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Local existence of solutions to randomized Gross-Pitaevskii hierarchies


Author: Vedran Sohinger
Journal: Trans. Amer. Math. Soc. 368 (2016), 1759-1835
MSC (2010): Primary 35Q55, 70E55
Published electronically: June 15, 2015
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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.


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Additional Information

Vedran Sohinger
Affiliation: Department of Mathematics, David Rittenhouse Lab, University of Pennsylvania, Office 3N4B, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Address at time of publication: Eidgenössische Technische Hochschule Zürich, Departement Mathematik, Rämistrasse 101, 8092 Zürich, Switzerland
Email: vedranso@math.upenn.edu, vedran.sohinger@math.ethz.ch

DOI: https://doi.org/10.1090/tran/6479
Keywords: Gross-Pitaevskii hierarchy, nonlinear Schr\"odinger equation, randomization, density matrices, collision operator, Duhamel iteration, local-in-time solutions
Received by editor(s): January 4, 2014
Published electronically: June 15, 2015
Additional Notes: The author was supported by a Simons Postdoctoral Fellowship.
Article copyright: © Copyright 2015 American Mathematical Society