PDE Seminar

Hartree equation in the Schatten class

  • 演讲者:Kenji Nakanishi(京都大学)

  • 时间:2022-11-08 14:30-16:00

  • 地点:Zoom ID 951 8222 8581, Passcode 221108


This is joint work with Sonae Hadama (Kyoto). We study an infinite system of Schrodinger equations with the Hartree interaction, which is a simplified mean-field model for fermions. Lewin and Sabin ('14-15) proved stability of some translation-invariant stationary solutions which are physically important, such as ideal Fermigas. One of the key tools was the Strichartz estimate for orthogonal sequences of solutions to the free equation, where the total mass (L^2) is not summable but merely p-th power summable with respect to the number of particles for some p>1. If the equation is rewritten for the orthogonal projection, then such solutions belong to the Schatten-p class. In this setting, the perturbation argument for the Duhamel integral is not so easy as in the scalar case, because the Schatten class is not simply embedded into or even compared with the space-time Lebesgue norms used for the Strichartz estimate. So we propose some framework to solve the nonlinear equation by the Duhamel formula. In particular, we introduce a norm for propagators corresponding to the best constants of the Strichartz estimate in the Schatten class, and a Schatten version of the Christ-Kiselev lemma for the Duhamel integral.