[{"scopus_import":"1","extern":"1","date_created":"2026-06-19T07:37:37Z","abstract":[{"text":"We prove global well-posedness of the Korteweg–de Vries equation for\r\ninitial data in the space H^−1(R). This is sharp in the class of H^s(R) spaces.\r\nEven local well-posedness was previously unknown for s < −3/4. The proof\r\nis based on the introduction of a new method of general applicability for the\r\nstudy of low-regularity well-posedness for integrable PDE, informed by the\r\nexistence of commuting flows. In particular, as we will show, completely\r\nparallel arguments give a new proof of global well-posedness for KdV with\r\nperiodic H−1 data, shown previously by Kappeler and Topalov, as well as\r\nglobal well-posedness for the fifth order KdV equation in L^2(R).\r\nAdditionally, we give a new proof of the a priori local smoothing bound\r\nof Buckmaster and Koch for KdV on the line. Moreover, we upgrade this\r\nestimate to show that convergence of initial data in H^−1(R) guarantees\r\nconvergence of the resulting solutions in L^2loc(R × R). Thus, solutions with\r\nH^−1(R) initial data are distributional solutions.","lang":"eng"}],"day":"05","volume":190,"language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1802.04851"}],"title":"KdV is well-posed in H^-1","intvolume":"       190","arxiv":1,"external_id":{"arxiv":["1802.04851"]},"citation":{"ama":"Killip R, Vişan M. KdV is well-posed in H^-1. <i>Annals of Mathematics</i>. 2019;190(1):249-305. doi:<a href=\"https://doi.org/10.4007/annals.2019.190.1.4\">10.4007/annals.2019.190.1.4</a>","apa":"Killip, R., &#38; Vişan, M. (2019). KdV is well-posed in H^-1. <i>Annals of Mathematics</i>. Annals of Mathematics. <a href=\"https://doi.org/10.4007/annals.2019.190.1.4\">https://doi.org/10.4007/annals.2019.190.1.4</a>","mla":"Killip, Rowan, and Monica Vişan. “KdV Is Well-Posed in H^-1.” <i>Annals of Mathematics</i>, vol. 190, no. 1, Annals of Mathematics, 2019, pp. 249–305, doi:<a href=\"https://doi.org/10.4007/annals.2019.190.1.4\">10.4007/annals.2019.190.1.4</a>.","ieee":"R. Killip and M. Vişan, “KdV is well-posed in H^-1,” <i>Annals of Mathematics</i>, vol. 190, no. 1. Annals of Mathematics, pp. 249–305, 2019.","short":"R. Killip, M. Vişan, Annals of Mathematics 190 (2019) 249–305.","chicago":"Killip, Rowan, and Monica Vişan. “KdV Is Well-Posed in H^-1.” <i>Annals of Mathematics</i>. Annals of Mathematics, 2019. <a href=\"https://doi.org/10.4007/annals.2019.190.1.4\">https://doi.org/10.4007/annals.2019.190.1.4</a>.","ista":"Killip R, Vişan M. 2019. KdV is well-posed in H^-1. Annals of Mathematics. 190(1), 249–305."},"publication":"Annals of Mathematics","type":"journal_article","_id":"22028","doi":"10.4007/annals.2019.190.1.4","page":"249-305","author":[{"full_name":"Killip, Rowan","first_name":"Rowan","last_name":"Killip"},{"last_name":"Visan","first_name":"Monica","full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca"}],"quality_controlled":"1","das_tickbox":"1","publisher":"Annals of Mathematics","date_updated":"2026-06-22T11:12:40Z","publication_status":"published","month":"07","date_published":"2019-07-05T00:00:00Z","article_type":"original","article_processing_charge":"No","oa":1,"OA_type":"green","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["0003-486X"]},"OA_place":"repository","issue":"1","year":"2019","status":"public"}]