Rankine–Hugoniot conditions

Remarks

Write $[\,\cdot\,]$ for the jump across the curve; then the condition is $[F(u)] = \sigma [u]$ with shock speed $\sigma = s'$. The proof tests the weak formulation with a test function supported near a point of $C$. On each side $u$ is a classical solution, so integrating by parts on $V_l$ and on $V_r$ leaves only boundary integrals over $C$. These combine to $\int_C \big( [F(u)] - s' [u] \big) \varphi \, dt = 0$ for all test functions, which gives the jump condition.

Example: for the scalar Burgers' equation, $F(u) = u^2/2$, the condition gives $s' = \dfrac{u_l^2/2 - u_r^2/2}{u_l - u_r} = \dfrac{u_l + u_r}{2}$. So a jump from $u_l = 2$ to $u_r = 0$ moves at speed $1$.

The conditions are necessary but do not select a unique solution. A jump from $0$ to $1$ for Burgers' equation also satisfies them and gives a weak solution, but not the physical one. An entropy condition is needed in addition. In several space dimensions the same argument gives $\sum_j [F_j(u)]\, \nu_{x_j} = - \nu_t [u]$, where $(\nu_x, \nu_t)$ is a normal to the surface of discontinuity.

Going deeper

For a student

The theorem shows that piecewise smooth weak solutions are exactly the piecewise classical solutions whose jumps obey the Rankine–Hugoniot relations. It is used to construct explicit solutions, for example of the Riemann problem, and to check proposed shock solutions. Since it allows non-physical jumps, it is always combined with an entropy condition such as the Lax condition, which leads to the notion of entropy solution. For a system, the set of states $u_r$ that can be joined to a given $u_l$ is the Hugoniot locus. For a strictly hyperbolic system, near $u_l$ it consists of $m$ curves through $u_l$.