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Local existence of solutions to randomized Gross-Pitaevskii hierarchies
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Sohinger, Vedran (2016) Local existence of solutions to randomized Gross-Pitaevskii hierarchies. Transactions of the American Mathematical Society, 368 (3). pp. 1759-1835. doi:10.1090/tran/6479 ISSN 0002-9947.
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Official URL: http://dx.doi.org/10.1090/tran/6479
Abstract
In this paper, we study the local-in-time existence of solutions to randomized forms of the Gross-Pitaevskii hierarchy on periodic domains. In particular, we study the independently randomized Gross-Pitaevskii hierarchy and the dependently randomized Gross-Pitaevskii hierarchy, which were first introduced in the author's joint work with Staffilani (2013). For these hierarchies, we construct local-in-time low-regularity solutions in spaces which contain a random component. The constructed density matrices will solve the full randomized hierarchies, thus extending the results from the author and Staffilani's joint work, where solutions solving arbitrarily long subhierarchies were given.
Our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of T. Chen and Pavlović (2013). The presence of randomization in the problem adds additional difficulties, most notably to estimating the Duhamel expansions that are crucial in the truncation argument. These difficulties are overcome by a detailed analysis of the Duhamel expansions. In the independently randomized case, we need to keep track of which randomization parameters appear in the Duhamel terms, whereas in the dependently randomized case, we express the Duhamel terms directly in terms of the initial data. In both cases, we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data.
Item Type: | Journal Article | ||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Journal or Publication Title: | Transactions of the American Mathematical Society | ||||||
Publisher: | American Mathematical Society | ||||||
ISSN: | 0002-9947 | ||||||
Official Date: | 2016 | ||||||
Dates: |
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Volume: | 368 | ||||||
Number: | 3 | ||||||
Page Range: | pp. 1759-1835 | ||||||
DOI: | 10.1090/tran/6479 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access |
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