I am trying to find the papers/books/notes that study 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 of solutions of the parabolic system $$(2) \hspace{1cm} u_t+A(u)\cdot u_x = \epsilon u_{xx},\\[2ex] $$ when $\epsilon\rightarrow0.$
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 let's not complicate this more than it needs to be).
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 papers where the authors used this technique:
- First they transform the initial data $(3)$ to the smooth one $u_0^{\epsilon}$ (this can be done for example by convolution of Riemann data $(3)$ with a mollifier).
- Than they solve the problem $(2)$ with the smooth data $u_0^{\epsilon}$ using the vanishing viscosity method.
- At the end they connect that solution to the solution of the original problem $(1), (3)$ by letting $\epsilon \rightarrow 0$.
The papers that I've found and that satisfy this conditions are given bellow (but none of them has solutions in the Sobolev spaces):
2."Vanishing viscosity solutions of nonlinear hyperbolic systems" - S. Bianchini, A. Bressan, 2005
Help with this would be nice (there are just too much references in the literature that use the vanishing viscosity method and I can't read them all).
