On the shift differentiability of the flow generated by a hyperbolic system of conservation laws

We consider the notion of shift tangent vector introduced in [7] for real valued BV functions and introduced in [9] for vector valued BV functions. These tangent vectors act on a function $u\in L^1$ shifting horizontally the points of its graph at different rates, generating in such a way a continuous path in $L^1$. The main result of [7] is that if the semigroup $\mathcal S$ generated by a scalar strictly convex conservation law is shift differentiable, i.e. paths generated by shift tangent vectors at $u_0$ are mapped in paths generated by shift tangent vectors at $\mathcal S_t u_0$ for almost every $t\geq 0$. This leads to the introduction of a sort of differential, the "shift differential", of the map $u_0 \to \mathcal S_t u_0$.
In this paper, using a simple decomposition of $u\in $BV in terms of its derivative, we extend the results of [9] and we give a unified definition of shift tangent vector, valid both in the scalar and vector case. This extension allows us to study the shift differentiability of the flow generated by a hyperbolic system of conservation laws.