= Solution
A bounded <weak solution> satisfies
$$
\boxed{\int_0^\infty\!\int_{\mathbb R}\bigl(u\varphi_t+f(u)\varphi_x\bigr)dx\,dt+\int_{\mathbb R}u_0(x)\varphi(0,x)dx=0}
$$
for every compactly supported $C^1$ <test function>, of either sign, allowed to meet $t=0$. Across a moving jump $x=s(t)$ with distinct left/right constant states, the distributional delta coefficient vanishes exactly when the <Rankine-Hugoniot condition> holds:
$$
\boxed{s'(t)(u_R-u_L)=f(u_R)-f(u_L).}
$$
It is required on every interface of a piecewise constant solution; the initial trace must also agree with the data.
For nonuniqueness respecting the paper's global bound on $f'$, use the <bounded-derivative Burgers flux> $f(z)=z^2/2$ on $|z|\le1$ and $f(z)=|z|-1/2$ outside. It is $C^1$ and $|f'|\le1$. Take data $-1$ for $x<0$, $1$ for $x>0$. A stationary jump obeys the condition because $f(-1)=f(1)$. Another weak solution is the <rarefaction wave>
$$
u(t,x)=\begin{cases}-1,&x\le-t,\\x/t,&-t<x<t,\\1,&x\ge t.\end{cases}
$$
The fan solves the equation inside each smooth region, is continuous at its edges, and has the same local $L^1$ initial trace. Thus \b[weak solutions are not unique]. The stationary expansion shock violates the entropy inequality, whereas the fan is admissible; using the unmodified quadratic flux alone would miss the printed global bounded-derivative hypothesis.
Back to article page
