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April 2019, 12(2): 347-356. doi: 10.3934/krm.2019015

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

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

* Corresponding author: Peter Markowich

Received  February 2018 Published  November 2018

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.

Citation: Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic & Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015
References:
[1]

M. G. Ancona and G. J. Iafrate, Quantum correction to the equation of state of an electron gas in a semiconductor, Physical Review B, 39 (1989), 9536-9540.

[2]

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Communications in Mathematical Physics, 287 (2009), 657-686.

[3]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Archive for Rational Mechanics and Analysis, 203 (2012), 499-527. doi: 10.1007/s00205-011-0454-7.

[4]

P. DegondS. Gallego and F. Méhats, Isothermal quantum hydrodynamics: Derivation, asymptotic analysis, and simulation, Multiscale Modeling & Simulation, 6 (2007), 246-272. doi: 10.1137/06067153X.

[5]

P. DegondS. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models, Communications in Mathematical Sciences, 5 (2007), 887-908. doi: 10.4310/CMS.2007.v5.n4.a8.

[6]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, Journal of Statistical Physics, 112 (2003), 587-628. doi: 10.1023/A:1023824008525.

[7]

A. Dold, Lectures on Algebraic Topology, Springer-Verlag, New York-Berlin, 1972.

[8]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC press, 2015.

[9]

W. GangboJ. HaskovecP. Markowich and J. Sierra, An optimal transport approach for the kinetic bohmian equation, Zapiski Nauchnyh Seminarov POMI, 457 (2017), 114-167.

[10]

I. Gasser and P. Markowich, Quantum hydrodynamics, Wigner transforms, the classical limit, Asymptotic Analysis, 14 (1997), 97-116.

[11]

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

[12]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Birkhäuser, Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8334-4.

[13]

L. Landau and E. M. Lifschitz, Lehrbuch der Theoretischen Physik, III - Quantenmechanik, Akademie-Verlag, 1979.

[14]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge University Press, 2012. doi: 10.1017/CBO9781139108133.

[15]

P. MarkowichT. Paul and C. Sparber, Bohmian measures and their classical limit, Journal of Functional Analysis, 259 (2010), 1542-1576. doi: 10.1016/j.jfa.2010.05.013.

[16]

P. MarkowichT. Paul and C. Sparber, On the dynamics of Bohmian measures, Archive for Rational Mechanics and Analysis, 205 (2012), 1031-1054. doi: 10.1007/s00205-012-0528-1.

[17]

W. S. Massey, A Basic Course in Algebraic Topology, Springer Science & Business Media, 1991.

show all references

References:
[1]

M. G. Ancona and G. J. Iafrate, Quantum correction to the equation of state of an electron gas in a semiconductor, Physical Review B, 39 (1989), 9536-9540.

[2]

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Communications in Mathematical Physics, 287 (2009), 657-686.

[3]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Archive for Rational Mechanics and Analysis, 203 (2012), 499-527. doi: 10.1007/s00205-011-0454-7.

[4]

P. DegondS. Gallego and F. Méhats, Isothermal quantum hydrodynamics: Derivation, asymptotic analysis, and simulation, Multiscale Modeling & Simulation, 6 (2007), 246-272. doi: 10.1137/06067153X.

[5]

P. DegondS. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models, Communications in Mathematical Sciences, 5 (2007), 887-908. doi: 10.4310/CMS.2007.v5.n4.a8.

[6]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, Journal of Statistical Physics, 112 (2003), 587-628. doi: 10.1023/A:1023824008525.

[7]

A. Dold, Lectures on Algebraic Topology, Springer-Verlag, New York-Berlin, 1972.

[8]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC press, 2015.

[9]

W. GangboJ. HaskovecP. Markowich and J. Sierra, An optimal transport approach for the kinetic bohmian equation, Zapiski Nauchnyh Seminarov POMI, 457 (2017), 114-167.

[10]

I. Gasser and P. Markowich, Quantum hydrodynamics, Wigner transforms, the classical limit, Asymptotic Analysis, 14 (1997), 97-116.

[11]

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

[12]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Birkhäuser, Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8334-4.

[13]

L. Landau and E. M. Lifschitz, Lehrbuch der Theoretischen Physik, III - Quantenmechanik, Akademie-Verlag, 1979.

[14]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge University Press, 2012. doi: 10.1017/CBO9781139108133.

[15]

P. MarkowichT. Paul and C. Sparber, Bohmian measures and their classical limit, Journal of Functional Analysis, 259 (2010), 1542-1576. doi: 10.1016/j.jfa.2010.05.013.

[16]

P. MarkowichT. Paul and C. Sparber, On the dynamics of Bohmian measures, Archive for Rational Mechanics and Analysis, 205 (2012), 1031-1054. doi: 10.1007/s00205-012-0528-1.

[17]

W. S. Massey, A Basic Course in Algebraic Topology, Springer Science & Business Media, 1991.

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