Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a strictly convex function.
And $u:\mathbb{R}×[0,+\infty[ \rightarrow \mathbb{R}$
where $u(x,0)=u_l$ if $x \lt 0$ and $u(x,0)=u_r$ if $x \gt 0$
And consider the conservation law equation:
$\frac{\partial u}{\partial t}+\frac{\partial f(u)}{\partial x}$, $x \in \mathbb{R}$, $t \gt 0$.
We know that if u does not satisfy the entropy condition $u_r \lt u_l$, then the rarefaction wave occur, and it is continuous.
So my question is why are rarefaction waves considered as
1)entropic solutions 2)unique
although the case of rarefaction waves came after the solution being non-entropic!!