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

The present paper first aims to study the BV-type regularity for viscosity solutions of the Hamilton-Jacobi equation $$u_t(t,x)+H(D_{x}u(t,x))=0\mathrm{\forall }(t,x)\in ]0,\mathrm{\infty }[\times {\mathbb{R}}^d$$ 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 ${\mathbf{W}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}({\mathbb{R}}^d)$ for the map $S_t$ that associates to every given initial data $u_0\in{\bf Lip}\big($\mathbb{R}$^d\big)$,thecorrespondingsolution$S_tu_0$. Finally, a counter example is constructed to show that both $D_{x}u(t,\cdot )$ and $DH(D_{x}u(t,\cdot ))$ fail to be in $B{V}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ for a general strictly convex and coercive $H\in {\mathcal{C}}^2({\mathbb{R}}^d)$.