On uniqueness of measure-valued solutions to Liouville’s equation of Hamiltonian PDEs. (English) Zbl 1374.35404
Summary: In this paper, the Cauchy problem of classical Hamiltonian PDEs is recast into a Liouville’s equation with measure-valued solutions. Then a uniqueness property for the latter equation is proved under some natural assumptions. Our result extends the method of characteristics to Hamiltonian systems with infinite degrees of freedom and it applies to a large variety of Hamiltonian PDEs (Hartree, Klein-Gordon, Schrödinger, Wave, Yukawa \(\dots\)). The main arguments in the proof are a projective point of view and a probabilistic representation of measure-valued solutions to continuity equations in finite dimension.
MSC:
| 35Q82 | PDEs in connection with statistical mechanics |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q61 | Maxwell equations |
| 37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) |
| 28A33 | Spaces of measures, convergence of measures |
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