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 ∈ L1 shifting horizontally the points of its graph at different rates, generating in such a way a continuous path in L1. The main result of [7] is that if the semigroup S generated by a scalar strictly convex conservation law is shift differentiable, i.e. paths generated by shift tangent vectors at u0 are mapped in paths generated by shift tangent vectors at Stu0 for almost every t ≥ 0. This leads to the introduction of a sort of differential, the "shift differential", of the map u0 → Stu0. In this paper, using a simple decomposition of u ∈ 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.