Let $H \in C^{\infty}(\mathbb{R}^d;\mathbb{R})$ and $f \in W^{1,\infty}(\mathbb{R}_+\times \mathbb{R}^d ;\mathbb{R})$ be a Lipschitz function that satisfies $$ \partial_t f - H(\nabla f) = 0 \qquad \text{almost everywhere in } \mathbb{R}_+ \times \mathbb{R}^d. $$ Assume furthermore that $f$ is semiconvex and that the initial condition $f(0,\cdot)$ is smooth. Can we find a short time $t_0 > 0$ such that $f$ and the classical solution to the equation coincide on $[0,t_0] \times \mathbb{R}^d$?
For the sake of clarity: the assumption that $f$ is semiconvex means that, for some constant $C < \infty$, the mapping $(t,x) \mapsto f(t,x) + C t^2 + C |x|^2$ is convex. The assumption that $f(0,\cdot)$ is smooth means that, say, derivatives of arbitrarily high order are uniformly bounded.
Under the stated assumptions, it is known that the equation has a classical smooth solution for a short time. The answer to the question is negative if the semiconvexity assumption on $f$ is removed. It is positive if we ask that $f$ be convex in place of semiconvex, by Proposition A.2 of https://arxiv.org/abs/2104.05360, and in fact $f$ coincides with the viscosity solution for all times in this case. Under the assumptions of the question, there are examples of functions $f$ that do not coincide with the viscosity solution; but in the examples I know, the first time when they differ is after singularities have already emerged.
