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.
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).
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