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A Regularity Criterion for the Schrödinger Map

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Current Trends in Analysis and Its Applications

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

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Abstract

We prove a regularity criterion

$$\nabla u\in L^2\bigl(0,T;\mathit{BMO}\bigl(\mathbb{R}^n\bigr) \bigr) $$

with 2≤n≤5 for the Schrödinger map. Here BMO is the space of functions with bounded mean oscillations.

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References

  1. I. Bejenaru, Global results for Schrödinger maps in dimensions n≥3. Commun. Partial Differ. Equ. 33(1-3), 451–477 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. I. Bejenaru, On Schrödinger maps. Am. J. Math. 130(4), 1033–1065 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Bejenaru, A.D. Ionescu, C.E. Kenig, Global existence and uniqueness of Schrödinger maps in dimensions d≥4. Adv. Math. 215(1), 263–291 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Bejenaru, A.D. Ionescu, C.E. Kenig, On the stability of certain spin models in 2+1 dimensions. J. Geom. Anal. 21(1), 1–39 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Bejenaru, A.D. Ionescu, C.E. Kenig, D. Tataru, Global Schrödinger maps in dimensions d≥2: small data in the critical Sobolev spaces. Ann. Math. 173(3), 1443–1506 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. I. Bejenaru, A. Ionescu, C.E. Kenig, D. Tataru, Equivariant Schrödinger maps in two spatial dimensions. http://arxiv.org/abs/1112.6122

  7. I. Bejenaru, D. Tataru, Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions. http://arxiv.org/abs/1009.1608

  8. N.H. Chang, J. Shatah, K. Uhlenbeck, Schrödinger maps. Commun. Pure Appl. Math. 53(5), 590–602 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Fan, T. Ozawa, On regularity criterion for the 2D wave maps and the 4D biharmonic wave maps. GAKUTO international series. Math. Sci. Appl. 32, 69–83 (2010)

    MathSciNet  Google Scholar 

  10. B. Guo, Y. Han, Global regular solutions for Landau–Lifshitz equation. Front. Math. China 1(4), 538–568 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. S. Gustafson, K. Kang, T.P. Tsai, Asymptotic stability of harmonic maps under the Schrödinger flow. Duke Math. J. 145(3), 537–583 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Gustafson, E. Koo, Global well-posedness for 2D radial Schrödinger maps into the sphere. http://arxiv.org/abs/1105.5659

  13. S. Gustafson, K. Nakanishi, a.T.P. Tsai, Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau–Lifshitz, and Schrödinger maps on \(\mathbb{R}^{2}\). Commun. Math. Phys. 300(1), 205–242 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. A.D. Ionescu, C.E. Kenig, Low-regularity Schrödinger maps. Differ. Integral Equ. 19(11) 1271–1300 (2006)

    MATH  MathSciNet  Google Scholar 

  15. A.D. Ionescu, C.E. Kenig, Low-regularity Schrödinger maps. II. Global well-posedness in dimensions d≥3. Commun. Math. Phys. 271(2), 523–559 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Kato, Existence and uniqueness of the solution to the modified Schrödinger map. Math. Res. Lett. 12(2-3), 171–186 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Kato, H. Koch, Uniqueness of the modified Schrödinger map in H 3/4+ϵ(R 2). Commun. Partial Differ. Equ. 32(1-3), 415–429 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Kato, G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  19. C.E. Kenig, G. Ponce, L. Vega, On the initial value problem for the Ishimori system. Ann. Henri Poincaré 1(2), 341–384 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. C.E. Kenig, A.R. Nahmod, The Cauchy problem for the hyperbolic-elliptic Ishimori system and Schrödinger maps. Nonlinearity 18(5), 1987–2009 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. H. McGahagan, An approximation scheme for Schrödinger maps. Commun. Partial Differ. Equ. 32(1-3), 375–400 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. F. Merle, P. Raphaël, I. Rodnianski, Blow up dynamics for smooth data equivariant solutions to the energy critical Schrödinger map problem. http://arxiv.org/abs/1102.4308

  23. F. Merle, P. Raphaël, I. Rodnianski, Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map. C. R. Math. 349(5-6), 279–283 (2011)

    Article  MATH  Google Scholar 

  24. A. Nahmod, J. Shatah, L. Vega, C. Zeng, Schrödinger maps and their associated frame systems. Int. Math. Res. Not. 2007(21) (2007). Artical ID rnm088, 29 pages

    Google Scholar 

  25. A. Nahmod, A. Stefanov, K. Uhlenbeck, On Schrödinger maps. Commun. Pure Appl. Math. 56(1), 114–151 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow. SIAM J. Math. Anal. 34(6), 1318–1330 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. T. Ozawa, J. Zhai, Global existence of small classical solutions to nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré 25(2), 303–311 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. I. Rodnianski, Y.A. Rubinstein, G. Staffilani, On the global well-posedness of the one-dimensional Schrödinger map flow. Anal. PDE 2(2), 187–209 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. I. Rodnianski, J. Sterbenz, On the formation of singularities in the critial O(3)σ-model. Ann. Math. 172(1), 187–242 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. P. Smith, Conditional global regularity of Schrödinger maps: sub-threshold dispersed energy. http://arxiv.org/abs/1012.4048

  31. P. Smith, Global regularity of critical Schrödinger maps: subthreshold dispersedenergy. http://arxiv.org/abs/1112.0251

  32. P.L. Sulem, C. Sulem, C. Bardos, On the continous limit for a system of classical spins. Commun. Math. Phys. 107, 431–454 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  33. T. Tao, Gauges for the Schrödinger map. http://www.math.ucla.edu/~tao/preprints/Expository/smap.dvi

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Acknowledgement

J. Fan is partially supported by NSFC (No. 11171154).

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037, P.R. China

    Jishan Fan

  2. Department of Applied Physics, Waseda University, Tokyo, 169-8555, Japan

    Tohru Ozawa

Authors
  1. Jishan Fan
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  2. Tohru Ozawa
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Correspondence to Tohru Ozawa .

Editor information

Editors and Affiliations

  1. Dept. of Computer Sc. & Comp. Methods, Pedagogical University, Kraków, Poland

    Vladimir V. Mityushev

  2. Imperial College London, London, United Kingdom

    Michael V. Ruzhansky

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Fan, J., Ozawa, T. (2015). A Regularity Criterion for the Schrödinger Map. In: Mityushev, V., Ruzhansky, M. (eds) Current Trends in Analysis and Its Applications. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-12577-0_26

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