Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system

Peter Markowich; Jesús Sierra

doi:10.3934/krm.2019015

Article Contents

2019, Volume 12, Issue 2: 347-356. Doi: 10.3934/krm.2019015

This issue Previous Article On a Boltzmann equation for Haldane statistics Next Article A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data

Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system

Peter Markowich^, and
Jesús Sierra

CEMSE Division, King Abdullah University of Science and Technology, Box 4700, Thuwal 23955-6900, Saudi Arabia

* Corresponding author: Peter Markowich
* Corresponding author: Peter Markowich

Received: February 2018

Published: April 2019

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Abstract

We 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.

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Mathematics Subject Classification: Primary: 35A02; Secondary: 35A16.

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