 Solution

ID: past-exam-of-the-applied-partial-differential-equations-course-of-the-university-of-oxford/2023/2/a/solution

f (u) = u^{2} /2

gives, for distinct states,

\overset{s}{˙} = \frac{f ( u _{-} ) - f ( u _{+} )}{u _{-} - u _{+}} = \frac{u _{-} + u _{+}}{2} .

f^{'} (u) = u

. A causal entropy shock has characteristics entering from both sides, so

u_{-} > \overset{s}{˙} > u_{+}

, equivalently

u_{-} > u_{+} .

An upward jump instead resolves into a rarefaction wave; the jump condition alone does not select the physical weak solution.

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