Radon measure-valued solutions of first order scalar conservation laws

We study nonnegative solutions of the Cauchy problem ∂ t u + ∂ x [ φ (u) ] = 0 in × (0, T), u = u 0 ≥ 0 in × 0 , \left\\beginaligned &\displaystyle\partial-tu+\partial-x[\varphi(u)]=0&% &\displaystyle\phantom\textin \mathbbR\times(0,T),\\ &\displaystyle u=u-0\geq 0&&\displaystyle\phantom\textin \mathbbR% \times\0\,\endaligned\right. where u 0 u-0 is a Radon measure and φ: [ 0, ∞) → \varphi\colon[0,\infty)\mapsto\mathbbR is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on φ, we prove their uniqueness if the singular part of u 0 u-0 is a finite superposition of Dirac masses. Regarding the behavior of φ at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case φ (u) = u \varphi(u)=u this happens for all times). In the latter case, we describe the evolution of the singular parts.