Quantum hydrodynamics with nonlinear interactions

Previous Article
The path decomposition technique for systems of hyperbolic conservation laws
DCDS-S Home
This Issue
Next Article
The research of Paolo Secchi

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
Related Papers

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 and 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), 657-686. 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.

show all references

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.

[1]	Yongming Luo, Athanasios Stylianou. On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3455-3477. doi: 10.3934/dcdsb.2020239
[2]	Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781
[3]	P.G. Kevrekidis, Dimitri J. Frantzeskakis. Multiple dark solitons in Bose-Einstein condensates at finite temperatures. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1199-1212. doi: 10.3934/dcdss.2011.4.1199
[4]	Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119
[5]	Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic and Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901
[6]	Florian Méhats, Christof Sparber. Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5097-5118. doi: 10.3934/dcds.2016021
[7]	Xuguang Lu. Long time strong convergence to Bose-Einstein distribution for low temperature. Kinetic and Related Models, 2018, 11 (4) : 715-734. doi: 10.3934/krm.2018029
[8]	Weizhu Bao, Loïc Le Treust, Florian Méhats. Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime. Kinetic and Related Models, 2017, 10 (3) : 553-571. doi: 10.3934/krm.2017022
[9]	Weizhu Bao, Yongyong Cai. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic and Related Models, 2013, 6 (1) : 1-135. doi: 10.3934/krm.2013.6.1
[10]	Pedro J. Torres, R. Carretero-González, S. Middelkamp, P. Schmelcher, Dimitri J. Frantzeskakis, P.G. Kevrekidis. Vortex interaction dynamics in trapped Bose-Einstein condensates. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1589-1615. doi: 10.3934/cpaa.2011.10.1589
[11]	Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651
[12]	José A. Carrillo, Katharina Hopf, Marie-Therese Wolfram. Numerical study of Bose–Einstein condensation in the Kaniadakis–Quarati model for bosons. Kinetic and Related Models, 2020, 13 (3) : 507-529. doi: 10.3934/krm.2020017
[13]	Luis Caffarelli, Serena Dipierro, Enrico Valdinoci. A logistic equation with nonlocal interactions. Kinetic and Related Models, 2017, 10 (1) : 141-170. doi: 10.3934/krm.2017006
[14]	Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic and Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165
[15]	T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125
[16]	Raffaele Folino, Ramón G. Plaza, Delyan Zhelyazov. Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022133
[17]	Vadym Vekslerchik, Víctor M. Pérez-García. Exact solution of the two-mode model of multicomponent Bose-Einstein condensates. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 179-192. doi: 10.3934/dcdsb.2003.3.179
[18]	Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306
[19]	Brahim Alouini. Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1629-1643. doi: 10.3934/cpaa.2011.10.1629
[20]	Vladimir S. Gerdjikov. Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1181-1197. doi: 10.3934/dcdss.2011.4.1181

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Paolo Antonelli Pierangelo Marcati
on Google Scholar
Paolo Antonelli Pierangelo Marcati

[Back to Top]