 Scalar conservation law

A one-dimensional scalar conservation law describes the balance of a scalar density $u$ and its conservation law flux $f(u)$:The second identity follows from the first for smooth $u$: the quantity inside a fixed interval changes only through its boundary fluxes. For a classical solution, the chain rule also gives the quasilinear partial differential equation $u_t+f^{\prime }(u)u_x=0$, with characteristic speed $f^{\prime }(u)$.

The conservative form still defines a discontinuous weak solution. With initial condition $u_0$, it requiresfor every smooth compactly supported test function on $[0,\infty )\times \mathbb{R}$. For $f(u)=u^2/2$, a jump from $1$ to $0$ travels at speed $1/2$ by the Rankine-Hugoniot condition, between incoming characteristic speeds $1$ and $0$. Reversing those states still gives a conservative jump of speed $1/2$, but entropy solution admissibility selects a rarefaction wave. Thus conservation determines jump balance while an entropy condition selects the continuation after characteristic crossing.^[ref]