Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials

  • Paolo Antonelli
  • Alessandro Michelangeli
  • Raffaele Scandone


We prove the existence of weak solutions in the space of energy for a class of nonlinear Schrödinger equations in the presence of a external, rough, time-dependent magnetic potential. Under our assumptions, it is not possible to study the problem by means of usual arguments like resolvent techniques or Fourier integral operators, for example. We use a parabolic regularisation, and we solve the approximating Cauchy problem. This is achieved by obtaining suitable smoothing estimates for the dissipative evolution. The total mass and energy bounds allow to extend the solution globally in time. We then infer sufficient compactness properties in order to produce a global-in-time finite energy weak solution to our original problem.


Nonlinear Schrödinger equation Magnetic potentials Viscosity regularisation Strichartz estimates Weak solutions 

Mathematics Subject Classification

35D40 35H30 35Q41 35Q55 35K08 


  1. 1.
    Antonelli, P., d’Amico, M., Marcati, P.: Nonlinear Maxwell-Schrödinger system and quantum magneto-hydrodynamics in 3-D. Commun. Math. Sci. 15, 451–479 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carles, R.: Nonlinear Schrödinger equation with time dependent potential. Commun. Math. Sci. 9(4), 937–964 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York University Courant Institute of Mathematical Sciences, New York (2003)Google Scholar
  4. 4.
    Cazenave, T., Haraux, A.: An introduction to semilinear evolution equations, translated from the 1990 French original by Y. Martel and revised by the authors. Oxford Lecture Series in Mathematics and Applications, 13. Clarendon, Oxford University Press, New York (1998)Google Scholar
  5. 5.
    Cazenave, T., Weissler, F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in \(H^s\). Nonlinear Anal. 14, 807–836 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christ, M., Kiselev, A.: Maximal functions associated to filtrations. J. Funct. Anal. 179, 409–425 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in \({\mathbb{R}}^3\). Ann. Math. 167(2), 767–865 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D’Ancona, P., Fanelli, L.: Strichartz and smoothing estimates of dispersive equations with magnetic potentials. Commun. Partial Differential Equations 33, 1082–1112 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D’Ancona, P., Fanelli, L., Vega, L., Visciglia, N.: Endpoint Strichartz estimates for the magnetic Schrödinger equation. J. Funct. Anal. 258, 3227–3240 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    De Bouard, A.: Nonlinear Schrodinger equations with magnetic fields. Differ. Integral Equ. 4, 73–88 (1991)zbMATHGoogle Scholar
  11. 11.
    Dodson, B.: Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when \(n=3\) via a linear-nonlinear decomposition. Discrete Cont. Dyn. Syst. 33, 1905–1926 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Doi, S.-I.: On the Cauchy problem for Schrödinger type equations and the regularity of solutions. J. Math. Kyoto Univ. 34, 319–328 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Doi, S.-I.: Remarks on the Cauchy problem for Schrödinger-type equations. Commun. Partial Differ. Equ. 21, 163–178 (1996)CrossRefzbMATHGoogle Scholar
  14. 14.
    Erdoğan, M.B., Goldberg, M., Schlag, W.: Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in \({\mathbb{R}}^3\). J. Eur. Math. Soc. (JEMS) 10, 507–531 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Erdoğan, M.B., Goldberg, M., Schlag, W.: Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions. Forum Math. 21, 687–722 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Fanelli, L., Garcia, A.: Counterexamples to Strichartz estimates for the magnetic Schrödinger equation. Commun. Contemp. Math. 13, 213–234 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fang, D., Han, Z., Dai, J.: The nonlinear Schrödinger equations with combined nonlinearities of power-type and Hartree-type. Chin. Ann. Math. Ser. B 32, 435–474 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fujiwara, D.: A construction of the fundamental solution for the Schrödinger equation. J. Anal. Math. 35, 41–96 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Georgiev, V., Stefanov, A., Tarulli, M.: Smoothing-Strichartz estimates for the Schrödinger equation with small magnetic potential. Discrete Cont. Dyn. Syst. 17, 771–786 (2007)CrossRefzbMATHGoogle Scholar
