The focus of this article is on the geometric mechanism for the blow-up of solutions to the initial value problem for scalar conservation laws. We prove that the sufficient and necessary condition of blow-up is the formation of characteristics envelope. Whether the solution blows up or not relates to the topology structure of a set dominated by initial data. At last we take Burger’s equation as an example to verify our main theorem.
 H. A. Levine and M. H. Protter, “The Breakdown of Solutions of Quasilinear First Order Systems of Partial Differential Equations,” Archive for Rational Mechanics and Analysis, Vol. 95, No. 3, 1986, pp. 253-267. doi:10.1007/BF00251361