Bifurcations of shock waves for viscosity solutions of Hamilton-Jacobi equations of one space variable
The authors consider the Cauchy problem for Hamilton-Jacobi equations in one space dimension. It is well-known that, even if the initial data may be sufficiently smooth, the Cauchy problem may not have a classical solution in the large. This means that singularities may appear in solutions in finite time. Therefore the notion of ``viscosity solution'' was introduced to define ``solutions with singularities''. The existence of viscosity solutions was first proved by \textit{M. G. Crandall} and \textit{P. L. Lions} [C. R. Acad. Sci., Paris, Sér. I 292, 183-186 (1981; Zbl 0469.49023)], and \textit{M. G. Crandall}, \textit{L. C. Evans} and \textit{P. L. Lions} [Trans. Am. Math. Soc. 282, 487-502 (1984; Zbl 0543.35011)]. The aim of this paper is to study singularities of viscosity solutions. For this aim, the authors introduce the notion of ``geometric solution'' for Hamilton-Jacobi equations and study the generic properties of singularities of geometric solutions. Next, projecting the geometric solutions to the base space, they construct the singularities of viscosity solutions. This procedure was first achieved by \textit{M. Tsuji} [J. Math. Kyoto Univ. 26, 299-308 (1986; Zbl 0655.35009)]. The principal part of this paper is how to construct the singularities of solutions in the case where the equations are not convex.
generic properties of singularities of geometric solutions
scientific article; zbMATH DE number 1121329