Structure of Entropy Solutions to General Scalar Conservation Laws in One Space Dimension

In this paper, we show that the entropy solution u(t) of a scalar conservation law enjoys the properties: • u is continuous outside a 1-rectifiable set Ξ⊂ R+ × R, • up to a countable set, for each point (t̄, x̄) ∈ Ξ there exists two cone shaped regions arbitrarily close to half planes where u is left/right continuous at (t̄, x̄). We provide examples showing that these estimates are nearly optimal. In order to achieve these regularity results, we extend the wave representation of the wavefront approximate solutions to entropy solution. This representation can be interpreted as some sort of Lagrangian representation of the solution to the nonlinear scalar PDE, and implies a fine structure on the level sets of u(t). Preprint SISSA 11/2014/MATE