The question that I hope to find some answer here is: do the results from

Bianchini, Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, 2005

paper still apply if we change a little some assumptions. Let me explain. The main results of this paper are given in Chapter XV of the book

Dafermos, Hyperbolic conservation laws in continuum physics, 2016

(in the Theorem 15.1.1 to be precise), and for the sake of simplicity I will use that source. In there we have the Cauchy Problem: $$ \begin{align} &U_t + \mathrm{div} F(U)=0, \tag{1}\label{1}\\ &U(x,0) = U_0(x),\tag{2}\label{2} \end{align} $$ where $-\infty < x < \infty$, $0<t<\infty$ and $U(x,t)\in \mathbb{R}^n$. This system is strictly hyperbolic in a ball $\mathcal{O}\subset \mathbb{R}^n$, centered at a certain state $U^*$, and initial data $U_0$ is of bounded variation on $(-\infty, \infty)$ such that $U_0 (-\infty)=U^*$.

We also have the parabolic problem: $$ U_t + \ F(U)=\mu U_{xx}\tag{3}\label{3}$$

The aim is to construct BV solutions to \eqref{1}, \eqref{2} as the $\mu \downarrow 0$ limit of solutions to \eqref{3}, \eqref{2}. This is done in the Theorem 15.1.1 where we could find unique solution of \eqref{3}, \eqref{2} and the convergence when $\mu \downarrow 0$.

I was wondering, could we apply this theorem and its results in the following cases:

• If instead of $-\infty < x < \infty$ we have $x \in [a,b] \subset \mathbb{R}$? In this case we would have $U_0(a)=U^*$ I assume?
• If we have a finite number of points where we don't have a strict hyperbolicity? For example a system that has just one point where the strict hyperbolicity is lost.
• If we combine two previous questions: Let's say that $-\infty < x < \infty$, but the support of $U_0$ is $[-a,a]$ - so for $x \notin [-a,a]$ the initial conditions are zero. And let's say that the system is not strictly hyperbolic in the origin i.e. in $U=0$. In this case $U^*$ would be equal to zero and at the same time the zero is the point where we do not have strict hyperbolicity.

This three questions are concerned with some work of mine during past month on a few different systems. I noticed that Bianchini and Bressan's result would be perfect fit for if I could use some of these modifications. If something wasn't clear, let me know and I will change it. Thanks in advance for the help.