What are the Rankine Hugoniot Jump Conditions for quasilinear equations?

Question

When we consider a conservation law $$ q_t + f(q)_x = 0, $$ the Rankine-Hugonoit conditions are given by $$ s(q^+ - q^-) = f(q^+) - f(q^-) . $$ However, how does this change if we are given some quasilinear equation $$ q_t + g(q)q_x = 0, $$ that cannot be written in conservation form? In particular, say I have a system of equations $$ q_t + (q-s)_x = 0 $$ $$ s_t + s(q)_x = 0 $$ The usual derivation of the RH conditions involves integration by parts of the conservation term, but we see that $$ \int (sq)_x dx = [[sq]] = \int sq_x dx + \int qs_x dx , $$ and there is no obvious way to deal with the $\int qs_x dx$ term.

Answer 1

Shock wave solutions require that we introduce the concept of weak solutions. Then, the Rankine-Hugoniot conditions which link the shock speed $\sigma$ to the jumps of $f$ and $q$ can be derived. However, due to the nonconservative (quasilinear) form of the PDE, the standard notion of weak solution in the sense of distributions does not apply. In facts, the nonconservative product $g(q) q_x$ is the product between a jump discontinuity and a Dirac delta, which isn't a well-defined distribution. For functions of bounded variation, Dal Maso-Le Floch-Murat [1] proposed a notion of weak solution to quasilinear equations, see this post. They derive a generalized Rankine-Hugoniot jump relation of the form $$ \sigma (q^+ - q^-) = \int_0^1 g(\phi(\tau)) \frac{\partial \phi}{\partial \tau} \text d \tau $$ where $\phi: [0,1] \to \Bbb R$ is a Lipschitz path connecting $q^-$ and $q^+$, i.e. $\phi(0) = q^-$ and $\phi(1) = q^+$. For instance, let's consider the linear path $\phi: \tau\mapsto q^- + \tau\, (q^+ - q^-)$. If $g = f'$ is derived from a conservative equation, then the traditional Rankine-Hugoniot relation is recovered.

The same kind of equation can be written for nonconservative systems of the form ${\bf q}_t + {\bf g}({\bf q}){\bf q}_x = {\bf 0}$, where $$ \sigma ({\bf q}^+ - {\bf q}^-) = \int_0^1 {\bf g}({\boldsymbol\phi}(\tau)) \frac{\partial {\boldsymbol \phi}}{\partial \tau} \text d \tau \, . $$ Again, if ${\bf g} = \partial {\bf f}/\partial {\bf q}$ is derived from a conservative formulation with flux $\bf f$, then the above equation recovers the traditional Rankine-Hugoniot jump condition for systems of conservation laws.

[1] G Dal Maso, P Le Floch, F Murat: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483-548 (1995)