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The present paper first aims to study the BV-type regularity for viscositysolutions 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 directionallyconvex Hamiltonian $H\in\mathcal{C}^{1}(\mathbb{R}^d)$. More precisely, weestablish 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 themetric entropy in ${\bf W}^{1,1}_{\mathrm{loc}}(\mathbb{R}^d)$ for the map$S_t$ that associates to every given initial data $u_0\in{\bfLip}\big(\mathbb{R}^d\big)$, the corresponding solution $S_tu_0$. Finally, acounter 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 strictlyconvex and coercive $H\in\mathcal{C}^2(\mathbb{R}^d)$.

Metric entropy for Hamilton-Jacobi equation with uniformly directionally convex Hamiltonian