A note on singular limits to hyperbolic systems of conservation laws

The paper concerns with a hyperbolic system of conservation laws in one space variable $$ u_t + f(u)_x = 0,\qquad u(0,x) = u_0(x), $$ where $ u \in \Bbb R^n$, $f:\Omega \subseteq \Bbb R^n \rightarrow \Bbb R^n.$ Let $ K_0 \subset \Omega $ be a compact and let $\delta_1 > 0 $ be sufficiently small such that $K_1 = \{ u \in \Bbb R^n: \text{dist}(u,K_0) \leq \delta_1\}\subset \Omega.$ \par Assuming that the Jacobian matrix $A = Df$ is uniformly strictly hyperbolic in $K_1, u_0(-\infty) \in K_0$ and that the total variation of $u_0$ is sufficiently small, then there exists a unique ``entropic" solution $u: [0,+\infty) \rightarrow BV(\Bbb R,\Bbb R^n).$