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. 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

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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. 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

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