A Riemann shock and a rarefaction of the scalar model: explicit weak solutions, entropy coefficients and initial trace

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the cited Lebesgue weak/entropy formulation.

Let $F(u)=\frac{1}{2}{u}^2$ and use the weak and entropy formulations of Weak and entropy solutions of the scalar conservation law $u_t+(u^2\mathrm{/}2{)}_x=0$, with the scalar field $u$ read as a dimensionless model variable. The coordinates $t,x$, the states $u_L,u_R$ and flux $F(u)$ are all dimensionless; no assignment to a measured density, velocity, length or time is made. Then:

1. (Shock.) For the Riemann data $u_L=2$, $u_R=1$ the piecewise constant field $u(t,x)=2$ for $x<\frac{3}{2}t$ and $u(t,x)=1$ for $x>\frac{3}{2}t$ $(t>0)$, taking the right-state value on the shock line, is a weak solution of $u_t+(F(u){)}_x=0$: its interface residual is $-s[u]+[F(u)]=-\frac{3}{2}\cdot (1-2)+\frac{1}{2}(1^2-2^2)=\frac{3}{2}-\frac{3}{2}=0$ with $s=\frac{3}{2}$, so the weak identity (W) is satisfied for every test function by the split integration over the two half-planes.
2. (Entropy of the shock.) For every $k\in (1,2)$ the interface coefficient of the pair $({\eta }_k,q_k)=(\mid u-k\mid ,\mathrm{sign}(u-k)(u^2-k^2)\mathrm{/}2)$ is $s[{\eta }_k]-[q_k]=-(k-2)(k-1)=(2-k)(k-1)>0$, so the shock satisfies the entropy inequality with strictly positive interface production on this range of $k$; the integrated inequality is strict only for nonnegative tests with ${\int }_0^{\mathrm{\infty }}\phi (t,st)dt>0$; for $k\mathrm{\notin }[1,2]$ the coefficient vanishes and the inequality holds with equality.
3. (Rarefaction.) For the Riemann data $u_L=1$, $u_R=2$ the field $u(t,x)=1$ for $x\le t$, $u(t,x)=x\mathrm{/}t$ for $t<x<2t$ and $u(t,x)=2$ for $x\ge 2t$ $(t>0)$ is continuous on $(0,\mathrm{\infty })\times \mathbb{R}$, satisfies $u_t+(F(u){)}_x=0$ classically inside the fan, and has initial trace $f$: ${\int }_{\mathbb{R}}\mid u(t,x)-f(x)\mid dx\le \frac{3}{2}t\to 0$ as $t\downarrow 0$.
4. (Weak kinks are unavoidable here.) The shock profile is discontinuous across $x=\frac{3}{2}t$ and the rarefaction is not differentiable across $x=t$ and $x=2t$; the classical derivation of the transport and balance identities of the continuum page does not apply at such sets, which is exactly why the weak formulation of the model is used. The admissibility of the shock and the equality case of the rarefaction are those of Riemann shock and rarefaction waves for $u_t+(u^2\mathrm{/}2{)}_x=0$ are weak solutions, and the shock is an entropy solution.

• Physical reading. The dimensionless scalar field $u$ is a model variable of an idealized one-dimensional transport problem; it is not claimed to be the velocity or density of a three-dimensional compressible fluid, and no derivation of the scalar equation from the momentum and mass balances of this page is asserted. The example shows that a scalar continuum model can have nondifferentiable weak solutions whose admissibility is controlled by entropy inequalities, in contrast with the smooth motions required by the transport and balance theorems of this page.

• Admissibility is not automatic. Both the shock and the fan are weak solutions; only the entropy inequalities select the physically admissible one among weak solutions of this model family. The computed coefficients show the shock is admissible for the stated data and the fan is the equality case. No statement is made about entropy solutions for general initial data or for systems of conservation laws.