Solution
= Solution
The <Rankine-Hugoniot condition> for the flux $f(u)=u^2/2$ gives, for distinct states,
$$
\boxed{\dot s=\frac{f(u_-)-f(u_+)}{u_--u_+}=\frac{u_-+u_+}{2}}.
$$
The <characteristic speed> is $f'(u)=u$. A causal <entropy shock> has characteristics entering from both sides, so $u_->\dot s>u_+$, equivalently
$$
\boxed{u_->u_+}.
$$
An upward jump instead resolves into a <rarefaction wave>; the jump condition alone does not select the physical weak solution.