  The present paper first aims to study the BV-type regularity for viscosity
solutions of the Hamilton-Jacobi equation \[
  u_t(t,x)+H\big(D_{x} u(t,x)\big)~=~0\qquad\forall (t,x)\in
]0,\infty[\times\mathbb{R}^d \] with a coercive and uniformly directionally
convex Hamiltonian $H\in\mathcal{C}^{1}(\mathbb{R}^d)$. More precisely, we
establish a BV bound on the slope of backward characteristics $DH(u(t,\cdot))$
starting at a positive time $t>0$. Relying on the BV bound, we quantify the
metric entropy in ${\bf W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^d)$ for the map
$S_t$ that associates to every given initial data $u_0\in{\bf
Lip}\big(\mathbb{R}^d\big)$, the corresponding solution $S_tu_0$. Finally, a
counter example is constructed to show that both $D_xu(t,\cdot)$ and
$DH(D_xu(t,\cdot))$ fail to be in $BV_{\mathrm{loc}}$ for a general strictly
convex and coercive $H\in\mathcal{C}^2(\mathbb{R}^d)$.
