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I studied the following theorem: (Rankine-Hugoniot condition)

Let $u:\mathbb{R} \times [0,+\infty) \to \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution of the conservation law if and only if the two of the following conditions are satisfied:

  1. $u$ is a classical solution of in the domain where $u$ is $C^1$

  2. $u$ satisfies the jump condition

$$(u_+−u_−)\eta_t+\sum_{j=1}^d f_j(u_+)−f_j(u_−)\eta_x=0$$

My doubts:

  1. Can we get the RH conditions for weaker assumptions on the solutions?

  2. If the solution $u\in BV$ then we know that $u_+$ and $u_-$ exist (though may not be equal); in such a case, can we apply RH condition? For example, let $u$ be a measurable function which is a weak solution of the conservation law, can we say that for any real number $a$, $u(a+,t)=u(a-,t)$ for a.e $t>0$?

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