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