American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[6]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued function,, Comm. Pure Appl. Math., 44 (1991), 375.  doi: 10.1002/cpa.3160440402.  Google Scholar

[7]

T. Cazenave, Semilinear Schödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Feynman, Superfluidity and Superconductivity,, Rev. Mod. Phys., 29 (1957).  doi: 10.1103/RevModPhys.29.205.  Google Scholar

[13]

C. Gardner, The quantum hydrodynamic model for semincoductor devices,, SIAM J. Appl. Math., 54 (1994), 409.  doi: 10.1137/S0036139992240425.  Google Scholar

[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.   Google Scholar

[15]

A. Griffin, T. Nikuni and E. Zaremba, Bose-Condensed Gases at Finite Temperatures,, Cambridge University Press, (2009).  doi: 10.1017/CBO9780511575150.  Google Scholar

[16]

A. Jüngel, Dissipative quantum fluid models,, Riv. Mat. Univ. Parma, 3 (2012), 217.   Google Scholar

[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.  Google Scholar

[18]

M. Keel and T. Tao, Enpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[19]

M. Kostin, On the Schrödinger-Langevin equation,, J. Chem. Phys., 57 (1972), 3589.   Google Scholar

[20]

L. Landau, Theory of the superfluidity of helium II,, Phys. Rev., 60 (1941).   Google Scholar

[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.  Google Scholar

[22]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Springer-Verlag, (2009).   Google Scholar

[23]

E. Madelung, Quantuentheorie in hydrodynamischer form,, Z. Physik, 40 (1927).   Google Scholar

[24]

P. Marcati, P. Markowich and R. Natalini, Mathematical Problems in Semiconductor Physics,, Pitman Res. Notices in Math. Series, (1996).   Google Scholar

[25]

P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[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.  Google Scholar

[27]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar

[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.  Google Scholar

[29]

E. Zaremba, T. Nikuni and A. Griffin, Dynamics of trapped Bose gases at finite temperatures,, J. Low Temp. Phys., 116 (1999), 277.   Google Scholar

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.  doi: 10.1007/s00220-008-0632-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[6]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued function,, Comm. Pure Appl. Math., 44 (1991), 375.  doi: 10.1002/cpa.3160440402.  Google Scholar

[7]

T. Cazenave, Semilinear Schödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Feynman, Superfluidity and Superconductivity,, Rev. Mod. Phys., 29 (1957).  doi: 10.1103/RevModPhys.29.205.  Google Scholar

[13]

C. Gardner, The quantum hydrodynamic model for semincoductor devices,, SIAM J. Appl. Math., 54 (1994), 409.  doi: 10.1137/S0036139992240425.  Google Scholar

[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.   Google Scholar

[15]

A. Griffin, T. Nikuni and E. Zaremba, Bose-Condensed Gases at Finite Temperatures,, Cambridge University Press, (2009).  doi: 10.1017/CBO9780511575150.  Google Scholar

[16]

A. Jüngel, Dissipative quantum fluid models,, Riv. Mat. Univ. Parma, 3 (2012), 217.   Google Scholar

[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.  Google Scholar

[18]

M. Keel and T. Tao, Enpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[19]

M. Kostin, On the Schrödinger-Langevin equation,, J. Chem. Phys., 57 (1972), 3589.   Google Scholar

[20]

L. Landau, Theory of the superfluidity of helium II,, Phys. Rev., 60 (1941).   Google Scholar

[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.  Google Scholar

[22]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Springer-Verlag, (2009).   Google Scholar

[23]

E. Madelung, Quantuentheorie in hydrodynamischer form,, Z. Physik, 40 (1927).   Google Scholar

[24]

P. Marcati, P. Markowich and R. Natalini, Mathematical Problems in Semiconductor Physics,, Pitman Res. Notices in Math. Series, (1996).   Google Scholar

[25]

P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[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.  Google Scholar

[27]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006).   Google Scholar

[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.  Google Scholar

[29]

E. Zaremba, T. Nikuni and A. Griffin, Dynamics of trapped Bose gases at finite temperatures,, J. Low Temp. Phys., 116 (1999), 277.   Google Scholar

[1]

Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781
[2]

P.G. Kevrekidis, Dimitri J. Frantzeskakis. Multiple dark solitons in Bose-Einstein condensates at finite temperatures. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1199-1212. doi: 10.3934/dcdss.2011.4.1199
[3]

Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119
[4]

Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic & Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901
[5]

Xuguang Lu. Long time strong convergence to Bose-Einstein distribution for low temperature. Kinetic & Related Models, 2018, 11 (4) : 715-734. doi: 10.3934/krm.2018029
[6]

Florian Méhats, Christof Sparber. Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5097-5118. doi: 10.3934/dcds.2016021
[7]

Weizhu Bao, Loïc Le Treust, Florian Méhats. Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime. Kinetic & Related Models, 2017, 10 (3) : 553-571. doi: 10.3934/krm.2017022
[8]

Weizhu Bao, Yongyong Cai. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic & Related Models, 2013, 6 (1) : 1-135. doi: 10.3934/krm.2013.6.1
[9]

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 & Applied Analysis, 2011, 10 (6) : 1589-1615. doi: 10.3934/cpaa.2011.10.1589
[10]

Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651
[11]

Luis Caffarelli, Serena Dipierro, Enrico Valdinoci. A logistic equation with nonlocal interactions. Kinetic & Related Models, 2017, 10 (1) : 141-170. doi: 10.3934/krm.2017006
[12]

Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic & Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165
[13]

T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125
[14]

J. F. Toland. Non-existence of global energy minimisers in Stokes waves problems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3211-3217. doi: 10.3934/dcds.2014.34.3211
[15]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[16]

Vadym Vekslerchik, Víctor M. Pérez-García. Exact solution of the two-mode model of multicomponent Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 179-192. doi: 10.3934/dcdsb.2003.3.179
[17]

Brahim Alouini. Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1629-1643. doi: 10.3934/cpaa.2011.10.1629
[18]

Vladimir S. Gerdjikov. Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1181-1197. doi: 10.3934/dcdss.2011.4.1181
[19]

Kui Li, Zhitao Zhang. A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 851-860. doi: 10.3934/dcds.2016.36.851
[20]

Anne de Bouard, Reika Fukuizumi, Romain Poncet. Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2793-2817. doi: 10.3934/dcdsb.2015.20.2793

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences