Past exam of the applied partial differential equations course of the University of Oxford / 2023 / 2 / a / Solution

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