Abstract
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.
Highlights
Liouville’s equation is a fundamental equation of statistical mechanics which describes the time evolution of phase-space distribution functions
Consider a rigged Hilbert space Z1 ⊂ Z0 ⊂ Z1′ such that (Z1, Z0) is a pair of complex separable Hilbert spaces, Z1 is densely continuously embedded in Z0 and Z1′ is the dual of Z1 with respect to the duality bracket extending the inner product ·, · Z0
A significant example is provided by Sobolev spaces Hs(Rd) ⊂ L2(Rd) ⊂ H−s(Rd) with s > 0
Summary
Liouville’s equation is a fundamental equation of statistical mechanics which describes the time evolution of phase-space distribution functions. There is an interesting approach [5, Chapter 8] related to optimal transport theory that improves the standard characteristics method by using a regularization argument and the differential structure of spaces of probability measures This approach avoids the use of a reference measure and it is suitable for generalization to systems with infinite degrees of freedom. This was exploited in [12, Appendix C] to prove a uniqueness property for Liouville’s equation considered in a weak sense,. For reader’s convenience a short appendix collecting useful notions in measure theory is provided (tightness, equi-integrability, Dunford-Pettis theorem, disintegration)
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