Riemann shock and rarefaction waves for $u_t+(u^2\mathrm{/}2{)}_x=0$ are weak solutions, and the shock is an entropy solution

On each of the two open regions $x<st$ and $x>st$ the function $u$ of (a) is constant, so the equation holds classically there. The traces across $x=st$ are $u_{-}=u_L$ (left) and $u_{+}=u_R$ (right), and the initial trace is the Riemann data $f$. By [F1], the weak residual of (a) integrates $\phi (t,st)$ against $s[u]-[F(u)]=s(u_R-u_L)-\frac{1}{2}(u_R^2-u_L^2)=(u_R-u_L)(s-\frac{u_L+u_R}{2})=0.$ Hence the weak identity (W) holds for every test function, and $u$ is a weak solution.

For (c), first integrate on $\epsilon <t<T$, splitting the truncated domain into the left constant region, the fan, and the right constant region, with $T$ and the spatial cutoff outside the support of the test function. There $u=x\mathrm{/}t$ is $C^1$ on the closed fan and satisfies $u_t+u{u}_x=0$ by [F3]; the constant pieces do too. Apply the divergence theorem to each piece. On a shared fan edge the outward normals are opposite and the traces of $u$, $F(u)$ and $\phi$ agree, so the two boundary integrals cancel. The bottom term is $-\int u(\epsilon ,x)\phi (\epsilon ,x)dx$. As $\epsilon \downarrow 0$, $u(\epsilon ,x)$ differs from $f(x)$ only between the rays $x=u_L\epsilon$ and $x=u_R\epsilon$, an interval of length $(u_R-u_L)\epsilon$, and it is uniformly bounded by $\max (\mathrm{\mid }{u}_L\mathrm{\mid },\mathrm{\mid }{u}_R\mathrm{\mid })$. Thus the difference in the bottom integral tends to zero; continuity of $\phi$ on its compact support replaces $\phi (\epsilon ,x)$ by $\phi (0,x)$. The integrands over $0<t<\epsilon$ are bounded on a fixed compact spatial support, so their integrals also tend to zero. Taking the limit gives (W) with the actual initial term.

For (b), apply the same computation to the pair $({\eta }_k,q_k)$ in place of $(u,F(u))$: on each open region the function $u$ is constant, so $({\eta }_k(u){)}_t+(q_k(u){)}_x=0$ classically there, and [F1] gives ${\int }_0^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathbb{R}}[{\eta }_k{\phi }_t+q_k{\phi }_x]+{\int }_{\mathbb{R}}{\eta }_k(f)\phi (0)={\int }_0^{\mathrm{\infty }}\phi (t,st)(s[{\eta }_k]-[q_k])dt\mathrm{.}$ For $u_R<k<u_L$ a direct evaluation gives $[{\eta }_k]=2k-u_L-u_R$ and $[q_k]=(2k^2-u_L^2-u_R^2)\mathrm{/}2$, whence $s[{\eta }_k]-[q_k]=-(k-u_L)(k-u_R)\ge 0$. For $k\ge u_L$ or $k\le u_R$ the same expressions give $s[{\eta }_k]-[q_k]=0$. In every case the right-hand side is $\ge 0$ for nonnegative test functions, which is (E).

Let $(\eta ,q)$ be any convex $C^1$ entropy pair. On either side of the shock the entropy residual vanishes classically, and [F1] applied to $(\eta (u),q(u))$ gives the residual coefficient $s[\eta ]-[q]={\int }_{u_R}^{u_L}(z-s){\eta }^{\mathrm{\prime }}(z)dz={\int }_{u_R}^{u_L}(z-s)({\eta }^{\mathrm{\prime }}(z)-{\eta }^{\mathrm{\prime }}(s))dz\ge 0.$ The middle equality uses $s=(u_L+u_R)\mathrm{/}2$, hence ${\int }_{u_R}^{u_L}(z-s)dz=0$. For any $a<b$, convexity gives $\eta (a+h)\le (1-h\mathrm{/}(b-a))\eta (a)+(h\mathrm{/}(b-a))\eta (b)$ for $0<h<b-a$; dividing by $h$ after subtracting $\eta (a)$ and taking $h\downarrow 0$ yields ${\eta }^{\mathrm{\prime }}(a)\le (\eta (b)-\eta (a))\mathrm{/}(b-a)$. Applying convexity to $b-h$ and taking the same limit gives $(\eta (b)-\eta (a))\mathrm{/}(b-a)\le {\eta }^{\mathrm{\prime }}(b)$. Thus ${\eta }^{\mathrm{\prime }}$ is nondecreasing, so $(z-s)({\eta }^{\mathrm{\prime }}(z)-{\eta }^{\mathrm{\prime }}(s))\ge 0$ for every $z$. This proves (E) for every declared convex $C^1$ pair, in addition to the Kružkov computation of step 2.1.

On the rarefaction fan every $C^1$ entropy pair has ${\mathrm{\partial }}_t\eta (u)+{\mathrm{\partial }}_{x}q(u)={\eta }^{\mathrm{\prime }}(u)(u_t+u{u}_x)=0$; its traces are continuous across the fan edges. For a Kružkov pair split additionally along $x=kt$ when that line lies inside the fan; the two entropy and flux traces agree there too. Apply the truncated-domain argument of step 1.2 to these entropy fields. Since $u$ ranges in the compact interval $[u_L,u_R]$, $\eta (u)$ and $q(u)$ are bounded and the initial difference is supported on an interval of length $(u_R-u_L)\epsilon$; the same limit establishes (E) with equality and its nonzero initial trace for every stated pair. Both profiles are therefore entropy solutions for the full class of the Definition, with the asserted additional Kružkov identities.