[{"article_type":"original","OA_place":"repository","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"full_name":"Killip, Rowan","first_name":"Rowan","last_name":"Killip"},{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","last_name":"Visan","first_name":"Monica","full_name":"Visan, Monica"}],"oa":1,"quality_controlled":"1","extern":"1","date_created":"2026-06-19T07:32:05Z","publication":"Analysis & PDE","_id":"22024","publisher":"Mathematical Sciences Publishers","oa_version":"Preprint","article_processing_charge":"No","language":[{"iso":"eng"}],"das_tickbox":"1","day":"27","publication_status":"published","external_id":{"arxiv":["1102.1192"]},"abstract":[{"text":"We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.","lang":"eng"}],"scopus_import":"1","fulldoi":"https://doi.org/10.2140/apde.2012.5.855","doi":"10.2140/apde.2012.5.855","OA_type":"green","month":"11","arxiv":1,"type":"journal_article","status":"public","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1102.1192"}],"date_published":"2012-11-27T00:00:00Z","title":"Global well-posedness and scattering for the defocusing quintic NLS in three dimensions","intvolume":"         5","citation":{"ista":"Killip R, Vişan M. 2012. Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. Analysis &#38; PDE. 5(4), 855–885.","ama":"Killip R, Vişan M. Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. <i>Analysis &#38; PDE</i>. 2012;5(4):855-885. doi:<a href=\"https://doi.org/10.2140/apde.2012.5.855\">10.2140/apde.2012.5.855</a>","apa":"Killip, R., &#38; Vişan, M. (2012). Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/apde.2012.5.855\">https://doi.org/10.2140/apde.2012.5.855</a>","mla":"Killip, Rowan, and Monica Vişan. “Global Well-Posedness and Scattering for the Defocusing Quintic NLS in Three Dimensions.” <i>Analysis &#38; PDE</i>, vol. 5, no. 4, Mathematical Sciences Publishers, 2012, pp. 855–85, doi:<a href=\"https://doi.org/10.2140/apde.2012.5.855\">10.2140/apde.2012.5.855</a>.","short":"R. Killip, M. Vişan, Analysis &#38; PDE 5 (2012) 855–885.","ieee":"R. Killip and M. Vişan, “Global well-posedness and scattering for the defocusing quintic NLS in three dimensions,” <i>Analysis &#38; PDE</i>, vol. 5, no. 4. Mathematical Sciences Publishers, pp. 855–885, 2012.","chicago":"Killip, Rowan, and Monica Vişan. “Global Well-Posedness and Scattering for the Defocusing Quintic NLS in Three Dimensions.” <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers, 2012. <a href=\"https://doi.org/10.2140/apde.2012.5.855\">https://doi.org/10.2140/apde.2012.5.855</a>."},"date_updated":"2026-06-22T10:20:04Z","publication_identifier":{"eissn":["1948-206X"],"issn":["2157-5045"]},"volume":5,"page":"855-885","keyword":["energy critical","nonlinear Schrödinger"],"issue":"4","year":"2012"},{"publication_status":"published","external_id":{"arxiv":["math/0511070"]},"abstract":[{"lang":"eng","text":"We undertake a comprehensive study of the nonlinear Schrödinger equation (mathematical formular) where u(t, x) is a complex-valued function in spacetime R, xRn/x, λ1 and λ2 are nonzero real constants, and (mathematical formular). We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H^1(ℝ n ) and in the pseudoconformal space Σ := {f ∈ H^1(ℝ^n); xf ∈ L^2(ℝ^n)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L2/x-critical, respectively H1/x-critical NLS, that is, λ1, λ2 > 0 and (mathematical formular) . The results at the endpoint p1= 4/n are conditional on a conjectured global existence and spacetime estimate for the L2/x-critical nonlinear Schrödinger equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint).\r\nAs an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H1/x for solutions to the nonlinear Schrödinger equation (mathematical formular) with 4/n < p < 4/n-2, which was first obtained by Ginibre and Velo (Citation1985)."}],"scopus_import":"1","fulldoi":"https://doi.org/10.1080/03605300701588805","doi":"10.1080/03605300701588805","date_updated":"2026-06-25T08:04:20Z","mathsc":["35Q55"],"publication_identifier":{"issn":["0360-5302"],"eissn":["1532-4133"]},"volume":32,"page":"1281-1343","issue":"8","keyword":["Energy-critical","Mass-critical","Nonlinear Schrödinger equation","Wellposedness"],"year":"2007","OA_type":"green","arxiv":1,"type":"journal_article","month":"08","status":"public","date_published":"2007-08-29T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.math/0511070","open_access":"1"}],"title":"The nonlinear Schrödinger equation with combined power-type nonlinearities","intvolume":"        32","citation":{"ieee":"T. Tao, M. Vişan, and X. Zhang, “The nonlinear Schrödinger equation with combined power-type nonlinearities,” <i>Communications in Partial Differential Equations</i>, vol. 32, no. 8. Informa UK Limited, pp. 1281–1343, 2007.","chicago":"Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities.” <i>Communications in Partial Differential Equations</i>. Informa UK Limited, 2007. <a href=\"https://doi.org/10.1080/03605300701588805\">https://doi.org/10.1080/03605300701588805</a>.","mla":"Tao, Terence, et al. “The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities.” <i>Communications in Partial Differential Equations</i>, vol. 32, no. 8, Informa UK Limited, 2007, pp. 1281–343, doi:<a href=\"https://doi.org/10.1080/03605300701588805\">10.1080/03605300701588805</a>.","short":"T. Tao, M. Vişan, X. Zhang, Communications in Partial Differential Equations 32 (2007) 1281–1343.","ama":"Tao T, Vişan M, Zhang X. The nonlinear Schrödinger equation with combined power-type nonlinearities. <i>Communications in Partial Differential Equations</i>. 2007;32(8):1281-1343. doi:<a href=\"https://doi.org/10.1080/03605300701588805\">10.1080/03605300701588805</a>","apa":"Tao, T., Vişan, M., &#38; Zhang, X. (2007). The nonlinear Schrödinger equation with combined power-type nonlinearities. <i>Communications in Partial Differential Equations</i>. Informa UK Limited. <a href=\"https://doi.org/10.1080/03605300701588805\">https://doi.org/10.1080/03605300701588805</a>","ista":"Tao T, Vişan M, Zhang X. 2007. The nonlinear Schrödinger equation with combined power-type nonlinearities. Communications in Partial Differential Equations. 32(8), 1281–1343."},"oa":1,"quality_controlled":"1","extern":"1","date_created":"2026-06-19T07:49:46Z","article_type":"original","OA_place":"repository","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"full_name":"Tao, Terence","first_name":"Terence","last_name":"Tao"},{"first_name":"Monica","full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","last_name":"Visan"},{"full_name":"Zhang, Xiaoyi","first_name":"Xiaoyi","last_name":"Zhang"}],"language":[{"iso":"eng"}],"das_tickbox":"1","day":"29","publication":"Communications in Partial Differential Equations","_id":"22047","oa_version":"Preprint","publisher":"Informa UK Limited","article_processing_charge":"No"}]