Quantum hydrodynamics with nonlinear interactions

Paolo  Antonelli; Pierangelo  Marcati

doi:10.3934/dcdss.2016.9.1

Article Contents

2016, Volume 9, Issue 1: 1-13. Doi: 10.3934/dcdss.2016.9.1

This issue Previous Article The research of Alberto Valli Next Article The path decomposition technique for systems of hyperbolic conservation laws

Quantum hydrodynamics with nonlinear interactions

Paolo Antonelli^1, and
Pierangelo Marcati^2,

1.
Gran Sasso Science Institute, viale F. Crispi, 7, 67100 L'Aquila
2.
DISIM - Università de L'Aquila, via Vetoio, 67010 Coppito (AQ)

Received: September 2014

Revised: February 2015

Published: February 2016

Abstract Related Papers Cited by

Abstract

In this paper we prove the global existence of large amplitude finite energy solutions for a system describing Quantum Fluids with nonlinear nonlocal interaction terms. The system may also (but not necessarily) include dissipation terms which do not provide any help to get the global existence. The method is based on the polar factorization of the wave function (which somehow generalizes the WKB method), the construction of approximate solutions via a fractional step argument and the deduction of Strichartz type estimates for the approximate solutions. Finally local smoothing and a compactness argument of Lions Aubin type allow to show the convergence.

Keywords:

Mathematics Subject Classification: Primary: 35Q40; Secondary: 35Q55, 35Q35, 82D37, 82C10.

Citation:

Related Papers

Cited by

References

[1]	P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys., 287 (2009), 657-686.doi: 10.1007/s00220-008-0632-0.
[2]	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.
[3]	P. Antonelli, R. Carles and C. Sparber, On nonlinear Schrödinger type equations with nonlinear damping, Int. Math. Res. Notices, 2015 (2015), 740-762.doi: 10.1093/imrn/rnt217.
[4]	G. Baccarani and M. Wordeman, An investignation of steady state velocity overshoot effects in Si and GaAs devices, Solid State Electron., ED-29 (1982), 970-977.
[5]	N. Berloff, Quantum vortices, travelling coherent structures and superfluid turbulence, in Stationary and Time Dependent Gross-Pitaevskii Equations (eds. A. Farina and J.-C. Saut), Contemp. Math., 473, AMS, 2006.
[6]	Y. Brenier, Polar factorization and monotone rearrangement of vector-valued function, Comm. Pure Appl. Math., 44 (1991), 375-417.doi: 10.1002/cpa.3160440402.
[7]	T. Cazenave, Semilinear Schödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, AMS, 2003.
[8]	P. Constantin and J.-C. Saut, Local smoothing properties of dispsersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.doi: 10.1090/S0894-0347-1988-0928265-0.
[9]	F. Dalfovo, S. Giorgini, L. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512.doi: 10.1103/RevModPhys.71.463.
[10]	D. Donatelli, E. Feireisl and P. Marcati, Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Comm. PDEs, 40 (2015), 1314-1335.doi: 10.1080/03605302.2014.972517.
[11]	H. Federer and W. P. Ziemer, The Lebesgue set of a function whose distribution derivatives are $p^{th}$ power summable, Indiana Univ. Math. J., 22 (1972), 139-158.doi: 10.1512/iumj.1973.22.22013.
[12]	R. Feynman, Superfluidity and Superconductivity, Rev. Mod. Phys., 29 (1957), p205.doi: 10.1103/RevModPhys.29.205.
[13]	C. Gardner, The quantum hydrodynamic model for semincoductor devices, SIAM J. Appl. Math., 54 (1994), 409-427.doi: 10.1137/S0036139992240425.
[14]	J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equations rivisited, Ann. Inst. H. Poincaré Anal. Non Lin., 2 (1985), 309-327.
[15]	A. Griffin, T. Nikuni and E. Zaremba, Bose-Condensed Gases at Finite Temperatures, Cambridge University Press, 2009.doi: 10.1017/CBO9780511575150.
[16]	A. Jüngel, Dissipative quantum fluid models, Riv. Mat. Univ. Parma, 3 (2012), 217-290.
[17]	A. Jüngel, M. Mariani and D. Rial, Local existence of solutions to the transient quantum hydrodynamics equations, Math. Models Methods Appl. Sci., 12 (2002), 485-495.doi: 10.1142/S0218202502001751.
[18]	M. Keel and T. Tao, Enpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.doi: 10.1353/ajm.1998.0039.
[19]	M. Kostin, On the Schrödinger-Langevin equation, J. Chem. Phys., 57 (1972), 3589-3591.
[20]	L. Landau, Theory of the superfluidity of helium II, Phys. Rev., 60 (1941), p356.
[21]	H. L. Li and P. Marcati, Existence and asymptotic behavior of multi-dimensional quanntum hydrodynamic model for semiconductors, Comm. Math. Phys., 245 (2004), 215-247.doi: 10.1007/s00220-003-1001-7.
[22]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Springer-Verlag, New York, 2009.
[23]	E. Madelung, Quantuentheorie in hydrodynamischer form, Z. Physik, 40 (1927), p322.
[24]	P. Marcati, P. Markowich and R. Natalini, Mathematical Problems in Semiconductor Physics, Pitman Res. Notices in Math. Series, 1996.
[25]	P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.doi: 10.1007/978-3-7091-6961-2.
[26]	J. M. Rakotoson and R. Temam, An optimal compactness theorem and application to elliptic-parabolic systems, Appl. Math. Letters, 14 (2001), 303-306.doi: 10.1016/S0893-9659(00)00153-1.
[27]	T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, vol. 106, AMS, 2006.
[28]	M. Tsubota, Quantized vortices in superfluid helium and Bose-Einstein condensates, J. Phys.: Conf. Ser., 31 (2006), 88-94.doi: 10.1088/1742-6596/31/1/014.
[29]	E. Zaremba, T. Nikuni and A. Griffin, Dynamics of trapped Bose gases at finite temperatures, J. Low Temp. Phys., 116 (1999), 277-345.