ON QUANTITATIVE COMPACTNESS ESTIMATES FOR HYPERBOLIC CONSERVATION LAWS

We are concerned with the compactness in L-loc(1) of the semigroup (S-t)(t >= 0) of entropy weak solutions generated by hyperbolic conservation laws in one space dimension. This note provides a survey of recent results establishing upper and lower estimates for the Kolmogorov epsilon-entropy of the image through the mapping S-t of bounded sets in L-1 boolean AND L-infinity, both in the case of scalar and of systems of conservation laws. As suggested by Lax [16], these quantitative compactness estimates could provide a measure of the order of "resolution" of the numerical methods implemented for these equations.