Formation of singularities for Hamilton-Jacobi equation. I (Q788186)

This note is concerned with the singularities of global solutions of the Hamilton-Jacobi equation in two space dimensions: \((*)\quad \partial u/\partial t=f(\partial u/\partial x)\) in \(\{t>0\), \(x\in R^ 2\}\). Assume that f(p) is uniformly convex, and that the initial data are sufficiently smooth. We construct the solution of (*) by a characteristic method. Then we can not get the smooth global solutions generally. The reason is due to the fact that a smooth mapping from \(R^ 2\) to \(R^ 2\) can not have a smooth inverse mapping uniquely in a neighborhood of a point where its Jacobian vanishes, i.e., that the inverse becomes many valued there. Therefore the solution generally takes many values after some time. The aim of this note is to show how to choose up the reasonable value of many values so that the solution is one-valued, continuous, and semi-concave.