TITLE: The Rankine-Hugoniot jump conditions for conservation and balance laws
QUESTION [9 upvotes]: Consider two problems:
$$(1) \hspace{1cm}    
u_t+f(u)_x  = 0,
$$
$$(2) \hspace{1cm}    
u_t+f(u)_x  = g(u).
$$
Problem (1) represents system of conservation laws, and problem (2) represents system of balance laws (or conservation laws with source term). If we have discontinuous initial data such as
$$(3)\hspace{1cm}  u(x,0)= \begin{cases}
u_l, x<0 \\[2ex]
u_r, x>0
\end{cases}$$
than we talk about Riemann problems. One type of solutions of (1),(3) and (2),(3) are shock waves given by:
$$(4)\hspace{1cm}  u(x,t)= \begin{cases}
u_l, x<s \cdot t \\[2ex]
u_r, x>s \cdot t
\end{cases}$$
where $s$ is speed of the shock. We calculate it from the Rankine-Hugoniot conditions. In order not to complicate things, here $u_l$ and $u_r$ are constants, so the shock speed $s$ is a constant. 
For problem (1),(3) Rankine-Hugoniot conditions are given by:
$$s\cdot (u_r - u_l) = f(u_r)- f(u_l)$$
I am interested in problem (2),(3). By  "C. Dafermos, Hyperbolic conservation laws in continuum physics, 2016",
the fact that we have a source doesn't change nothing, i.e. Rankine-Hugoniot conditions are again given by 
$$s\cdot (u_r - u_l) = f(u_r)- f(u_l)$$
and speed is the same than. (See start of a Chapter 3)
How is that possible? In my head it sounds reasonable that source affects the speed of the shock. At least I would expeced that speed  changes with t, i.e. now we would have $s(t)$. 
Also, in this paper
the authors included source in the Rankine-Hugoniot conditions (see (12),(13),(14) in this paper). Similar things could be found in various papers that do numerical solving of pdes.
So my questions are:
How do Rankine-Hugoniot conditions look in the case of balance laws given above and how do we calculate the speed of the shock wave in that case?
To be honest I maybe have misunderstood something in the C. Dafermos book. Maybe the author just wanted to say that the form of the Rankine-Hugoniout conditions stays the same. And the speed is changing because the source affects left and right states. (That sounds reasonable to me)
Additionaly, I would like to know also what happens in the problems where source in $(2)$ isn't given with $g(u)$. Instead of $g(u)$ it could be written anything alse (e.g. $g(u)$=constant or $g(u)=\partial_{xt}^2 u$ or whatever). Of course we talk about Rankine-Hugoniot conditions only in the case of discontinous solutions. If we take $g(u)=\triangle u$, as I recall, we have smooth solutions and we do can't talk about Rankine-Hugoniot conditions. 
I would really appreciate a help with this.

REPLY [2 votes]: As specified in this post the Rankine-Hugoniot condition which locally links the jumps to the shock speed is unchanged when an arbitrary r.h.s. $g(u)$ is included in the balance law. Now, let us consider the Riemann problem -- i.e. the initial value problem (3) -- and assume that the admissible solution is a shock wave. The method of characteristics makes it possible to identify the values of $u$ on both sides of the shock path $x = s(t)$. To illustrate this feature analytically, consider $g(u) = a u$. The method of characteristics gives
$$
u(x,t) = \left\lbrace
\begin{aligned}
& u_l e^{at}, & & x< s(t) ,\\
& u_r e^{at}, & & x> s(t) .
\end{aligned}\right.
$$
The first line is the left state in the Rankine-Hugoniot (RH) condition, and the second line is the right state in the RH condition. Thus, the RH condition yields
$$
s'(t) = \frac{f(u_r e^{at}) - f(u_l e^{at})}{u_r - u_l}
$$
with the initial condition $s(0) = 0$. Note that the speed $s'$ is constant if $a=0$.
If the r.h.s. consists of a differential operator, then it may be possible that the Riemann solution can't be a shock wave. The viscous Burgers equation $u_t + u u_x = \epsilon u_{xx}$ is such an example.