# entropy – Rarefaction wave is an entropic solution!!

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.

So my question is why are rarefaction waves considered as

1)entropic solutions
2)unique

Although the case of rarefaction waves came after the function being non-entropic!!