Remarks on the derivation of finite energy weak solutions to the QHD system

Antonelli, Paolo

doi:10.1090/proc/14502

Remarks on the derivation of finite energy weak solutions to the QHD system
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Proc. Amer. Math. Soc. 149 (2021), 1985-1997 Request permission

Abstract:

In this note we give an alternative proof of existence of finite energy weak solutions to the quantum hydrodynamics (QHD) system. The main novelty in our approach is that no regularization procedure or approximation is needed, as it is only based on the integral formulation of NLS equation and the a priori bounds given by the Strichartz estimates. The main advantage of this proof is that it can be applied to a wider class of QHD systems.

References

Paolo Antonelli, Michele d’Amico, and Pierangelo Marcati, Nonlinear Maxwell-Schrödinger system and quantum magneto-hydrodynamics in 3-D, Commun. Math. Sci. 15 (2017), no. 2, 451–479. MR 3620565, DOI 10.4310/CMS.2017.v15.n2.a7
P. Antonelli, L.E. Hientzsch, P. Marcati, and H. Zheng, On some results for quantum hydrodynamical models, RIMS Kokyuroku n. 2070 (Mathematical Analysis in Fluid and Gas Dynamics), 86–107 (to appear), available online at http://www.kurims.kyoto-u.ac.jp/ kyodo/kokyuroku/contents/pdf/2070-08.pdf.
Paolo Antonelli and Pierangelo Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys. 287 (2009), no. 2, 657–686. MR 2481754, DOI 10.1007/s00220-008-0632-0
Paolo Antonelli and Pierangelo Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Ration. Mech. Anal. 203 (2012), no. 2, 499–527. MR 2885568, DOI 10.1007/s00205-011-0454-7
Paolo Antonelli and Pierangelo Marcati, Some results on systems for quantum fluids, Recent advances in partial differential equations and applications, Contemp. Math., vol. 666, Amer. Math. Soc., Providence, RI, 2016, pp. 41–54. MR 3537457, DOI 10.1090/conm/666/13237
Paolo Antonelli and Pierangelo Marcati, Quantum hydrodynamics with nonlinear interactions, Discrete Contin. Dyn. Syst. Ser. S 9 (2016), no. 1, 1–13. MR 3461643, DOI 10.3934/dcdss.2016.9.1
Rémi Carles, Raphaël Danchin, and Jean-Claude Saut, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity 25 (2012), no. 10, 2843–2873. MR 2979973, DOI 10.1088/0951-7715/25/10/2843
Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
Kazumasa Fujiwara and Hayato Miyazaki, The derivation of conservation laws for nonlinear Schrödinger equations with power type nonlinearities, Regularity and singularity for partial differential equations with conservation laws, RIMS Kôkyûroku Bessatsu, B63, Res. Inst. Math. Sci. (RIMS), Kyoto, 2017, pp. 13–21. MR 3751978
Carl L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math. 54 (1994), no. 2, 409–427. MR 1265234, DOI 10.1137/S0036139992240425
Ingenuin Gasser and Peter A. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asymptot. Anal. 14 (1997), no. 2, 97–116. MR 1451208, DOI 10.3233/ASY-1997-14201
J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309–327 (English, with French summary). MR 801582, DOI 10.1016/S0294-1449(16)30399-7
Feimin Huang, Hai-Liang Li, and Akitaka Matsumura, Existence and stability of steady-state of one-dimensional quantum hydrodynamic system for semiconductors, J. Differential Equations 225 (2006), no. 1, 1–25. MR 2228690, DOI 10.1016/j.jde.2006.02.002
Feimin Huang, Hailiang Li, Akitaka Matsumura, and Shinji Odanaka, Well-posedness and stability of quantum hydrodynamics for semiconductors in $\Bbb R^3$, Some problems on nonlinear hyperbolic equations and applications, Ser. Contemp. Appl. Math. CAM, vol. 15, Higher Ed. Press, Beijing, 2010, pp. 131–160. MR 2816449, DOI 10.1142/9789814322898_{0}006
Ansgar Jüngel, Hai-Liang Li, and Akitaka Matsumura, The relaxation-time limit in the quantum hydrodynamic equations for semiconductors, J. Differential Equations 225 (2006), no. 2, 440–464. MR 2225796, DOI 10.1016/j.jde.2005.11.007
Ansgar Jüngel, Maria Cristina Mariani, and Diego Rial, Local existence of solutions to the transient quantum hydrodynamic equations, Math. Models Methods Appl. Sci. 12 (2002), no. 4, 485–495. MR 1899838, DOI 10.1142/S0218202502001751
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
I. M. Khalatnikov, An introduction to the theory of superfluidity, Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. Translated from the Russian by Pierre C. Hohenberg; Translation edited and with a foreword by David Pines; Reprint of the 1965 edition. MR 1084373
L. Landau, Theory of the superfluidity of helium II, Phys. Rev. 60 (1941), 356.
Hailiang Li and Pierangelo Marcati, Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors, Comm. Math. Phys. 245 (2004), no. 2, 215–247. MR 2039696, DOI 10.1007/s00220-003-1001-7
Felipe Linares and Gustavo Ponce, Introduction to nonlinear dispersive equations, Universitext, Springer, New York, 2009. MR 2492151
E. Madelung, Quantentheorie in hydrodynamischer form, Z. Phys. 40 (1927), 322.
Shinya Nishibata and Masahiro Suzuki, Initial boundary value problems for a quantum hydrodynamic model of semiconductors: asymptotic behaviors and classical limits, J. Differential Equations 244 (2008), no. 4, 836–874. MR 2391346, DOI 10.1016/j.jde.2007.10.035
T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations 25 (2006), no. 3, 403–408. MR 2201679, DOI 10.1007/s00526-005-0349-2
L. Pitaevskii and S. Stringari, Bose-Einstein condensation and superfluidity, Clarendon Press, Oxford, 2016.
Hayato Miyazaki, The derivation of the conservation law for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, J. Math. Anal. Appl. 417 (2014), no. 2, 580–600. MR 3194504, DOI 10.1016/j.jmaa.2014.03.055
Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106