I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the second order quasilinear wave equations:

Suppose a quasilinear wave equation defined on a manifold $(M,g(u))$, \begin{equation} \Box_{g(u)} u =F(u,u^\prime) \end{equation} where $\Box_{g(u)} :=(g^{-1}(u))^{\mu\nu}\nabla_\nu\nabla_\nu$ and the connnection $\nabla$ is compatible with the metric $g(u)$.

If we want to study its shocks, how should we define its weak solution? There might be several possible definitions of the weak solution with minor differences? But how to decide which one is best? I am confusing about this and hoping your comments and answers. Thanks in advance.

I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the second order quasilinear wave equations:

Suppose a quasilinear wave equation defined on a manifold $(M,g(u))$, \begin{equation} \Box_{g(u)} u =F(u,u^\prime) \end{equation} where $\Box_{g(u)} :=(g^{-1}(u))^{\mu\nu}\nabla_\nu\nabla_\nu$ and the connnection $\nabla$ is compatible with the metric $g(u)$.

If we want to study its shocks, how should we define its weak solution? There might be several possible definitions of the weak solution with minor differences. But how to decide which one is best? and how to decide the Rankine-Hugoniot condition and entropy condition?

I am confused about this and am hoping for your comments and answers. Thanks in advance.