Systems of Conservation Laws: A Challenge for the XXIst Century

In applied partial differential equations, people distinguish between at least three important families, named hyperbolic, elliptic and parabolic. Their paradigms are, respectively, the wave, the Laplace and the heat equations, whose qualitative properties differ drastically from each other. Linear theories have been designed for each class, involving appropriate functional spaces. By Duhamel’s principle, these theories have been extended to most semi-linear problems, where the principal part is still linear. However, fully nonlinear problems, or even quasilinear ones, are more difficult to deal with and require other ideas. Solutions of reasonable problems might have poor regularity, in which case they are called weak. Then, an ‘entropy’ criterion may be needed to select a unique, relevant solution among the weak ones. In most scalar cases, a comparison principle holds, and monotonicity encodes the entropy condition; here is the realm of nonlinear semigroup theories.