Dynamics inside singularities of viscosity solutions to the Hamilton – Jacobi equation

It is well known that, after finite time, solutions to Hamilton–Jacobi equations develop singularities where smoothness is lost; these solutions can be constructed, e.g., using the method of characteristics. Extending classical characteristics into the singularities is, of course, not possible, and a proper generalization of this notion becomes necessary. The commonly known definition of generalized characteristics according to $P$. Cannarsa suffers from lack of uniqueness: in general, the velocity of a generalized characteristic issued from a given point of singularity takes values from a continuous set, and the uniqueness is only restored if either the space is one-dimensional or the Hamiltonian is quadratic. We propose an approach based on the regularization by “vanishing viscosity”, which utilizes the convex structure of the Hamiltonian more economically and leads to a unique definition of the velocity in any dimension. Interestingly, a similar regularization by “weak noise” does not ensure uniqueness when the dimension is three or more.