I studied the following theorem:(Rankine-Hugoniot condition)
Let u:ℝ×[0,+∞)→ℝ u : R × [ 0 , + ∞ ) → 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:
i) u is a classical solution of in the domain where u u is $C^1$
ii) u satisfies the jump condition
$(u_+−u_−)\eta_t+∑_{j=1}^d f_j(u+)−f_j(u−)η_x=0$
My Doubts: i) Can we get the RH conditions for weaker assumptions on the solutions?
ii)If the solution $u\in BV$ then we know that $u_+$ and $u_-$ exists(though may not be equal) in such 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 $t>0$