Lower compactness estimates for scalar balance laws

We study the compactness in Lloc of the semigroup (St)t≥0 of entropy weak solutions to strictly convex scalar conservation laws in one space dimension. The compactness of St for each t > 0 was established by P. D. Lax [1]. Upper estimates for the Kolmogorov’s ε-entropy of the image through St of bounded sets C in L ∩ L∞ which is denoted by Hε(St(C) | L(R)) := log2 Nε(St(C)). where Nε(St(C)) is the minimal number of sets in a cover of St(C) by subsets of L(R) having diameter no larger than 2ε, were given by C. De Lellis and F. Golse [3]. Here, we provide lower estimates on this ε-entropy of the same order as the one established in [3], thus showing that such an ε-entropy is of size ≈ (1/ε). Moreover, we extend these estimates of compactness to the case of convex balance laws. (Joint work with Fabio Ancona and Olivier Glass)