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Hamiltonian ODEs on a space of deficient measures. (English) Zbl 1334.37091

Summary: We continue the study (initiated in [L. Ambrosio and the second author, Commun. Pure Appl. Math. 61, No. 1, 18–53 (2008; Zbl 1132.37028)]) of Borel measures whose time evolution is provided by an interacting Hamiltonian structure. Here, the principal focus is the development and advancement of deficiency in the measure caused by displacement of mass to infinity in finite time. We introduce – and study in its own right – a regularization scheme based on a dissipative mechanism which naturally degrades mass according to distance traveled (in phase space). Our principal results are obtained based on some dynamical considerations in the form of a condition which forbids mass to return from infinity.

MSC:

37L55 Infinite-dimensional random dynamical systems; stochastic equations
49Q20 Variational problems in a geometric measure-theoretic setting
28A33 Spaces of measures, convergence of measures
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)

Citations:

Zbl 1132.37028
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