Scalar conservation laws, fluxes and Cauchy data

Let $n\ge 1$ and let $\mathrm{\Pi }=O\times (a,b)\subseteq {\mathbb{R}}^n\times \mathbb{R}$ be an open space--time cylinder. The equation $u_t+{\mathrm{div}}_{x}f(u)=0\text{in }\mathrm{\Pi }$ is the scalar conservation law in conservation form, where the unknown $u\text{\hspace{0.17em}⁣}:\mathrm{\Pi }\to \mathbb{R}$ is the conserved quantity and the flux is a map $f\text{\hspace{0.17em}⁣}:\mathbb{R}\to {\mathbb{R}}^n$. The classical expression is used when $u$ and $f\circ u$ are $C^1$; the distributional expression is used whenever $u,f(u)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Pi })$. In particular, if $u$ is bounded measurable and $f$ is continuous, then $f(u)$ is locally integrable and its distributional divergence is defined (A locally integrable function on ${\mathbb{R}}^n$, The space $L^p(\mu )$ as the quotient by null functions).

A Cauchy problem is posed separately on ${\mathrm{\Pi }}_T={\mathbb{R}}^n\times (0,T)$ and has initial datum $u_0\in L^{\mathrm{\infty }}({\mathbb{R}}^n)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)$, understood as an equivalence class; the weak formulation records it in the initial boundary term and the entropy formulation uses a strong local $L^1$ trace. If $f\in C^1(\mathbb{R};{\mathbb{R}}^n)$ and $u\in C^1$, the chain rule gives the equivalent quasilinear equation $u_t+f^{\mathrm{\prime }}(u)\cdot \mathrm{\nabla }u=0$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$, $C^k$ maps and multi-index derivative notation in Euclidean space, The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, Divergence and curl of a $C^1$ vector field). This equivalence is not asserted for discontinuous $u$: the product of $f^{\mathrm{\prime }}(u)$ with a distributional gradient is not generally defined, while ${\mathrm{div}}_{x}f(u)$ is defined distributionally whenever $f(u)\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$.

The one-dimensional case is $u_t+f(u{)}_x=0$ with scalar flux $f\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$. Whenever the Riemann theory, the Rankine--Hugoniot condition or the characteristic formula below is invoked, $f$ is assumed at least $C^1$ (and $C^2$ where $f^{\mathrm{\prime }}$ or $f^{\mathrm{\prime }\mathrm{\prime }}$ is differentiated); adding a constant vector to $f$ does not change the equation because its divergence is zero.