Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system
KAUST DepartmentApplied Mathematics and Computational Science
Applied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Preprint Posting Date2018-05-10
Online Publication Date2018-11-27
Print Publication Date2019
Embargo End Date2020-04-01
Permanent link to this recordhttp://hdl.handle.net/10754/632524
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AbstractWe investigate the non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. This form of ill-posedness is related to the change of the number of connected components of the support of the position density (called nodal domains) of the weak solution throughout its time evolution. We start by considering a scenario consisting of initial and final time, showing that if there is a decrease in the number of connected components, then we have non-uniqueness. This result relies on the Brouwer invariance of domain theorem. Then we consider the case in which the results involve a time interval and a full trajectory (position-current densities). We introduce the concept of trajectory-uniqueness and its characterization.
CitationMarkowich, P., & Sierra, J. (2019). Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic & Related Models, 12(2), 347–356. doi:10.3934/krm.2019015
SponsorsThe authors acknowledge in-depth discussions with Paolo Antonelli and Pierangelo Marcati on the mathematical analysis of the QHD system.
JournalKinetic & Related Models