One Perspective on Open Problems in Multi-Dimensional Conservation Laws

It is evident from the lectures at this meeting that the subject of systems of hyperbolic conservation laws is flourishing as one of the prototypical examples of the modern mode of applied mathematics. Research in this area often involves strong and close interdisciplinary interactions among diverse areas of applied mathematics including (1)Large (and small) scale computing(2)Asymptotic modelling(3)Qualitative modelling(4)Rigorous proofs for suitable prototype problems combined with careful attention to experimental data when possible. In fact, the subject is developing at such a rapid rate that new predictions of phenomena through a combination of theory and computations can be made in regimes which are not readily accessible to experimentalists. Pioneering examples of this type of interaction can be found in the papers of Grove, Glaz, and Colella in this volume as well as the recent work of Woodward, Artola, and the author ([1], [2], [3], [4], [5], [6]). In this last work, novel mechanisms of nonlinear instability in supersonic vortex sheets have been documented and explained very recently through a sophisticated combination of numerical experiments and mathematical theory.