The Riemann problem for Burgers' equation $u_t +(f(u))_x = 0$, where $f(u)=\frac{1}{2}u^2$, has a shock solution: $$ u(x,t) = \left\lbrace \begin{aligned} &u_L &&\text{if}\quad x<st \\ &u_R &&\text{if}\quad x>st \, . \end{aligned} \right. $$ with Rankine-Hugoniot jump condition for $u_L > u_R$ $$s = \frac{f (u_R) − f (u_L)}{u_R - u_L} $$
My question is that does this solution satisfy Riemann problem for any general Scalar Conservation Law $u_t +(f(u))_x = 0$?
Yes, it does, since it is obviously a solution on either side of the shock and satisfies the Rankine-Hugoniot condition. The nonlinearity $f$ only affects the shock. Of course this is not the case for non-piecewise constant solutions.