3.1.6 Expansion waves

Smooth regions with

$$\frac{\partial v}{\partial x} > 0$$ (171)

produce a rarefaction wave or expansion wave.

• What about steps with $v_{\mbox{\footnotesize left}} < v_{\mbox{\footnotesize right}}$?
• Expansion shock with characteristics running out of it is weak solution.
• But: any small but non-zero viscosity would smooth the step and cause a rarefaction wave (or rarefaction fan).
• Only solutions that fulfil an entropy condition are allowed.

Lax entropy condition: For a convex scalar conservation law, a discontinuity propagating with speed $s$ satisfies the Lax entropy condition if

$$v_{\mbox{\footnotesize left}} > s > v_{\mbox{\footnotesize right}} \enspace .$$ (172)

• Therefore: expansion shocks are not allowed.
• An entropy condition destroys time-reversibility.