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