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The Quantum Hydrodynamics System in Two Space Dimensions

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Abstract

In this paper we study global existence of weak solutions for the quantum hydrodynamics system in two-dimensional energy space. We do not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property.

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Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge, CB3 0WA, England

    Paolo Antonelli

  2. Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, Via Vetoio, Coppito (AQ), 67010, Italy

    Pierangelo Marcati

Authors
  1. Paolo Antonelli
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  2. Pierangelo Marcati
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Correspondence to Paolo Antonelli.

Additional information

Communicated by C. Dafermos

Paolo Antonelli is supported by Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST).

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Antonelli, P., Marcati, P. The Quantum Hydrodynamics System in Two Space Dimensions. Arch Rational Mech Anal 203, 499–527 (2012). https://doi.org/10.1007/s00205-011-0454-7

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  • DOI: https://doi.org/10.1007/s00205-011-0454-7

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