AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Near soliton evolution for equivariant Schrödinger Maps in two spatial dimensions
About this Title
Ioan Bejenaru, Department of Mathematics, University of Chicago, Chicago, Illinois 60637 and Daniel Tataru, Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 228, Number 1069
ISBNs: 978-0-8218-9215-2 (print); 978-1-4704-1481-8 (online)
DOI: https://doi.org/10.1090/memo/1069
Published electronically: July 11, 2013
MSC: Primary 58J35; Secondary 35B65
Table of Contents
Chapters
- 1. Introduction
- 2. An outline of the paper
- 3. The Coulomb gauge representation of the equation
- 4. Spectral analysis for the operators $H$, $\tilde H$; the $X,L X$ spaces
- 5. The linear $\tilde H$ Schrödinger equation
- 6. The time dependent linear evolution
- 7. Analysis of the gauge elements in $X,LX$
- 8. The nonlinear equation for $\psi$
- 9. The bootstrap estimate for the $\lambda$ parameter.
- 10. The bootstrap argument
- 11. The $\dot H^1$ instability result
Abstract
We consider the Schrödinger Map equation in $2+1$ dimensions, with values into $\mathbb {S}^2$. This admits a lowest energy steady state $Q$, namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. We prove that $Q$ is unstable in the energy space $\dot H^1$. However, in the process of proving this we also show that within the equivariant class $Q$ is stable in a stronger topology $X \subset \dot H^1$.- Ioan Bejenaru, On Schrödinger maps, Amer. J. Math. 130 (2008), no. 4, 1033–1065. MR 2427007, DOI 10.1353/ajm.0.0014
- Ioan Bejenaru, Global results for Schrödinger maps in dimensions $n\geq 3$, Comm. Partial Differential Equations 33 (2008), no. 1-3, 451–477. MR 2398238, DOI 10.1080/03605300801895225
- I. Bejenaru, A. D. Ionescu, and C. E. Kenig, Global existence and uniqueness of Schrödinger maps in dimensions $d\geq 4$, Adv. Math. 215 (2007), no. 1, 263–291. MR 2354991, DOI 10.1016/j.aim.2007.04.009
- I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru, Global Schrödinger maps in dimensions $d\geq 2$: small data in the critical Sobolev spaces, Ann. of Math. (2) 173 (2011), no. 3, 1443–1506. MR 2800718, DOI 10.4007/annals.2011.173.3.5
- Ioan Bejenaru and Daniel Tataru, Large data local solutions for the derivative NLS equation, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 957–985. MR 2443925, DOI 10.4171/JEMS/136
- Nai-Heng Chang, Jalal Shatah, and Karen Uhlenbeck, Schrödinger maps, Comm. Pure Appl. Math. 53 (2000), no. 5, 590–602. MR 1737504, DOI 10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.3.CO;2-I
- S. Gustafson, K. Kang, and T.-P. Tsai, Schrödinger flow near harmonic maps, Comm. Pure Appl. Math. 60 (2007), no. 4, 463–499. MR 2290708, DOI 10.1002/cpa.20143
- Stephen Gustafson, Kyungkeun Kang, and Tai-Peng Tsai, Asymptotic stability of harmonic maps under the Schrödinger flow, Duke Math. J. 145 (2008), no. 3, 537–583. MR 2462113, DOI 10.1215/00127094-2008-058
- Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai, Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on $\Bbb R^2$, Comm. Math. Phys. 300 (2010), no. 1, 205–242. MR 2725187, DOI 10.1007/s00220-010-1116-6
- Alexandru D. Ionescu and Carlos E. Kenig, Low-regularity Schrödinger maps, Differential Integral Equations 19 (2006), no. 11, 1271–1300. MR 2278007
- Alexandru D. Ionescu and Carlos E. Kenig, Low-regularity Schrödinger maps. II. Global well-posedness in dimensions $d\geq 3$, Comm. Math. Phys. 271 (2007), no. 2, 523–559. MR 2287916, DOI 10.1007/s00220-006-0180-4
- Martin Hadac, Sebastian Herr, and Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, 917–941. MR 2526409, DOI 10.1016/j.anihpc.2008.04.002
- Jun Kato, Existence and uniqueness of the solution to the modified Schrödinger map, Math. Res. Lett. 12 (2005), no. 2-3, 171–186. MR 2150874, DOI 10.4310/MRL.2005.v12.n2.a3
- Jun Kato and Herbert Koch, Uniqueness of the modified Schrödinger map in $H^{3/4+\epsilon }(\Bbb R^2)$, Comm. Partial Differential Equations 32 (2007), no. 1-3, 415–429. MR 2304155, DOI 10.1080/03605300600910332
- Carlos E. Kenig and Andrea R. Nahmod, The Cauchy problem for the hyperbolic-elliptic Ishimori system and Schrödinger maps, Nonlinearity 18 (2005), no. 5, 1987–2009. MR 2164729, DOI 10.1088/0951-7715/18/5/007
- J. Krieger, W. Schlag, and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543–615. MR 2372807, DOI 10.1007/s00222-007-0089-3
- Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58 (2005), no. 2, 217–284. MR 2094851, DOI 10.1002/cpa.20067
- Helena McGahagan, An approximation scheme for Schrödinger maps, Comm. Partial Differential Equations 32 (2007), no. 1-3, 375–400. MR 2304153, DOI 10.1080/03605300600856758
- Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal. 11 (2001), no. 1, 74–123. MR 1829643, DOI 10.1007/PL00001673
- Andrea Nahmod, Atanas Stefanov, and Karen Uhlenbeck, On Schrödinger maps, Comm. Pure Appl. Math. 56 (2003), no. 1, 114–151. MR 1929444, DOI 10.1002/cpa.10054
- Andrea Nahmod, Atanas Stefanov, and Karen Uhlenbeck, On Schrödinger maps, Comm. Pure Appl. Math. 56 (2003), no. 1, 114–151. MR 1929444, DOI 10.1002/cpa.10054
- Andrea Nahmod, Jalal Shatah, Luis Vega, and Chongchun Zeng, Schrödinger maps and their associated frame systems, Int. Math. Res. Not. IMRN 21 (2007), Art. ID rnm088, 29. MR 2352219, DOI 10.1093/imrn/rnm088
- Pierre Raphaël and Igor Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 1–122. MR 2929728, DOI 10.1007/s10240-011-0037-z
- Igor Rodnianski and Jacob Sterbenz, On the formation of singularities in the critical $\textrm {O}(3)$ $\sigma$-model, Ann. of Math. (2) 172 (2010), no. 1, 187–242. MR 2680419, DOI 10.4007/annals.2010.172.187
- P.-L. Sulem, C. Sulem, and C. Bardos, On the continuous limit for a system of classical spins, Comm. Math. Phys. 107 (1986), no. 3, 431–454. MR 866199
- Daniel Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008), no. 3, 571–634. MR 2418923, DOI 10.1353/ajm.0.0000