Semi-local classification of geometric singularities for Hamilton-Jacobi equations (Q1891310)

The authors consider the Cauchy problem for Hamilton-Jacobi equations and study the generation and propagation of singularities appearing in the solutions. To consider this problem, the first author [Adv. Stud. Pure Math. 22, 89-100 (1993; reviewed above)] defined a notion of ``geometric solution'' in a framework of one parameter Legendrian unfoldings. On the other hand, the theory of viscosity solutions assured the global existence of weak solutions which satisfy the supplementary condition. Singularities of viscosity solutions correspond to the intersection of branches of a graph of multivalued geometric solutions. The principal part of this paper is as follows: To study the bifurcation of the branches of its graph, they formulated the problem in terms of multi- Legendrian unfoldings and gave the generic list of the bifurcations of the branches of the multivalued graph. The construction of singularities of weak solutions is not mentioned here. But they announce to publish a paper on this subject.