Multidimensional Conservation Laws
The term 'hyperbolic conservation laws' refers to systems of quasilinear hyperbolic partial differential equations. Such systems model a variety of phenomena from compressible fluid flow to tumour aggregation, and give rise to a rich repertory of algorithms in computational fluid dynamics. Despite the ubiquity of conservation laws in applications and the extensive use of numerical approximations, little mathematical theory exists for systems in more than one space variable.
During the past fifteen years, a concerted effort has been underway to formulate a theory of multidimensional conservation laws. Several approaches appear promising, including the study of functions with very weak regularity properties, and the analysis of self-similar problems, described in some detail in this talk.
Recent numerical simulations present tantalizing evidence of unexpected singularities in solutions of self-similar problems.
Many questions, both for analysts and computational scientists, remain open.
Barbara Lee Keyfitz, Fields Institute, Canada, and University of Houston