Abstract
In this paper we consider the global existence of weak solutions to a class of Quantum Hydrodynamics (QHD) systems with initial data, arbitrarily large in the energy norm. These type of models, initially proposed by Madelung [24], have been extensively used in Physics to investigate Supefluidity and Superconductivity phenomena [10], [19] and more recently in the modeling of semiconductor devices [11]. Our approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method which iterates a SchrÖodinger Madelung picture with a suitable wave function updating mechanism. Therefore several a priori bounds of energy, dispersive and local smoothing type allow us to prove the compactness of the approximating sequences. No uniqueness result is provided. A more detailed exposition of the results is given in [2].
AMS(MOS) subject classifications. 35Q40 (Primary); 35Q55, 35Q35, 82D37, 82C10, 76Y05 (Secondary).
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References
Ambrosio L., Transport Equations and Cauchy Problem for BV Vector Fields, Inventiones Mathematicae 158: 227–260 (2004).
P. Antonelli and P. Marcati, On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics, Comm. Math. Phys., 287(2): 657–686 (2009).
Antonelli P. and Marcati P., to appear.
Carlen E., Conservative Diffusions, Comm. Math. Phys. 94: 293–315 (1984).
Cazenave T., Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, AMS, 2003.
Constantin P. and Saut J.C., Local Smoothing Properties of Dispersive Equations, J. Amer. Math. Soc. 1: 413–439 (1988).
Dalfovo F., Giorgini S., Pitaevskii L., and Stringari S., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71: 463–512 (1999).
Di Perna R.J. and Lions P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98: 511–547 (1989).
Degond P. and Ringhofer C., Quantum moment hydrodynamics entropy principle, J. Stat. Phys. 112: 587–628 (2003).
Feynman R.P., Superfluidity and Superconductivity, Rev. Mod. Phys., 29(2): 205 (1957).
Gardner C., The Quantum Hydrodynamic Model for Semiconductor Devices SIAM J. Appl. Math. 54: 409–427 (1994).
Gasser I. and Markowich P., Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptot. Anal. 14(2): 97–116 (1997).
Guerra F. and Morato L., Quantization of Dynamical Systems and Stochastic Control Theory, Phys. Rev. D 27: 1771–1786 (1983).
Jüngel A., Mariani M.C., and Rial D., Local Existence of Solutions to the Transient Quantum Hydrodynamic Equations, Math. Models Methods Appl. Sci. 12(4): 485–495 (2002).
Khalatnikov I.M., An introduction to the Theory of Superfluidity, Benjamin N.Y., 1965.
Keel M. and Tao T., Endpoint Strichartz Estimates, Amer. J. Math. 120: 955–980 (1998).
Kirkpatrick T.R. and Dorfman J.R., Transport theory for a weakly interacting condensed Bose gas, Phys. Rev. A 28: 2576 (1983).
Kostin M., On the Schrödinger-Langevin equation, J. Chem. Phys. 57: 3589–3591 (1972).
Landau L.D., Theory of the Superfluidity of Helium II, Phys. Rev. 60: 356 (1941).
Li H. and Lin C.-K., Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems, EJDE 93: 1–17 (2003).
Li H. and Marcati P., Existence and asymptotic behavior of multi-dimensional quanntum hydrodynamic model for semiconductors, Comm. Math. Phys. 245(2): 215–247 (2004).
Nelson E., Quantum Fluctuations, Princeton University Press, 1984.
Lifshitz E.M. and Pitaevskii L., Physical Kinetics, Pergamon, Oxford (1981).
Madelung E., Quantentheorie in hydrodynamischer form, Z. Physik, 40: 322 (1927).
Rakotoson J.M. and Temam R., An Optimal Compactness Theorem and Application to Elliptic-Parabolic Systems, Appl. Math. Letters 14: 303–306 (2001).
Teufel S. and Tumulka R., Simple proof for global existence of bohmian trajectories, Comm. Math. Phys. 258: 349–365 (2005).
Weigert S., How to determine a quantum state by measurements: the Pauli problem for a particle with arbitrary potential, Phys. Rev. A 53: 4, 2078–2083 (1996).
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Antonelli, P., Marcati, P. (2011). Finite Energy Weak Solutions to the Quantum Hydrodynamics System. In: Bressan, A., Chen, GQ., Lewicka, M., Wang, D. (eds) Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications, vol 153. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9554-4_9
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DOI: https://doi.org/10.1007/978-1-4419-9554-4_9
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