SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one Space Dimension - Bianchini, S; Caravenna, L (2012)
We prove that if t {mapping} u(t) ∈ BV (ℝ) is the entropy solution to a N × 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∈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, 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 μ 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 μ i,jump is positive. © 2012 Springer-Verlag.