-
Previous Article
The path decomposition technique for systems of hyperbolic conservation laws
- DCDS-S Home
- This Issue
-
Next Article
The research of Paolo Secchi
Quantum hydrodynamics with nonlinear interactions
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 |
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] |
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 |
[17] |
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 |
[18] |
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 |
[19] |
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 |
[20] |
Kui Li, Zhitao Zhang. A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 851-860. doi: 10.3934/dcds.2016.36.851 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]