  We study quantitative compactness estimates in $\mathbf{W}^{1,1}_{loc}$ for
the map $S_t$, $t>0$ that associates to every given initial data $u_0\in
Lip(\mathbb{R}^N)$ the corresponding solution $S_t u_0$ of a Hamilton-Jacobi
equation $$ u_t+H\big(\nabla_{/!x} u\big)=0\,, \qquad t\geq 0,\quad x\in
\mathbb{R}^N, $$ with a uniformly convex Hamiltonian $H=H(p)$. We provide upper
and lower estimates 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. Estimates of this type
are inspired by a question posed by P.D. Lax within the context of conservation
laws, andcould provide a measure of the order of "resolution" of a numerical
method implemented for this equation.
