The Burgers rarefaction Riemann solution

Specialisation of the solver. By [F2], $f^{\mathrm{\prime }}(u)=u$ and $(f^{\mathrm{\prime }}{)}^{-1}(\xi )=\xi$; the three branches of the strictly convex Riemann solver [F1] read $u_L$ for $x\mathrm{/}t\le u_L$, $x\mathrm{/}t$ for $u_L<x\mathrm{/}t<u_R$, and $u_R$ for $x\mathrm{/}t\ge u_R$, which is exactly the displayed centred rarefaction. Hence by [F1] it is the unique Kruzhkov entropy solution of the Riemann problem, satisfies the weak conservation law, all Kruzhkov entropy inequalities, and attains the Riemann datum in the strong local $L^1$ sense.

Direct check of the middle branch and the interfaces. On the fan, $u(t,x)=x\mathrm{/}t$, so $u_t=-x\mathrm{/}{t}^2$, $u_x=1\mathrm{/}t$ and $u_t+u{u}_x=-x\mathrm{/}{t}^2+(x\mathrm{/}t)(1\mathrm{/}t)=0$ by the chain rule [F3]; on the two outer regions $u$ is constant, so both derivatives vanish there. At $x=u_{L}t$ the three-branch formula gives $u_L$ from both the first and middle branches, and at $x=u_{R}t$ it gives $u_R$ from both the middle and last branches; the traces of $u$ and of $f(u)$ therefore agree across the rays, so by [F3] no interface terms arise in the weak residual and the profile is a distributional weak solution.

Entropy production and regularity. For any convex $C^2$ pair $(\eta ,q)$ with $q^{\mathrm{\prime }}={\eta }^{\mathrm{\prime }}{f}^{\mathrm{\prime }}$, the chain rule gives ${\mathrm{\partial }}_t\eta (u)+{\mathrm{\partial }}_{x}q(u)={\eta }^{\mathrm{\prime }}(u)(u_t+f^{\mathrm{\prime }}(u)u_x)={\eta }^{\mathrm{\prime }}(u)(u_t+u{u}_x)$ on each smooth branch; this vanishes on the fan by step 1.2 and on the outer branches because $u$ is constant. Across the rays the traces of $\eta (u)$ and $q(u)$ agree because $u$ is continuous there, so no interface measure arises: the entropy production is identically $0$ on $\mathbb{R}\times (0,\mathrm{\infty })$ for every convex $C^2$ pair. (For the non-smooth Kruzhkov pairs, the entropy inequalities are supplied by the solver [F1].) On the fan $u_x=1\mathrm{/}t$, so the profile is locally Lipschitz on every compact subset of the open strip $t>0$; no uniform Lipschitz bound as $t\downarrow 0$ is claimed.

Initial trace and the special case. The discrepancy from the initial step is supported between $\min \{0,u_L\}t$ and $\max \{0,u_R\}t$, and is bounded by $u_R-u_L$. Thus ${\int }_K\mathrm{\mid }u(t)-u_0\mathrm{\mid }\le (u_R-u_L)(\max \{0,u_R\}-\min \{0,u_L\})t\to 0$. When $u_L=0$, $u_R=1$, the fan is $x\mathrm{/}t$ on $0<x<t$, with outer states $0$ and $1$. For nonsmooth convex pairs, smooth convex approximation and uniform convergence of the integral fluxes pass the zero-production identity of step 2.1 to the limit; thus production is zero, not merely nonpositive.