Stationary shocks and contact discontinuities in conservation laws 1)What is a stationary shock or contact discontinuity?
I mean If I have $u_r<u_l$, then the entropy condition is satisfied and there exists a shock curve.
And if $u_r>u_l$, then the entropy condition is not satisfied unless I separated them by a rarefaction wave  $x/t$.
But when does a stationary shock exist?
2)Why in the scalar case the conservation law problem has a unique entropy solution, if it exists?
 A: *

*These are different concepts. Consider Burgers' equation $u_t + uu_x=0$, and a discontinuity with left/right states $u_l$, $u_r$ propagating at the speed (Rankine-Hugoniot) $$
s =\tfrac12(u_l+u_r) \, .
$$ $\blacksquare$ A stationary shock is a discontinuity located at constant position. Thus, the shock speed $s$ vanishes. For such a solution to be admissible, we need to check the Lax entropy condition $$
u_l > s > u_r
$$
which leads to the condition $u_l = -u_r > 0 $. These stationary shocks are a particular case of shock waves. $\blacksquare$ A contact discontinuity has its shock speed $s$ given by the Rankine-Hugoniot condition, and the characteristic speeds $u_l$, $u_r$ on both sides of the discontinuity are equal. Thus, the Lax entropy condition is not satisfied, so that contact discontinuities are not admissible in the case of Burgers' equation. However, such solutions can be obtained in the case of the linear advection equation $u_t+cu_x=0$, for which the characteristic speed $c$ is the same on both sides of the discontinuity.


*That's a theorem, see related literature where the proof is given.
