I am trying to find the paper/book which studies problem (1),(3) given bellow using the vanishing viscosity method. I am especially interested in solutions in Sobolev spaces. 
	
**More detailed 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), we use the same initial data $u_0$, if they are smooth enough. Otherwise, if they are not smooth enough, we take smooth enough version $u_0^{\epsilon}$ (of course here it could be some other paramether instead of $\epsilon$; but that would be more complicated case). 

Riemann initial data are given with:
	
$$(3) \hspace{1cm}
u(x,0)= \begin{cases}
u_l, x<0 \\[2ex]
u_r, x\geq 0.
\end{cases}$$
	

**I am interested in** problems where we change initial data (3) to the smooth one, solve problem (2) with the smooth data (we get them for example, by convolution of Riemann data with a mollifier). At the end we connect it with the original problem (1), (3) by letting $\epsilon \rightarrow 0$. 


So far I have found three works that study this problem but the solutions are not in the Sobolev spaces: 

1.["Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data" - D. Hoff, 1995][1]

2.["Vanishing viscosity solutions of nonlinear hyperbolic systems" - S. Bianchini, A. Bressan, 2005][2]

3.["On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding" - W. Shen, 2017][3]


  [1]: https://www.sciencedirect.com/science/article/pii/S0022039685711114
  [2]: http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n1-p06.pdf
  [3]: https://link.springer.com/content/pdf/10.1007%2Fs00030-017-0461-y.pdf