Regularity estimates for Hamilton–Jacobi equations and hyperbolic conservation laws

Consider the Hamilton–Jacobi equation
$$ u_t+H(\nabla u)=0 $$
with convex Hamiltonian. In spite of the fact that the Hamiltonian is only convex, and thus the characteristic vector field $d$ is in general not differentiable, we will show that the vector field $d$ has enough regularity to allow a change of variable formula.
Applications of this fact are a proof of the Sudakov theorem in optimal transportation theory and a solution of a conjecture of Cellina. In the case where $H$ is uniformly convex, we will show that the solution is not only semiconcave, but its first derivative is SBV.
We will also consider the hyperbolic system
$$ u_t+f(u)_x=0 $$
and show that the direction of the characteristics are SBV.