In the book by R. LeVeque: "Numerical methods for conservation laws", Birkhauser, (1992), 2nd edition, in the Subsection 3.1.2 called "Nonsmooth data", the author talks about possibilities for finding generalized solutions of the conservation laws when we have nonsmooth initial data. On $22^{nd}$ page he says: "One possibility is to approximate nonsmooth data $u_0(x)$ by a sequence of smooth functions $u^{\epsilon}_0(x)$, with $\parallel u^{\epsilon}_0(x)- u_0 (x) \parallel_{L^1} <\epsilon \: \mbox{as} \: \epsilon \rightarrow 0$... Unfortunately, this approach of smoothing the initial data will not work for nonlinear problems."
My (nonlinear) problem is this: let's say we have conservation law $$\begin{cases} u_t+f(u)_x=0, \\[2ex] u(x,0)=u_0 (x), \end{cases}$$
where $u_0$ is nonsmooth Riemann initial data $$u_0(x)= \begin{cases} u_l, x<0, \\[2ex] u_r, x>0, \end{cases}$$ where $u_l$ and $u_r$ are constants.
We approximate the Riemann data with a smooth data that depends on some parameter $\epsilon$, i.e. we change $u_0$ with $u^{\epsilon}_0(x)$. In my case, the new initial data $u^{\epsilon}_0(x)$ are in the Sobolev space $H^s(\mathbb{R})$, where $s$ is integer bigger than five. This is now a Cauchy problem with smooth data that we can solve. We find solutions of the approximate problem in some Sobolev-valued $H^s(\mathbb{R})$ space (e.g. $C([0,T];H^s(\mathbb{R}))$, $L^p(0,T,H^s(\mathbb{R}))$ or similar). And now we have a solution of approiximate problem that depends on parameter $\epsilon$. In order to get back on the initial problem we should let $\epsilon\rightarrow 0$.
My question is: what do we need in order to pass to the limit $\epsilon\rightarrow 0$ (and then get a connection with a solution of the Riemann problem)? Generally I've always thought that this is possible either directly, by using some compactness lemma or with some energy estimates, but I am not sure how to do it in this case. Here on the one side we have a classical smooth solution, and on the other side, after we let $\epsilon\rightarrow 0$, we should get discontinuous (weak) solution possibly in some other space. So we have different type of solutions.
Also if anyone know any paper/book where pde problem similar to this is studied let me know. I am interested in problems where we change initial condition to the smoooth one, than solve the "approximation problem" and than connected that solution with the solution of the original problem. It could be by letting $\epsilon\rightarrow 0$ directly or using some theorem.