Kolmogorov $\epsilon$-entropy of numerical solutions for scalar conservation laws with convex flux

Building on the information-theoretic perspective of P.~D.~Lax [$\mathit{\text{Proc.}}\mathit{\text{Sympos.,}}\mathit{\text{Math.}}\mathit{\text{Res.}}\mathit{\text{Center,}}\mathit{\text{Univ.}}\mathit{\text{Wisconsin}}$, 1978], we establish a two-sided quantitative compactness estimate for numerical solutions of scalar conservation laws with a uniformly convex flux, expressed in terms of Kolmogorov $\epsilon$-entropy. We prove that, under specific grid constraints, conservative, monotone finite-difference schemes satisfying a discrete one-sided Lipschitz condition (OSLC) preserve the $1\mathrm{/}\epsilon$ Kolmogorov entropy scaling of the corresponding exact entropy solution set, matching the bounds obtained by De~Lellis and Golse [$\mathit{\text{Comm.}}\mathit{\text{Pure}}\mathit{\text{Appl.}}\mathit{\text{Math.}}$\ $\mathbf{\text{58}}$ (2005)] and by Ancona, Glass, and Nguyen [$\mathit{\text{Comm.}}\mathit{\text{Pure}}\mathit{\text{Appl.}}\mathit{\text{Math.}}$\ $\mathbf{\text{65}}$ (2012)]. Specifically, the upper bound follows from the discrete OSLC, while the lower bound relies on a uniform approximation argument on a bounded-variation precursor class. Our results show that prototypical first-order methods are high-resolution in Lax’s sense. Finally, we abstract the lower bound mechanism into a general transfer principle, discuss implications for information recovery via post-processing, and indicate directions for future work.