Quantum hydrodynamics with nonlinear interactions

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February 2016, 9(1): 1-13. doi: 10.3934/dcdss.2016.9.1

Quantum hydrodynamics with nonlinear interactions

Paolo Antonelli ^1, and Pierangelo Marcati ^2,

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

Received September 2014 Revised February 2015 Published December 2015

Abstract
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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: Quantum Hydrodynamics, Bose Einstein., nonlocal interactions, global existence, finite energy.

Mathematics Subject Classification: Primary: 35Q40; Secondary: 35Q55, 35Q35, 82D37, 82C1.

Citation: Paolo Antonelli, Pierangelo Marcati. Quantum hydrodynamics with nonlinear interactions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 1-13. doi: 10.3934/dcdss.2016.9.1

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. 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. 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. 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.*
[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)*, (2006).
[6]	Y. Brenier, Polar factorization and monotone rearrangement of vector-valued function,, Comm. Pure Appl. Math., 44 (1991), 375. doi: 10.1002/cpa.3160440402.
[7]	T. Cazenave, Semilinear Schödinger Equations,, Courant Lecture Notes in Mathematics, (2003).
[8]	P. Constantin and J.-C. Saut, Local smoothing properties of dispsersive equations,, J. Amer. Math. Soc., 1 (1988), 413. 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. 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. 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. doi: 10.1512/iumj.1973.22.22013.
[12]	R. Feynman, Superfluidity and Superconductivity,, Rev. Mod. Phys., 29 (1957). doi: 10.1103/RevModPhys.29.205.
[13]	C. Gardner, The quantum hydrodynamic model for semincoductor devices,, SIAM J. Appl. Math., 54 (1994), 409. 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.
[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.
[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. doi: 10.1142/S0218202502001751.
[18]	M. Keel and T. Tao, Enpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039.
[19]	M. Kostin, On the Schrödinger-Langevin equation,, J. Chem. Phys., 57 (1972), 3589.
[20]	L. Landau, Theory of the superfluidity of helium II,, Phys. Rev., 60 (1941).
[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. doi: 10.1007/s00220-003-1001-7.
[22]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Springer-Verlag, (2009).
[23]	E. Madelung, Quantuentheorie in hydrodynamischer form,, Z. Physik, 40 (1927).
[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, (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. doi: 10.1016/S0893-9659(00)00153-1.
[27]	T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006).
[28]	M. Tsubota, Quantized vortices in superfluid helium and Bose-Einstein condensates,, J. Phys.: Conf. Ser., 31 (2006), 88. 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.

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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. 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. 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. 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.*
[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)*, (2006).
[6]	Y. Brenier, Polar factorization and monotone rearrangement of vector-valued function,, Comm. Pure Appl. Math., 44 (1991), 375. doi: 10.1002/cpa.3160440402.
[7]	T. Cazenave, Semilinear Schödinger Equations,, Courant Lecture Notes in Mathematics, (2003).
[8]	P. Constantin and J.-C. Saut, Local smoothing properties of dispsersive equations,, J. Amer. Math. Soc., 1 (1988), 413. 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. 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. 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. doi: 10.1512/iumj.1973.22.22013.
[12]	R. Feynman, Superfluidity and Superconductivity,, Rev. Mod. Phys., 29 (1957). doi: 10.1103/RevModPhys.29.205.
[13]	C. Gardner, The quantum hydrodynamic model for semincoductor devices,, SIAM J. Appl. Math., 54 (1994), 409. 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.
[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.
[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. doi: 10.1142/S0218202502001751.
[18]	M. Keel and T. Tao, Enpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039.
[19]	M. Kostin, On the Schrödinger-Langevin equation,, J. Chem. Phys., 57 (1972), 3589.
[20]	L. Landau, Theory of the superfluidity of helium II,, Phys. Rev., 60 (1941).
[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. doi: 10.1007/s00220-003-1001-7.
[22]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Springer-Verlag, (2009).
[23]	E. Madelung, Quantuentheorie in hydrodynamischer form,, Z. Physik, 40 (1927).
[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, (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. doi: 10.1016/S0893-9659(00)00153-1.
[27]	T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006).
[28]	M. Tsubota, Quantized vortices in superfluid helium and Bose-Einstein condensates,, J. Phys.: Conf. Ser., 31 (2006), 88. 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.

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