  20. 20.
    Ginibre, J., Velo, G., Scattering theory in the energy space for a class of nonlinear Schrödinger equations. In: Semigroups, Theory and Applications, Vol. I (Trieste, vol. 141 of Pitman Res. Notes Math. Ser., Longman Sci. Tech. Harlow 1986, pp. 110–120 (1984)Google Scholar
  21. 21.
    Ginibre, J., Velo, G.: Smoothing properties and retarded estimates for some dispersive evolution equations. Commun. Math. Phys. 144, 163–188 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of Hartree equations. In: Nonlinear Wave Equations (Providence, RI, vol. 263 of Contemp. Math., Amer. Math. Soc., Providence, RI 2000, pp. 29–60 (1998)Google Scholar
  23. 23.
    Guo, Y., Nakamitsu, K., Strauss, W.: Global finite-energy solutions of the Maxwell-Schrödinger system. Commun. Math. Phys. 170, 181–196 (1995)CrossRefzbMATHGoogle Scholar
  24. 24.
    Haas, F.: Quantum plasmas, vol. 65 of Springer Series on Atomic, Optical, and Plasma Physics, Springer, New York. An hydrodynamic approach (2011)Google Scholar
  25. 25.
    Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Killip, R., Vişan, M.: Nonlinear Schrödinger equations at critical regularity. In: Evolution Equations, vol. 17 of Clay Math. Proc., , pp. 325–437. Amer. Math. Soc., Providence (2013)Google Scholar
  27. 27.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  28. 28.
    Linares, F., Ponce, G.: Introduction to Nonlinear Dispersive Equations, Universitext, 2nd edn. Springer, New York (2015)zbMATHGoogle Scholar
  29. 29.
    Miao, C., Xu, G., Zhao, L.: The Cauchy problem of the Hartree equation. J. Partial Differ. Equ. 21, 22–44 (2008)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Michel, L.: Remarks on non-linear Schrödinger equation with magnetic fields. Commun. Partial Differ. Equ. 33, 1198–1215 (2008)CrossRefzbMATHGoogle Scholar
  31. 31.
    Michelangeli, A.: Role of scaling limits in the rigorous analysis of Bose-Einstein condensation. J. Math. Phys. 48, 102102 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Michelangeli, A.: Global well-posedness of the magnetic Hartree equation with non-Strichartz external fields. Nonlinearity 28, 2743 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mizutani, H.: Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Commun. Pure Appl. Anal. 13, 2177–2210 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Naibo, V., Stefanov, A.: On some Schrödinger and wave equations with time dependent potentials. Math. Ann. 334, 325–338 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Nakamura, M., Wada, T.: Global existence and uniqueness of solutions to the Maxwell-Schrödinger equations. Commun. Math. Phys. 276, 315–339 (2007)CrossRefzbMATHGoogle Scholar
  36. 36.
    Nakamura, Y., Shimomura, A.: Local well-posedness and smoothing effects of strong solutions for nonlinear Schrödinger equations with potentials and magnetic fields. Hokkaido Math. J. 34, 37–63 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Olgiati, A.: Remarks on the derivation of Gross–Pitaevskii equation with magnetic Laplacian. In: Dell’Antonio, G., Michelangeli, A. (eds.) Advances in Quantum Mechanics: Contemporary Trends and Open Problems, vol. 18 of Springer INdAM Series, pp. 257–266. Springer International Publishing (2017)Google Scholar
  38. 38.
    Roubíček, T.S.: Nonlinear partial differential equations with applications, vol. 153 of International Series of Numerical Mathematics, 2nd edn. Birkhäuser/Springer Basel AG, Basel (2013)Google Scholar
  39. 39.
    Schlein, B.: Derivation of effective evolution equations from microscopic quantum dynamics (2008). arXiv:0807.4307
  40. 40.
    Stefanov, A.: Strichartz estimates for the magnetic Schrödinger equation. Adv. Math. 210, 246–303 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Wang, B., Huo, Z., Hao, C., Guo, Z.: Harmonic Analysis Method for Nonlinear Evolution Equations. I. World Scientific Publishing Co. Pte. Ltd., Hackensack (2011)Google Scholar
  42. 42.
    Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987)CrossRefzbMATHGoogle Scholar
  43. 43.
    Yajima, K.: Schrödinger evolution equations with magnetic fields. J. Anal. Math. 56, 29–76 (1991)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Paolo Antonelli
    • 1
  • Alessandro Michelangeli
    • 2
  • Raffaele Scandone
    • 2
  1. 1.Gran Sasso Science Institute – GSSIL’AquilaItaly
  2. 2.SISSA – International School for Advanced StudiesTriesteItaly

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