Invariance property of a conservation law without convexity Mijoung KimYong-Jung Kim 35K5735K6535L6535B65asymptotics in timecharacteristicsconvergence order$N$-wavepotential comparisonsimilarity The main goal of this paper is to investigate the mechanism of a conservation law that gives the $N$-wave like asymptotics. It turns out that the positivity of the flux function provides a certain invariance of solution which singles out the right asymptotics among two parameter family of $N$-waves. Two kinds of long time asymptotic convergence orders in $L^1$-norm to this $N$-wave are proved using a potential comparison technique. The first one is of the magnitude of the $N$-wave itself and the second one is of order $1/t$. We observe that these asymptotic convergence orders are related to space and time translations of potentials. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3495 10.1512/iumj.2009.58.3495 en Indiana Univ. Math. J. 58 (2009) 733 - 750 state-of-the-art mathematics http://iumj.org/access/