An $L^p$ shock admissibility condition for conservation laws

We estimate the $L^p$ ($p>0$) local distance between piecewise constant solutions to the Cauchy problem of conservation laws and propose a shock admissibility condition for having an $L^p$ local contraction of such solutions. Moreover, as an application, we prove that there exist $L^p$ locally contractive solutions on some set of initial functions, to the Cauchy problem of conservation laws with convex or concave flux functions. As a result, for conservation laws with convex or concave flux functions, we see that rarefaction waves have an $L^q$ ($q\geq 1$) local contraction and shock waves have an $L^r$ ($0<r\leq 1$) local contraction.