[{"extern":"1","article_type":"original","oa":1,"language":[{"iso":"eng"}],"date_published":"2024-04-02T00:00:00Z","doi":"10.1017/fmp.2024.4","publisher":"Cambridge University Press","year":"2024","quality_controlled":"1","das_tickbox":"1","external_id":{"arxiv":["2003.05011"]},"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"_id":"22079","arxiv":1,"scopus_import":"1","month":"04","citation":{"short":"B. Harrop-Griffiths, R. Killip, M. Vişan, Forum of Mathematics, Pi 12 (2024).","mla":"Harrop-Griffiths, Benjamin, et al. “Sharp Well-Posedness for the Cubic NLS and MKdV in H^s(R).” <i>Forum of Mathematics, Pi</i>, vol. 12, e6, Cambridge University Press, 2024, doi:<a href=\"https://doi.org/10.1017/fmp.2024.4\">10.1017/fmp.2024.4</a>.","ista":"Harrop-Griffiths B, Killip R, Vişan M. 2024. Sharp well-posedness for the cubic NLS and mKdV in H^s(R). Forum of Mathematics, Pi. 12, e6.","chicago":"Harrop-Griffiths, Benjamin, Rowan Killip, and Monica Vişan. “Sharp Well-Posedness for the Cubic NLS and MKdV in H^s(R).” <i>Forum of Mathematics, Pi</i>. Cambridge University Press, 2024. <a href=\"https://doi.org/10.1017/fmp.2024.4\">https://doi.org/10.1017/fmp.2024.4</a>.","ieee":"B. Harrop-Griffiths, R. Killip, and M. Vişan, “Sharp well-posedness for the cubic NLS and mKdV in H^s(R),” <i>Forum of Mathematics, Pi</i>, vol. 12. Cambridge University Press, 2024.","ama":"Harrop-Griffiths B, Killip R, Vişan M. Sharp well-posedness for the cubic NLS and mKdV in H^s(R). <i>Forum of Mathematics, Pi</i>. 2024;12. doi:<a href=\"https://doi.org/10.1017/fmp.2024.4\">10.1017/fmp.2024.4</a>","apa":"Harrop-Griffiths, B., Killip, R., &#38; Vişan, M. (2024). Sharp well-posedness for the cubic NLS and mKdV in H^s(R). <i>Forum of Mathematics, Pi</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/fmp.2024.4\">https://doi.org/10.1017/fmp.2024.4</a>"},"publication_status":"published","type":"journal_article","volume":12,"OA_place":"publisher","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","PlanS_conform":"1","status":"public","main_file_link":[{"url":"https://doi.org/10.1017/fmp.2024.4","open_access":"1"}],"date_created":"2026-06-19T08:26:10Z","publication":"Forum of Mathematics, Pi","DOAJ_listed":"1","publication_identifier":{"eissn":["2050-5086"]},"mathsc":["35Q55","35Q53","37K10"],"OA_type":"gold","has_accepted_license":"1","title":"Sharp well-posedness for the cubic NLS and mKdV in H^s(R)","day":"02","ddc":["500"],"intvolume":"        12","date_updated":"2026-06-30T12:16:50Z","article_number":"e6","oa_version":"Published Version","abstract":[{"lang":"eng","text":"We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in H^s(R) for any regularity s > −1/2. Well-posedness has long been known for s ≥ 0, see [55], but not previously for any s < 0. The scaling-critical value s = −1/2 is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in H^s(R) for any s > −1/2. The best regularity achieved previously was s ≥ 1/4 (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the nonperturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity. "}],"author":[{"full_name":"Harrop-Griffiths, Benjamin","last_name":"Harrop-Griffiths","first_name":"Benjamin"},{"full_name":"Killip, Rowan","last_name":"Killip","first_name":"Rowan"},{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica","last_name":"Visan","full_name":"Visan, Monica"}]}]