Nonconvex Riemann data can require a composite shock--rarefaction wave

The shock is admissible with speed $3\mathrm{/}4$. For the descending jump $u^{-}=1\to u^{+}=-\frac{1}{2}$, the chord slope is $s=(f(-\frac{1}{2})-f(1))\mathrm{/}(-\frac{1}{2}-1)=(-\frac{1}{8}-1)\mathrm{/}(-\frac{3}{2})=\frac{3}{4}$, so $s[u]=\frac{3}{4}\cdot (-\frac{3}{2})=-\frac{9}{8}$ equals $[f]=-\frac{9}{8}$: Rankine--Hugoniot holds. The residual is $F(z)=z^3-1-\frac{3}{4}(z-1)=(z-1)(z^2+z+\frac{1}{4})=(z-1)(z+\frac{1}{2}{)}^2$, which is $\le 0$ on $[-\frac{1}{2},1]$; since $u^{+}-u^{-}=-\frac{3}{2}<0$, the product $F(z)(u^{+}-u^{-})\ge 0$ and the chord criterion of [F1] makes the shock entropy-admissible.

The rarefaction branch solves the equation. By [F2], $\psi (\xi )=-\sqrt{\xi \mathrm{/}3}$ is the inverse of $f^{\mathrm{\prime }}$ on $[-1,-\frac{1}{2}]$; on the fan $u(t,x)=\psi (x\mathrm{/}t)$ with $\xi =x\mathrm{/}t$, so [F3] gives $u_t+f^{\mathrm{\prime }}(u)u_x=\frac{1}{t}{\psi }^{\mathrm{\prime }}(\xi )(f^{\mathrm{\prime }}(\psi (\xi ))-\xi )=\frac{1}{t}{\psi }^{\mathrm{\prime }}(\xi )(\xi -\xi )=0$ for $\frac{3}{4}<\xi <3$. At the right edge $\xi =3$, $\psi (3)=-1$ matches the constant state $-1$; at the left edge $\xi =\frac{3}{4}$, $\psi (\frac{3}{4})=-\frac{1}{2}$ is the right state of the shock.

The profile is a weak solution. Splitting the test integral into the constant left region, the fan, the constant right region, and the interfaces: the outer regions contribute only boundary terms; the interface at $x=\frac{3}{4}t$ is handled by the Rankine--Hugoniot computation of step 1.1, so its coefficient $[f]-s[u]$ vanishes; and at $x=3t$ the traces of $u$ (hence of $f(u)$) match continuously, so by [F3] no interface term arises. Adding the pieces, the weak residual vanishes, so $u$ is a distributional weak solution; the discrepancy with the initial step datum is supported in $(0,3t)$ with amplitude at most $2$, so the strong local $L^1$ trace holds.

All entropy inequalities hold. For a convex $C^2$ pair $(\eta ,q)$ with $q^{\mathrm{\prime }}={\eta }^{\mathrm{\prime }}{f}^{\mathrm{\prime }}$, the production is computed piecewise: it vanishes on the constant regions; on the fan it equals ${\eta }^{\mathrm{\prime }}(u)(u_t+f^{\mathrm{\prime }}(u)u_x)=0$ by step 1.2, with no interface term at $x=3t$ because the traces of $\eta (u),q(u)$ match there; and at the shock it is the measure with coefficient $[q]-s[\eta ]$, which is $\le 0$ by the chord condition of step 1.1. Hence the entropy production is a nonpositive measure supported on $x=\frac{3}{4}t$ for every convex $C^2$ pair. The Kruzhkov pairs ${\eta }_k(s)=\mathrm{\mid }s-k\mathrm{\mid }$ are obtained by uniform approximation on the bounded range $[-1,1]$ by the smooth convex pairs ${\eta }^{\delta }(s)=\sqrt{(s-k{)}^2+{\delta }^2}-\delta \to \mathrm{\mid }s-k\mathrm{\mid }$ with fluxes $q^{\delta }(s)={\int }_k^s({\eta }^{\delta }{)}^{\mathrm{\prime }}(z)f^{\mathrm{\prime }}(z)dz\to \mathrm{sgn}(s-k)(f(s)-f(k))$, so all Kruzhkov inequalities hold and, with the weak equation and trace of step 2.1, $u$ is a Kruzhkov entropy solution.

Uniqueness and the concave hull. By [F4], uniqueness in the entropy class identifies the constructed profile as the entropy solution of these Riemann data, even though the single-jump weak solution of the same data exists and is non-entropic. The concave hull of $f$ on $[-1,1]$ follows the cubic arc on $[-1,-\frac{1}{2}]$ (where $f^{\mathrm{\prime }\mathrm{\prime }}=6u<0$, so $f$ is concave) and then the chord of slope $\frac{3}{4}$ from $(-\frac{1}{2},-\frac{1}{8})$ to $(1,1)$; the fan and shock of the profile are exactly the entropy waves corresponding to this arc and chord.