  In this paper, we derive the lower bounds for the gradients of viscosity
solutions to the Hamilton--Jacobi equation, where the convex Hamiltonian
depends on the unknown function. We obtain gradient estimates using two
different methods. First, we utilize the equivalence between viscosity
solutions and Barron--Jensen solutions to study the properties of the
inf-convolution. Second, we examine the Lie equation to understand how initial
gradients propagate along its solutions.
