive; two primitives differ by a constant; and ${\int }_a^{b}f=G(b)-G(a)$ for any primitive $G$, The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$); second-derivative identities require $\eta \in C^2$ with ${\eta }^{\mathrm{\prime }\mathrm{\prime }}\ge 0$. Conversely, under Countable Choice and Dependent Choice, any locally Lipschitz $q$ satisfying $q^{\mathrm{\prime }}={\eta }^{\mathrm{\prime }}{f}^{\mathrm{\prime }}$ almost everywhere has this formula, by the fundamental theorem for absolutely continuous functions (Fundamental theorem of calculus for absolutely continuous functions, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). This converse analytic interface is separate from defining the explicit integral pairs used here.

Such a pair $(\eta ,q)$ is an entropy--entropy flux pair. For bounded measurable $u$ its entropy inequality is ${\mathrm{\partial }}_t\eta (u)+{\mathrm{div}}_{x}q(u)\le 0\text{in }{\mathcal{D}}^{\mathrm{\prime }}({\mathrm{\Pi }}_T),$ equivalently ${\int }_{{\mathrm{\Pi }}_T}(\eta (u){\phi }_t+q(u)\cdot \mathrm{\nabla }\phi )\ge 0$ for every nonnegative $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Pi }}_T)$ (Distribution, Distributional derivative). All compositions are locally integrable on this bounded range. Adding a constant vector to $q$ leaves the inequality unchanged. The weak conservation law implies equality for the affine pairs $(\eta ,q)=(s,f(s))$ and $(-s,-f(s))$; conversely their two entropy inequalities together imply that equality. The inequality for $(s,f(s))$ alone does not imply the weak equation (Scalar conservation laws, fluxes and Cauchy data).