The self-similar Riemann problem

Let $n=1$ and $f\in C^1(\mathbb{R})$ (Scalar conservation laws, fluxes and Cauchy data). The Riemann problem for $u_t+f(u{)}_x=0$ prescribes the two-state initial datum $u_0(x)=\{\begin{array}{ll}{u}_L,& x<0,\\ u_R,& x>0,\end{array}{u}_L,u_R\in \mathbb{R}\mathrm{.}$ One looks for self-similar solutions $u(t,x)=v(x\mathrm{/}t)$, $t>0$, that is, solutions invariant under the scaling $(t,x)\mapsto (\lambda t,\lambda x)$, $\lambda >0$; such a $u$ is determined by the single function $v\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$ and is constant on each ray $x=\xi t$.

The initial condition is read as the strong local $L^1$ trace $u(t,\cdot )\to u_0$ as $t\downarrow 0$ (Kruzhkov entropy solutions), and any jump or corner of $v$ occurs on a ray; admissibility is the entropy condition of Kruzhkov entropy solutions, not a further restriction on the self-similar ansatz. Self-similarity is an ansatz to be justified by the uniqueness theorem rather than an additional hypothesis. No choice principle occurs.