{"id":1473,"date":"2025-10-31T22:27:23","date_gmt":"2025-11-01T05:27:23","guid":{"rendered":"https:\/\/wp.math.berkeley.edu\/hades\/?p=1473"},"modified":"2025-10-31T22:27:23","modified_gmt":"2025-11-01T05:27:23","slug":"dispersive-estimates-for-non-integrable-1d-defocusing-cubic-nls-at-sharp-regularity","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/hades\/2025\/10\/31\/dispersive-estimates-for-non-integrable-1d-defocusing-cubic-nls-at-sharp-regularity\/","title":{"rendered":"Dispersive Estimates for Non-integrable 1D Defocusing Cubic NLS at Sharp Regularity"},"content":{"rendered":"<p>The HADES seminar on Tuesday, <b>November 4th<\/b>, will be at <strong>3:30pm<\/strong>\u00a0in\u00a0<strong>Room 740<\/strong>.<\/p>\n<p><strong>Speaker:\u00a0<\/strong>Ryan Martinez<\/p>\n<p><strong>Abstract: <\/strong>We present work, still in progress, with Mihaela Ifrim and Daniel Tataru, which proves global well-posedness, global $L^6$ based Strichartz estimates, and global bilinear spacetime $L^2$ estimates for non-integrable 1D defocusing cubic NLS at the sharp regularity $H^{-1\/2 + \\epsilon}$ with mild regularity assumptions on the nonlinearity; taking for granted a suitable local well-posedness theory.<\/p>\n<p>In $L^2$, this problem was well understood by Ifrim and Tataru, by using a modified energy method in a frequency localized setting. However, below $L^2$ there are several challenges. First, Christ, Colliander, and Tao show that the initial data-to-solution map fails to even be uniformly continuous locally in time below $L^2$. For the completely integrable problem Harrop-Griffiths, Killip, and Visan proved global (and local) well-posedness in the sense of continuous dependence and local smoothing estimates for the problem in the sharp space. Our work supplements their work by in addition providing global $L^6$ and bilinear $L^2$ estimates, but does not itself depend on complete integrability. To emphasize this, we prove the result for general nonlinearities, of course assuming the existence of a local theory, which at this time, seems out of reach.<\/p>\n<p>The main challenge of this work is that the modified energy method used by Ifrim and Tataru at $L^2$ fails at high frequency below $s = -1\/3$. To overcome this we use an infinite series of corrections.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The HADES seminar on Tuesday, November 4th, will be at 3:30pm\u00a0in\u00a0Room 740. Speaker:\u00a0Ryan Martinez Abstract: We present work, still in progress, with Mihaela Ifrim and Daniel Tataru, which proves global well-posedness, global $L^6$ based Strichartz estimates, and global bilinear spacetime $L^2$ estimates for non-integrable 1D defocusing cubic NLS at the sharp regularity $H^{-1\/2 + \\epsilon}$ [&hellip;]<\/p>\n","protected":false},"author":116,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25],"tags":[],"class_list":["post-1473","post","type-post","status-publish","format-standard","hentry","category-fall-2025"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1473","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/users\/116"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/comments?post=1473"}],"version-history":[{"count":1,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1473\/revisions"}],"predecessor-version":[{"id":1474,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1473\/revisions\/1474"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/media?parent=1473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/categories?post=1473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/tags?post=1473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}