ABSTRACT

In this lecture, we will first present a short historical review of the discovery of the Gauss Green formulas in the classical settings. We then discuss recent progress in developing a theory of divergence-measure fields (a class of weakly differentiable vector fields) over rough open sets and their intrinsic connections with entropy solutions (allowing discontinuities and singularities) of nonlinear conservation laws (i.e., nonlinear partial differential equations of divergence form).  In particular, the interior and exterior Gauss–Green formulas for divergence-measure fields over a general open set will be presented, by developing a representation of the interior (resp. exterior) normal trace of the field on the boundary of the open set as the limit of classical normal traces over the boundaries of interior (resp. exterior) smooth approximations of the set.  We will also show the connection of this theory to a longstanding open fundamental problem in the axiomatic foundation of continuum physics and employ it to provide a rigorous proof of the equivalence between the entropy solutions of multidimensional nonlinear systems of balance laws and the mathematical formulation of physical balance laws via the Cauchy flux. Further trends, perspectives, and open problems in this direction will also be addressed.