SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension

We prove that if $t \mapsto u(t) \in \BV(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields \[ u_t + f(u)_x = 0, \] then up to a countable set of times $\{t_n\}_{n \in \N}$ the function $u(t)$ is in SBV, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families \[ u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \R^N, \] and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure $\mu_{i,\jump}$ is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not $0$, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $\mu_{i,\jump}$ is positive.