I am trying to find the paper/book where they study problem (1),(3) given bellow using Vanishing viscosity method. I am especially interested in solutions in Sobolev spaces. 
	
A little bit of explanation: From Chapter 10 of "Hyperbolic partial differential equations", P. Lax, 2006:
	
The vanishing viscosity method for solving the initial value problem for hyperbolic systems of conservation laws consists of finding solutions of
	$$(1) \hspace{1cm}    
	u_t+A(u)\cdot u_x  = 0
	\hspace{1cm} A_{ij}=\frac{\partial{f_i}}{\partial{u_j}},
	\hspace{1cm}
	u(x,0)=u_{0}(x)\\[2ex] 
	$$
	as the limit $\epsilon\rightarrow0$ of solutions of the parabolic system
	$$(2) \hspace{1cm}    
	u_t+A(u)\cdot u_x  = \epsilon u_{xx}.\\[2ex] 
	$$
	
In literature, when solving problem (2), somethimes we have the same initial data $u_0$, sometimes smooth enough version $u_0^{\epsilon}$. But what about if data are not smooth? For example, if we have Riemann data:
	
$$(3) \hspace{1cm}
u(x,0)= \begin{cases}
u_l, x<0 \\[2ex]
u_r, x\geq 0
\end{cases}$$
	
or some other discontinuous initial data. 	

**For me this would be ideal:** 
In the reference paper authors started with Riemann data. Than they solve problem (2) with original piecewise continuous Riemann data OR with continuous/smooth approximation of the data (for example, convolve Riemann data with a mollifier; or just use other approximation of a initial data).
They use that to solve problem (1),(3). So they would have some continuous/smooth/discontinuous data that goes to piecewise continuous/discontinuous data. 

Of course I accept any other reference anyone knows. :)

So far I have found two works that study this problem but the solutions are not in the Sobolev spaces:
1.["Vanishing viscosity solutions of nonlinear hyperbolic systems" - S. Bianchini, A. Bressan, 2005][1]
2.["On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding" - W. Shen, 2017][2]


  [1]: http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n1-p06.pdf
  [2]: https://link.springer.com/content/pdf/10.1007%2Fs00030-017-0461-y.pdf