Skip to main content
SpringerLink
Log in

The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schrödinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bejenaru I. On Schrödinger maps. Amer J Math, 2008, 130: 1033–1065

    Article  MATH  MathSciNet  Google Scholar 

  2. Bejenaru I. Global results for Schrödinger maps in dimensions n ≥ 3. Comm Partial Differential Equations, 2008, 33: 451–477

    Article  MATH  MathSciNet  Google Scholar 

  3. Bejenaru I, Ionescu A D, Kenig C E. Global existence and uniqueness of Schrödinger maps in dimensions d ≥ 4. Adv Math, 2007, 215: 263–291

    Article  MATH  MathSciNet  Google Scholar 

  4. Bejenaru I, Ionescu A D, Kenig C E, et al. Global Schrödinger maps in dimensions d ≥ 2: Small data in the critical Sobolev spaces. Ann of Math (2), 2011, 173: 1443–1506

    Article  MATH  MathSciNet  Google Scholar 

  5. Chang N H, Shatah J, Uhlenbeck K. Schrödinger maps. Comm Pure Appl Math, 2000, 53: 590–602

    Article  MATH  MathSciNet  Google Scholar 

  6. Ding W, Wang Y. Local Schrödinger flow into Kähler manifolds. Sci China Ser A, 2001, 44: 1446–1464

    Article  MATH  MathSciNet  Google Scholar 

  7. Guo B, Ding S. Landau-Lifshitz Equations. Hackensack: World Scientific, 2008

    Book  MATH  Google Scholar 

  8. Guo Z. Spherically averaged maximal function and scattering for the 2D cubic derivative Schrödinger equation. Int Math Res Not IMRN, 2017, in press

    Google Scholar 

  9. Guo Z, Wang B. Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation. J Differential Equations, 2009, 246: 3864–3901

    Article  MATH  MathSciNet  Google Scholar 

  10. Han L, Wang B, Guo B. Inviscid limit for the derivative Ginzburg-Landau equation with small data in modulation and Sobolev spaces. Appl Comput Harmon Anal, 2012, 32: 197–222

    Article  MATH  MathSciNet  Google Scholar 

  11. Hmidi T, Keraani S. Inviscid limit for the two-dimensional N-S system in a critical Besov space. Asymptot Anal, 2007, 53: 125–138

    MATH  MathSciNet  Google Scholar 

  12. Huang C, Wang B. Inviscid limit for the energy-critical complex Ginzburg-Landau equation. J Funct Anal, 2008, 255: 681–725

    Article  MATH  MathSciNet  Google Scholar 

  13. Ionescu A D, Kenig C E. Low-regularity Schrödinger maps. Differential Integral Equations, 2006, 19: 1271–1300

    MATH  MathSciNet  Google Scholar 

  14. Ionescu A D, Kenig C E. Low-regularity Schrödinger maps, II: Global well-posedness in dimensions d ≥ 3. Comm Math Phys, 2007, 271: 523–559

    Article  MATH  MathSciNet  Google Scholar 

  15. Keel M, Tao T. Endpoint Strichartz estimates. Amer J Math, 1998, 120: 360–413

    Article  MATH  MathSciNet  Google Scholar 

  16. Lakshmanan M. The fascinating world of the Landau-Lifshitz-Gilbert equation: An overview. Philos Trans R Soc Lond Ser A Math Phys Eng Sci, 2011, 369: 1280–1300

    Article  MATH  MathSciNet  Google Scholar 

  17. Lakshmanan M, Nakamura K. Landau-Lifshitz equation of Ferromagnetism: Exact treatment of the Gilbert damping. Phys Rev Lett, 1984, 53: 2497–2499

    Article  Google Scholar 

  18. Landau L D, Lifshitz E M. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys Z Sovietunion, 1935, 8: 153–169

    MATH  Google Scholar 

  19. Tao T. Global regularity of wave maps II: Small energy in two dimensions. Comm Math Phys, 2001, 224: 443–544

    Article  MATH  MathSciNet  Google Scholar 

  20. Tataru D. Local and global results for the wave maps I. Comm Partial Differential Equations, 1998, 23: 1781–1793

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang B. The limit behavior of solutions for the Cauchy problem of the complex Ginzburg-Landau equation. Comm Pure Appl Math, 2002, 55: 481–508

    Article  MATH  MathSciNet  Google Scholar 

  22. Wang B, Huo Z, Hao C, et al. Harmonic Analysis Method for Nonlinear Evolution Equations, I. Singapore: World Scientific, 2011

    Book  MATH  Google Scholar 

  23. Wang B, Wang Y. The inviscid limit for the derivative Ginzburg-Landau equations. J Math Pures Appl (9), 2004, 83: 477–502

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by Australian Research Council Discovery Project (Grant No. DP170101060), National Natural Science Foundation of China (Grant No. 11201498) and the China Scholarship Council (Grant No. 201606495010).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Monash University, Melbourne, VIC, 3800, Australia

    ZiHua Guo

  2. School of Statistics and Mathematics, Central University of Finance and Economics, Beijing, 100081, China

    ChunYan Huang

Authors
  1. ZiHua Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. ChunYan Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ChunYan Huang.

Additional information

Dedicated to the memory of Professor CHENG MinDe on the occasion of the centenary of his birth

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, Z., Huang, C. The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space. Sci. China Math. 60, 2155–2172 (2017). https://doi.org/10.1007/s11425-017-9146-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-017-9146-x

Keywords

MSC(2010)

Search

Navigation