Inspired by a question posed by Lax, in recent years it has received an increasing attention the study of quantitative compactness estimates for the solution operator $S_t$, $t>0$ that associates to every given initial data $u_0$ the corresponding solution $S_t u_0$ of a conservation law or of a first order Hamilton-Jacobi equation. Estimates of this type play a central roles in various areas of information theory and statistics as well as of ergodic and learning theory. In the present setting, this concept could provide a measure of the order of ``resolution'' of a numerical method for the corresponding equation. In this talk we shall first review the results obtained in collaboration with O. Glass and K.T. Nguyen, concerning the compactness estimates for solutions to conservation laws. Next, we shall turn to the analysis of the Hamilton-Jacobi equation pursued in collaboration with P. Cannarsa and K.T.~Nguyen.