Hamilton–Jacobi flows with nowhere differentiable initial data

We deal with a sort of inverse problem. Namely, we aim at deriving features of initial data of an Hamilton-Jacobi flow starting from specific assumptions on the flow. To this scope, we introduce a class of nowhere differentiable functions, and characterize a function in this class through a property of the Hamilton–Jacobi flow issued from this function. This analysis shows that, despite the regularizing effect of the flow, nowhere differentiability of initial data can be nevertheless detected looking at the flow for positive times.