SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one Space Dimension

We prove that if is the entropy solution to a N x 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 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) = N Sigma(N)(i=1)v(i)(t)(r) over tilde (i)(t), v(i)(t) is an element of M(R), (r) over tilde (i)(t) is an element of 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.