{"id":1441,"date":"2025-10-08T16:11:31","date_gmt":"2025-10-08T23:11:31","guid":{"rendered":"https:\/\/wp.math.berkeley.edu\/hades\/?p=1441"},"modified":"2025-10-08T16:11:31","modified_gmt":"2025-10-08T23:11:31","slug":"long-time-behavior-of-rough-solutions-to-defocusing-nonlinear-schrodinger-equations","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/hades\/2025\/10\/08\/long-time-behavior-of-rough-solutions-to-defocusing-nonlinear-schrodinger-equations\/","title":{"rendered":"Long-time behavior of rough solutions to defocusing Nonlinear Schr\u00f6dinger Equations"},"content":{"rendered":"<p>The HADES seminar on Tuesday, <strong>October<\/strong><strong> 14th<\/strong>, will be at <strong>3:30pm<\/strong>\u00a0in\u00a0<strong>Room 740<\/strong>.<\/p>\n<p><strong>Speaker:\u00a0<\/strong>Zachary Lee<\/p>\n<p><strong>Abstract: <\/strong>The Nonlinear Schr\u00f6dinger Equation (NLS) arises in various physical contexts, notably in models of Bose\u2013Einstein condensation and nonlinear optics. I will begin by outlining these motivations and by presenting several heuristics\u2014scaling, dispersion, and symmetry\u2014that shed light on the qualitative behavior of its solutions. I will then turn to a rigorous analysis based on the Duhamel formulation of the equation, together with Strichartz estimates, conservation laws, and Morawetz inequalities, which provide global control for <img loading=\"lazy\" decoding=\"async\" id=\"m_1315583364954477363l0.8238768396626325\" class=\"CToWUd\" title=\"H^1\" src=\"https:\/\/ci3.googleusercontent.com\/meips\/ADKq_NZHhN-VxGUkAS50ChlEtzpUvc3W6AADuWI1LkP7FH35EE4M-fM02LXeUi5P-UNbmGfC9I0jFbAe_FgfYWWlCuoF8szsli7CC0m3qWU0iPC3dU67f6wmn5BTeoal=s0-d-e1-ft#https:\/\/s0.wp.com\/latex.php?zoom=3&amp;bg=ffffff&amp;fg=000000&amp;s=0&amp;latex=H%5E1\" alt=\"H^1\" width=\"18\" height=\"13\" data-bit=\"iit\" \/> data in the defocusing case. In the final part of the talk, I will describe how these techniques can be adapted below the energy space using almost conservation laws (the I-method), and present a new global existence result for the one-dimensional defocusing septic NLS for a class of discontinuous and unbounded initial data.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The HADES seminar on Tuesday, October 14th, will be at 3:30pm\u00a0in\u00a0Room 740. Speaker:\u00a0Zachary Lee Abstract: The Nonlinear Schr\u00f6dinger Equation (NLS) arises in various physical contexts, notably in models of Bose\u2013Einstein condensation and nonlinear optics. I will begin by outlining these motivations and by presenting several heuristics\u2014scaling, dispersion, and symmetry\u2014that shed light on the qualitative behavior [&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-1441","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\/1441","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=1441"}],"version-history":[{"count":1,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1441\/revisions"}],"predecessor-version":[{"id":1442,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1441\/revisions\/1442"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/media?parent=1441"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/categories?post=1441"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/tags?post=1441"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}