Browsing by Subject "Gross-Pitaevskii hierarchy"
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Item The dynamics of bose gases(2015-05) Taliaferro, Kenneth William; Chen, Thomas (Ph. D. in mechanical engineering and Ph. D. in mathematical physics); Maggi, Francesco; Pavlovic, Natasa; Tzirakis, Nikolaos; Vasseur, AlexisWe study the Gross-Pitaevskii (GP) hierarchy, which is an infinite sequence of coupled partial differential equations that models the dynamics of Bose gases and arises in the derivation of the cubic and quintic nonlinear Schrödinger equations from an N-body linear Schrödinger equation. In Chapter 2, we consider the cubic case in R³ and derive the GP hierarchy in the strong topology corresponding to the spaces used by Klainerman and Machedon in (82). We also prove that positive semidefiniteness of solutions is preserved over time and use this result to prove global well-posedness of solutions to the GP hierarchy. This is based on a joint work with Thomas Chen (24). In Chapters 3 and 4, we prove uniqueness of solutions to the GP hierarchy in R[superscript d] in a low regularity Sobolev type space in the cubic and quintic cases, respectively. These chapters are an extension of the work of Chen-Hainzl-Pavlović-Seiringer (17) and are based on joint works with Younghun Hong and Zhihui Xie (70,71).Item On the derivation and uniqueness of the Gross-Pitaevskii hierarchy for a Bose gas coupled to a heavy tracer particle(2021-08-05) Cheng, Yanlin; Chen, Thomas (Ph. D. in mechanical engineering and Ph. D. in mathematical physics); Patrizi, Stefania; Pavlovic, Natasa; Tran, Minh-BinhWe consider a system of N identical bosons coupled to a heavy tracer particle. We prove the convergence of the BBGKY hierarchy of marginals to the Gross-Pitaevskii (GP) hierarchy. The main tool we use to prove the uniqueness is the equivalence between the GP hierarchy and the Liouville equations on H⁻¹(R³) established by Ammari, Liard and Rouffort in [2].