Skip to main content
SpringerLink
Log in

Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case

  • Chapter
  • First Online:
New Tools for Nonlinear PDEs and Application

Part of the book series: Trends in Mathematics ((TM))

  • 675 Accesses

Abstract

The aim of this work is to give a complete picture concerning the asymptotic behaviour of the solutions to fractional Ginzburg-Landau equation. In previous works, we have shown global well-posedness for the past interval in the case where spatial dimension is less than or equal to 3. Moreover, we have also shown blow-up of solutions for the future interval in one dimensional case. In this work, we summarise the asymptotic behaviour in the case where spatial dimension is less than or equal to 3 by proving blow-up of solutions for a future time interval in multidimensional case. The result is obtained via ODE argument by exploiting a new weighted commutator estimate.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Bellazzini, V. Georgiev, N. Visciglia, Long time dynamics for semirelativistic NLS and half wave in arbitrary dimension. Math. Annalen. 371(1–2), 707–740 (2018)

    Article  MathSciNet  Google Scholar 

  2. J.P. Borgna, D.F. Rial, Existence of ground states for a one-dimensional relativistic Schödinger equation. J. Math. Phys. 53, 062301 (2012)

    Article  MathSciNet  Google Scholar 

  3. A.P. Calderón, A. Zygmund, On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)

    Article  MathSciNet  Google Scholar 

  4. T. Cazenave, Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10 (American Mathematical Society/New York University/Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, 2003)

    Google Scholar 

  5. T. Cazenave, F.B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in Nonlinear Semigroups, Partial Differential Equations and Attractors, ed. by T.L. Gill, W.W. Zachary (Washington, DC, 1987). Lecture Notes in Mathematics, vol. 1394 (Springer, Berlin, 1989), pp. 18–29

    Google Scholar 

  6. T. Cazenave, F.B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s. Nonlinear analysis. Theory, methods & applications. Int. Multidiscip. J. Ser. A Theory Methods 14, 807–836 (1990)

    Google Scholar 

  7. Y. Cho, T. Ozawa, Sobolev inequalities with symmetry. Commun. Contemp. Math. 11, 355–365 (2009)

    Article  MathSciNet  Google Scholar 

  8. E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  9. L. Forcella, K. Fujiwara, V. Georgiev, T. Ozawa, Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete Contin. Dyn. Syst. 39, 2661–2678 (2019). arXiv:1804.02524

    Google Scholar 

  10. K. Fujiwara, Remark on local solvability of the Cauchy problem for semirelativistic equations. J. Math. Anal. Appl. 432, 744–748 (2015)

    Article  MathSciNet  Google Scholar 

  11. K. Fujiwara, V. Georgiev, T. Ozawa, Blow-up for self-interacting fractional Ginzburg-Landau equation. Dyn. Partial Differ. Equ. 15, 175–182 (2018)

    Article  MathSciNet  Google Scholar 

  12. K. Fujiwara, V. Georgiev, T. Ozawa, On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases. arXiv: 1611.09674 (2016)

    Google Scholar 

  13. J. Ginibre, T. Ozawa, G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 60, 211–239 (1994)

    MathSciNet  MATH  Google Scholar 

  14. J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50–68 (1995)

    Article  MathSciNet  Google Scholar 

  15. L. Grafakos, S. Oh, The Kato-Ponce Inequality. Comm. Partial Differ. Equ. 39, 1128–1157 (2014)

    Article  MathSciNet  Google Scholar 

  16. M. Ikeda, T. Inui, Some non-existence results for the semilinear Schrödinger equation without gauge invariance. J. Math. Anal. Appl. 425, 758–773 (2015)

    Article  MathSciNet  Google Scholar 

  17. M. Ikeda, Y. Wakasugi, Small-data blow-up of L 2-solution for the nonlinear Schrödinger equation without gauge invariance. Differ. Integral Equ. 26, 11–12 (2013)

    MATH  Google Scholar 

  18. T. Inui, Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity. Proc. Am. Math. Soc. 144, 2901–2909 (2016)

    Article  Google Scholar 

  19. T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)

    Article  MathSciNet  Google Scholar 

  20. C. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71, 1–21 (1993)

    Article  MathSciNet  Google Scholar 

  21. S. Klainerman, M. Machedon, Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46, 1221–1268 (1993)

    Article  MathSciNet  Google Scholar 

  22. A. Kufner, B. Opic, Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series (Longman Scientific & Technical, Harlow, 1990)

    Google Scholar 

  23. N. Laskin, Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A. 268, 298–305 (2000)

    Article  MathSciNet  Google Scholar 

  24. E. Lenzmann, A. Schikorra, Sharp commutator estimates via harmonic extensions. arXiv: 1609.08547

    Google Scholar 

  25. D. Li, On Kato-Ponce and fractional Leibniz. Rev. Mat. Iberoamericana. (in press). arXiv:1609.01780v2. It appeared on arXiv in 2016 and revised in 2018

    Google Scholar 

  26. M. Nakamura, T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces. Publ. Res. Inst. Math. Sci. 37, 255–293 (2001)

    Article  MathSciNet  Google Scholar 

  27. T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 25, 403–408 (2006)

    Article  Google Scholar 

  28. W. Sickel, L. Skrzypczak, Radial subspaces of Besov and Lizorkin-Triebel classes: extended Strauss lemma and compactness of embeddings. J. Fourier Anal. Appl. 6, 639–662 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

V. Georgiev was supported in part by INDAM, GNAMPA – Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and Top Global University Project, Waseda University and the Project PRA 2018 – 49 of University of Pisa. The authors are grateful to the referees for their helpful comments.

Author information

Authors and Affiliations

  1. Bâtiment des Mathématiques, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Luigi Forcella

  2. Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore, Pisa, Italy

    Kazumasa Fujiwara

  3. Department of Mathematics, University of Pisa, Pisa, Italy

    Vladimir Georgiev

  4. Faculty of Science and Engineering, Waseda University, Shinjuku-ku, Tokyo, Japan

    Vladimir Georgiev

  5. IMI–BAS, Sofia, Bulgaria

    Vladimir Georgiev

  6. Department of Applied Physics, Waseda University, Shinjuku-ku, Tokyo, Japan

    Tohru Ozawa

Authors
  1. Luigi Forcella
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kazumasa Fujiwara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vladimir Georgiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tohru Ozawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazumasa Fujiwara .

Editor information

Editors and Affiliations

  1. Mathematics, University of Bari, Bari, Italy

    Marcello D'Abbicco

  2. Department of Computing and Mathematics, University of São Paulo, Ribeirão Preto, São Paulo, Brazil

    Marcelo Rempel Ebert

  3. Department of Mathematics, University of Pisa, Pisa, Italy

    Vladimir Georgiev

  4. Department of Applied Physics, Waseda University, Tokyo, Japan

    Tohru Ozawa

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Forcella, L., Fujiwara, K., Georgiev, V., Ozawa, T. (2019). Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case. In: D'Abbicco, M., Ebert, M., Georgiev, V., Ozawa, T. (eds) New Tools for Nonlinear PDEs and Application. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10937-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation