On the Lagrangian structure of quantum fluid models

Philipp Fuchs; Ansgar Jüngel; Max von Renesse

doi:10.3934/dcds.2014.34.1375

Article Contents

2014, Volume 34, Issue 4: 1375-1396. Doi: 10.3934/dcds.2014.34.1375

This issue Previous Article Gradient flow structures for discrete porous medium equations Next Article Optimal transport and large number of particles

On the Lagrangian structure of quantum fluid models

1.
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien
2.
Institute for Mathematics, Universität Leipzig, Augustusplatz 10, 04109 Leipzig

Received: December 31, 2012

Revised: April 30, 2013

Published: March 2014

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Abstract

Some quantum fluid models are written as the Lagrangian flow of mass distributions and their geometric properties are explored. The first model includes magnetic effects and leads, via the Madelung transform, to the electromagnetic Schrödinger equation in the Madelung representation. It is shown that the Madelung transform is a symplectic map between Hamiltonian systems. The second model is obtained from the Euler-Lagrange equations with friction induced from a quadratic dissipative potential. This model corresponds to the quantum Navier-Stokes equations with density-dependent viscosity. The fact that this model possesses two different energy-dissipation identities is explained by the definition of the Noether currents.

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Mathematics Subject Classification: Primary: 35Q40, 37J10, 58D30.

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