  We study quantitative estimates of compactness in $\mathbf{W}^{1,1}_{loc}$
for the map $S_t$, $t>0$ that associates to every given initial data $u_0\in
\mathrm{Lip}(\mathbb{R}^N)$ the corresponding solution $S_t u_0$ of a
Hamilton-Jacobi equation $$ u_t+H\big(x, \nabla_{\!x} u\big)=0\,, \qquad t\geq
0,\quad x\in \mathbb{R}^N, $$ with a convex and coercive Hamiltonian
$H=H(x,p)$. We provide upper and lower bounds of order $1/\varepsilon^N$ on the
the Kolmogorov $\varepsilon$-entropy in $\mathbf{W}^{1,1}$ of the image through
the map $S_t$ of sets of bounded, compactly supported initial data.
Quantitative estimates of compactness, as suggested by P.D. Lax, could provide
a measure of the order of "resolution" and of "complexity" of a numerical
method implemented for this equation. We establish these estimates deriving
accurate a-priori bounds on the Lipschitz, semiconcavity and semiconvexity
constant of a viscosity solution when the initial data is semiconvex. The
derivation of a small time controllability result is also fundamental to
establish the lower bounds on the $\varepsilon$-entropy.
