Systems of conservation laws: A challenge for the XXIst century

ory began in the 1950's, with the works by E. Hopf, O. Oleinik, I. Gelfand and P. Lax. Several important steps have been done since then, but obstacles have limited the theory to mainly weakly nonlinear problems and/or the single space dimension. The challenge for the XXIth century is therefore to solve large data and multi-d questions. The beauty of the field presented here lies in the variety of problems to be considered and the broad set of relevant mathematical techniques : functional analysis, dynamical systems, numerical analysis, pseudo-differential operators, differential geometry, etc... Besides, physical intuition is of great help in this kind of research. In this review, we first present the origin of conservation laws. We next describe their basic features with a non-exhaustive state-of-the-art. We conclude with a short list of open problems which we hope to be tractable in the next few decades. UMPA, CNRS # 5669, Ecole Normale Sup'erieure de Lyon