Shock wave

Remarks

Because $u$ is a weak solution, the shock speed is forced by the Rankine–Hugoniot condition $s'(t) = \dfrac{F(u_l) - F(u_r)}{u_l - u_r}$. The entropy condition says that characteristics run into the shock from both sides, so the shock does not create information. For uniformly convex $F$ it is equivalent to $u_l > u_r$. For systems a shock of the $k$-th family is admissible when $\lambda_k(u_l) > s' > \lambda_k(u_r)$ together with conditions on the other eigenvalues.

Example: for Burgers' equation, $F(u) = u^2/2$, the function $u = 1$ for $x < t/2$ and $u = 0$ for $x > t/2$ has an admissible shock along $s(t) = t/2$. Check: the speed $1/2 = (\tfrac12 - 0)/(1 - 0)$ satisfies the Rankine–Hugoniot condition, which makes $u$ a weak solution, and $F'(1) = 1 > \tfrac12 > 0 = F'(0)$.

Non-example: the function $u = 0$ for $x < t/2$ and $u = 1$ for $x > t/2$ is also a weak solution of Burgers' equation with a jump along $s(t) = t/2$. It fails the entropy condition, since $F'(0) = 0 < \tfrac12$. The admissible solution with these initial values is a rarefaction wave instead.