An 𝐿^{𝑝} shock admissibility condition for conservation laws

We estimate the L p L^p ( p > 0 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 L^p local contraction of such solutions. Moreover, as an application, we prove that there exist L p 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 L^q ( q ≥ 1 q\geq 1 ) local contraction and shock waves have an L r L^r ( 0 > r ≤ 1 0>r\leq 1 ) local contraction.