Ionescu, Alexandru D.; Kenig, Carlos E. Low-regularity Schrödinger maps. II: Global well-posedness in dimensions \(d \geq 3\). (English) Zbl 1137.35068 Commun. Math. Phys. 271, No. 2, 523-559 (2007). The authors prove that the Schrödinger map initial value problem is globally well-posed in dimensions \(d \geq 3\) for small data in the critical Besov spaces. Reviewer: Ömer Kavaklioglu (Izmir) Cited in 2 ReviewsCited in 42 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35B45 A priori estimates in context of PDEs 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35B65 Smoothness and regularity of solutions to PDEs Keywords:critical Besov spaces PDFBibTeX XMLCite \textit{A. D. Ionescu} and \textit{C. E. Kenig}, Commun. Math. Phys. 271, No. 2, 523--559 (2007; Zbl 1137.35068) Full Text: DOI References: [1] Bejenaru, I.: On Schrödinger maps. Preprint 2006, available at arxiv.org 0604255 · Zbl 1123.35060 [2] Chang N.-H., Shatah J. and Uhlenbeck K. (2000). Schrödinger maps. Comm. Pure Appl. Math. 53: 590–602 · Zbl 1028.35134 [3] Ding W.Y. and Wang Y.D. (2001). Local Schrödinger flow into Kähler manifolds. Sci. China Ser. A 44: 1446–1464 · Zbl 1019.53032 [4] Ionescu, A.D., Kenig, C.: Global well-posedness of the Benjamin–Ono equation in low-regularity spaces. J. Amer. Math. Soc. 50894-0347(06) 00551-0, published electronically 24 october 2006 [5] Ionescu, A.D., Kenig, C.: Low-regularity Schrödinger maps. Preprint 2006, available at arxiv.org 0605210 · Zbl 1212.35449 [6] Kato J. (2005). Existence and uniqueness of the solution to the modified Schrödinger map. Math. Res. Lett. 12: 171–186 · Zbl 1082.35140 [7] Kato, J., Koch, H.: Uniqueness of the modified Schrödinger map in \({H^{3/4+\epsilon}(\mathbb{R}^2)}\) . Preprint, 2005, available at arxiv.org 0508423 · Zbl 1387.35139 [8] Kenig C.E. and Nahmod A. (2005). The Cauchy problem for the hyperbolic-elliptic Ishimori system and Schrödinger maps. Nonlinearity 18: 1987–2009 · Zbl 1213.35358 [9] Kenig, C.E., Pollack, D., Staffilani, G., Toro, T.: The Cauchy problem for Schrödinger flows into Kähler manifolds. Preprint, 2005, available at arxiv.org 0511701 · Zbl 1193.35208 [10] Klainerman S. and Machedon M. (1993). Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46: 1221–1268 · Zbl 0803.35095 [11] Klainerman S. and Rodnianski I. (2001). On the global regularity of wave maps in the critical Sobolev norm. Internat. Math. Res. Notices 13: 655–677 · Zbl 0985.58009 [12] Klainerman S. and Selberg S. (1997). Remark on the optimal regularity for equations of wave maps type. Comm. Part. Differ. Eqs. 22: 901–918 · Zbl 0884.35102 [13] McGahagan, H.: An approximation scheme for Schrödinger maps. Preprint, 2005 · Zbl 1122.35138 [14] Nahmod A., Stefanov A. and Uhlenbeck K. (2003). On the well-posedness of the wave map problem in high dimensions. Comm. Anal. Geom. 11: 49–83 · Zbl 1085.58022 [15] Nahmod A., Stefanov A. and Uhlenbeck K. (2003). On Schrödinger maps. Comm. Pure Appl. Math. 56: 114–151 · Zbl 1028.58018 [16] Nahmod, A., Stefanov, A., Uhlenbeck, K.: Erratum: ”On Schrödinger maps” [Comm. Pure Appl. Math. 56 114–151 (2003)], Comm. Pure Appl. Math. 57, 833–839 (2004) · Zbl 1028.58018 [17] Nahmod, A., Shatah, J., Vega, L., Zeng, C.: Schrödinger maps into Hermitian symmetric spaces and their associated frame systems, available at arxiv.org 0612481 · Zbl 1142.35087 [18] Shatah J. and Struwe M. (2002). The Cauchy problem for wave maps. Int. Math. Res. Notices 11: 555–571 · Zbl 1024.58014 [19] Sulem P.L., Sulem C. and Bardos C. (1986). On the continuous limit for a system of classical spins. Commun. Math. Phys. 107: 431–454 · Zbl 0614.35087 [20] Soyeur A. (1992). The Cauchy problem for the Ishimori equations. J. Funct. Anal. 105: 233–255 · Zbl 0763.35077 [21] Tao T. (2001). Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Internat. Math. Res. Notices 6: 299–328 · Zbl 0983.35080 [22] Tao T. (2001). Global regularity of wave maps. II. Small energy in two dimensions. Commun. Math. Phys. 224: 443–544 · Zbl 1020.35046 [23] Tataru D. (1998). Local and global results for wave maps. I. Comm. Part. Differ. Eq. 23: 1781–1793 · Zbl 0914.35083 [24] Tataru D. (2001). On global existence and scattering for the wave maps equation. Amer. J. Math. 123: 37–77 · Zbl 0979.35100 [25] Tataru D. (2005). Rough solutions for the wave maps equation. Amer. J. Math. 127: 293–377 · Zbl 1330.58021 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.