Global well-posedness of the nonlinear Hartree equation for infinitely many particles with singular interaction

Abstract

The nonlinear Hartree equation (NLH) in the Heisenberg picture admits steady states of the form ${\gamma }_f=f(-\Delta )$ representing quantum states of infinitely many particles. In this article, we consider the time evolution of perturbations from a large class of such steady states via the three-dimensional NLH. We prove that if the interaction potential $w$ has finite measure and initial states have finite relative entropy, then solutions preserve the relative free energy, and they exist globally in time. This result extends the important work of Lewin and Sabin [arXiv:1310.0603] to singular interaction cases.

31 pages (first version); 32 pages (second version)

References in corpus (1)

• Asymptotic stability of a wide class of stationary solutions for the Hartree and Schrödinger equations for infinitely many particles

Cited by in corpus (1)

• Probabilistic Strichartz estimates in Schatten classes and their applications to the Hartree equation