Convex Duality and Entropy-Based Moment Closures: Characterizing Degenerate Densities

A common method for constructing a function from a finite set of moments is to solve a constrained minimization problem. The idea is to find, among all functions with the given moments, that function which minimizes a physically motivated, strictly convex functional. In the kinetic theory of gases, this functional is the kinetic entropy; the given moments are macroscopic densities; and the solution to the constrained minimization problem is used to formally derive a closed system of partial differential equations which describe how the macroscopic densities evolve in time. Moment equations are useful because they simplify the kinetic, phase-space description of a gas, and with entropy-based closures, they retain many of the fundamental properties of kinetic transport. Unfortunately, in many situations, macroscopic densities can take on values for which the constrained minimization problem does not have a solution. In this paper, we give a geometric description of these so-called degenerate densities in a very general setting. Our key tool is the complementary slackness condition that is derived from a dual formulation of a minimization problem with relaxed constraints. We show that the set of degenerate densities is a union of convex cones defined by the complementary slackness conditions and, under reasonable assumptions, that this set is small in both a topological and a measure-theoretic sense. This result is important for further assessment and implementation of entropy-based moment closures. An expanded version of this work can be found in [Hauck et al., SIAM J. Contr. Optim., Vol. 47, 2008, pp. 1977-2015].