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Abstract
We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones in a semi-classical limit and obtain as a by-product global well-posedness of the Vlasov equation in the (relative) energy space.
Notes
1 The Fermi gas at zero temperature also has a variational interpretation which we used in [Citation1], although a bit different. Here we only treat the positive temperature case.
Additional information
Funding
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant MDFT No 725528 of M.L.) and from the Agence Nationale de la Recherche10.13039/501100001665 (grant DYRAQ No ANR-17-CE40-001 of J.S.).