On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy
In this paper, we present a uniqueness result for solutions to the Gross-Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the...
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| Format: | Article |
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Elsevier
2018
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| Online Access: | http://hdl.handle.net/1721.1/115993 https://orcid.org/0000-0002-8220-4466 |
| _version_ | 1811077794538455040 |
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| author | Gressman, Philip Sohinger, Vedran Staffilani, Gigliola |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Gressman, Philip Sohinger, Vedran Staffilani, Gigliola |
| author_sort | Gressman, Philip |
| collection | MIT |
| description | In this paper, we present a uniqueness result for solutions to the Gross-Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear Schrödinger equation. In this way, we obtain a periodic analogue of the uniqueness result on R³ previously proved by Klainerman and Machedon [75], except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class H[superscript α] for α > 1. By constructing a specific counterexample, we show that, on T³, the existing techniques from the work of Klainerman and Machedon approach do not apply in the endpoint case α=1. This is in contrast to the known results in the non-periodic setting, where these techniques are known to hold for all α≥1. In our analysis, we give a detailed study of the crucial spacetime estimate associated to the free evolution operator. In this step of the proof, our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice. This method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius R, depending on the number-theoretic properties of the normal vector to the plane. We are hence able to obtain a sharp range of admissible Sobolev exponents for which the spacetime estimate holds. Keywords: Gross–Pitaevskii hierarchy;
Nonlinear Schrödinger equation; BBGKY hierarchy; Bose–Einstein condensation; Determinant of a lattice; U and V spaces; Factorized solutions; Multilinear estimates |
| first_indexed | 2024-09-23T10:48:29Z |
| format | Article |
| id | mit-1721.1/115993 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T10:48:29Z |
| publishDate | 2018 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/1159932022-09-27T15:09:26Z On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy Gressman, Philip Sohinger, Vedran Staffilani, Gigliola Massachusetts Institute of Technology. Department of Mathematics Staffilani, Gigliola In this paper, we present a uniqueness result for solutions to the Gross-Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear Schrödinger equation. In this way, we obtain a periodic analogue of the uniqueness result on R³ previously proved by Klainerman and Machedon [75], except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class H[superscript α] for α > 1. By constructing a specific counterexample, we show that, on T³, the existing techniques from the work of Klainerman and Machedon approach do not apply in the endpoint case α=1. This is in contrast to the known results in the non-periodic setting, where these techniques are known to hold for all α≥1. In our analysis, we give a detailed study of the crucial spacetime estimate associated to the free evolution operator. In this step of the proof, our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice. This method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius R, depending on the number-theoretic properties of the normal vector to the plane. We are hence able to obtain a sharp range of admissible Sobolev exponents for which the spacetime estimate holds. Keywords: Gross–Pitaevskii hierarchy; Nonlinear Schrödinger equation; BBGKY hierarchy; Bose–Einstein condensation; Determinant of a lattice; U and V spaces; Factorized solutions; Multilinear estimates 2018-05-30T18:57:29Z 2018-05-30T18:57:29Z 2014-02 2013-09 2018-05-30T17:37:35Z Article http://purl.org/eprint/type/JournalArticle 0022-1236 1096-0783 http://hdl.handle.net/1721.1/115993 Gressman, Philip et al. “On the Uniqueness of Solutions to the Periodic 3D Gross–Pitaevskii Hierarchy.” Journal of Functional Analysis 266, 7 (April 2014): 4705–4764 © 2014 Elsevier Inc https://orcid.org/0000-0002-8220-4466 http://dx.doi.org/10.1016/J.JFA.2014.02.006 Journal of Functional Analysis Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier arXiv |
| spellingShingle | Gressman, Philip Sohinger, Vedran Staffilani, Gigliola On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy |
| title | On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy |
| title_full | On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy |
| title_fullStr | On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy |
| title_full_unstemmed | On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy |
| title_short | On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy |
| title_sort | on the uniqueness of solutions to the periodic 3d gross pitaevskii hierarchy |
| url | http://hdl.handle.net/1721.1/115993 https://orcid.org/0000-0002-8220-4466 |
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