dc.contributor.author Hauck, Cory D. dc.contributor.author Levermore, C. David dc.contributor.author Tits, Andre L. dc.date.accessioned 2007-06-07T15:53:03Z dc.date.available 2007-06-07T15:53:03Z dc.date.issued 2007-05-30 dc.identifier.uri http://hdl.handle.net/1903/6630 dc.description.abstract 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. en Unfortunately, in many situations, macroscopic densities can take on values for which the constrained minimization problem does not have a solution. Essentially, this is because the moments are not continuous functionals with respect to the L1 topology. In this paper, we give a geometric description of these so-called degenerate densities in the most general possible 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 and, under reasonable assumptions, that this set is small in both a topological and measure theoretic sense. This result is important for further assessment and implementation of entropy-based moment closures. dc.format.extent 505486 bytes dc.format.mimetype application/pdf dc.language.iso en_US en dc.relation.ispartofseries ISR en dc.relation.ispartofseries TR 2007-9 en dc.subject convex duality en dc.subject convex optimization en dc.subject optimization in function spaces en dc.subject kinetic theory en dc.subject entropy-based closures en dc.subject moment equations en dc.subject gas dynamics en dc.title Convex Duality and Entropy-Based Moment Closures: Characterizing Degenerate Densities en dc.type Technical Report en dc.relation.isAvailableAt Institute for Systems Research en_us dc.relation.isAvailableAt Digital Repository at the University of Maryland en_us dc.relation.isAvailableAt University of Maryland (College Park, MD) en_us
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