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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 796368, 7 pages
http://dx.doi.org/10.5402/2012/796368
Research Article

Regularity Criteria for Hyperbolic Navier-Stokes and Related System

1Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

Received 10 July 2012; Accepted 2 August 2012

Academic Editors: S. Cingolani, P. Mironescu, L. Sanchez, and T. Tran

Copyright © 2012 Jishan Fan and Tohru Ozawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Rack and J. Saal, “Hyperbolic Navier-Stokes equations I: local well posedness,” Evolution Equations and Control Theory, vol. 1, no. 1, pp. 195–215, 2012. View at Publisher · View at Google Scholar
  2. H. Kozono, T. Ogawa, and Y. Taniuchi, “The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,” Mathematische Zeitschrift, vol. 242, no. 2, pp. 251–278, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. I. Bejenaru, A. Ionescu, C. E. Kenig, and D. Tataru, “Equivariant Schrödinger maps in two spatial dimensions,” http://arxiv.org/abs/1112.6122.
  5. I. Bejenaru, A. D. Ionescu, and C. E. Kenig, “Global existence and uniqueness of Schrödinger maps in dimensions d4,” Advances in Mathematics, vol. 215, no. 1, pp. 263–291, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. I. Bejenaru, A. D. Ionescu, and C. E. Kenig, “On the stability of certain spin models in 2+1 dimensions,” Journal of Geometric Analysis, vol. 21, no. 1, pp. 1–39, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru, “Global Schrödinger maps in dimensions d2: small data in the critical Sobolev spaces,” Annals of Mathematics, vol. 173, no. 3, pp. 1443–1506, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. I. Bejenaru and D. Tataru, “Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions,” http://arxiv.org/abs/1009.1608.
  9. I. Bejenaru, “Global results for Schrödinger maps in dimensions n3,” Communications in Partial Differential Equations, vol. 33, no. 1–3, pp. 451–477, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. I. Bejenaru, “On Schrödinger maps,” American Journal of Mathematics, vol. 130, no. 4, pp. 1033–1065, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. N. H. Chang, J. Shatah, and K. Uhlenbeck, “Schrödinger maps,” Communications on Pure and Applied Mathematics, vol. 53, no. 5, pp. 590–602, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. S. Gustafson and E. Koo, “Global well-posedness for 2D radial Schrödinger maps into the sphere,” http://arxiv.org/abs/1105.5659.
  13. S. Gustafson, K. Kang, and T. P. Tsai, “Asymptotic stability of harmonic maps under the Schrödinger flow,” Duke Mathematical Journal, vol. 145, no. 3, pp. 537–583, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. Gustafson, K. Nakanishi, and T. P. Tsai, “Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on 2,” Communications in Mathematical Physics, vol. 300, no. 1, pp. 205–242, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. A. D. Ionescu and C. E. Kenig, “Low-regularity Schrödinger maps,” Differential and Integral Equations, vol. 19, no. 11, pp. 1271–1300, 2006. View at Zentralblatt MATH
  16. A. D. Ionescu and C. E. Kenig, “Low-regularity Schrödinger maps. II. Global well-posedness in dimensions d3,” Communications in Mathematical Physics, vol. 271, no. 2, pp. 523–559, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J. Kato, “Existence and uniqueness of the solution to the modified Schrödinger map,” Mathematical Research Letters, vol. 12, no. 2-3, pp. 171–186, 2005. View at Zentralblatt MATH
  18. J. Kato and H. Koch, “Uniqueness of the modified Schrödinger map in H3/4+ϵ(R2),” Communications in Partial Differential Equations, vol. 32, no. 1–3, pp. 415–429, 2007. View at Publisher · View at Google Scholar
  19. C. E. Kenig, G. Ponce, and L. Vega, “On the initial value problem for the Ishimori system,” Annales Henri Poincaré, vol. 1, no. 2, pp. 341–384, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. C. E. Kenig and A. R. Nahmod, “The Cauchy problem for the hyperbolic-elliptic Ishimori system and Schrödinger maps,” Nonlinearity, vol. 18, no. 5, pp. 1987–2009, 2005. View at Publisher · View at Google Scholar
  21. H. McGahagan, “An approximation scheme for Schrödinger maps,” Communications in Partial Differential Equations, vol. 32, no. 1–3, pp. 375–400, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. F. Merle, P. Raphaël, and 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, and I. Rodnianski, “Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map,” Comptes Rendus Mathématique, vol. 349, no. 5-6, pp. 279–283, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. A. Nahmod, J. Shatah, L. Vega, and C. Zeng, “Schrödinger maps and their associated frame systems,” International Mathematics Research Notices, vol. 2007, no. 21, Article ID rnm088, 29 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. A. Nahmod, A. Stefanov, and K. Uhlenbeck, “On Schrödinger maps,” Communications on Pure and Applied Mathematics, vol. 56, no. 1, pp. 114–151, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. I. Rodnianski, Y. A. Rubinstein, and G. Staffilani, “On the global well-posedness of the one-dimensional Schrödinger map flow,” Analysis & PDE, vol. 2, no. 2, pp. 187–209, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. I. Rodnianski and J. Sterbenz, “On the formation of singularities in the critical O(3)σ-model,” Annals of Mathematics, vol. 172, no. 1, pp. 187–242, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. P. Smith, “Conditional global regularity of Schrödinger maps: sub-threshold dispersed energy,” http://arxiv.org/abs/1012.4048.
  29. P. Smith, “Global regularity of critical Schrödinger maps: subthreshold dispersedenergy,” http://arxiv.org/abs/1112.0251.
  30. T. Tao, “Gauges forthe Schrödinger map”.
  31. T. Ozawa and J. Zhai, “Global existence of small classical solutions to nonlinear Schrödinger equations,” Annales de l'Institut Henri Poincaré, vol. 25, no. 2, pp. 303–311, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. B. Guo and Y. Han, “Global regular solutions for Landau-Lifshitz equation,” Frontiers of Mathematics in China, vol. 1, no. 4, pp. 538–568, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. X. Pu and B. Guo, “Well-posedness for the fractional Landau-Lifshitz equation without Gilbert damping,” Calculus of Variations and Partial Differential Equations. In press. View at Publisher · View at Google Scholar