{"id":1372,"date":"2025-09-10T18:37:32","date_gmt":"2025-09-11T01:37:32","guid":{"rendered":"https:\/\/wp.math.berkeley.edu\/hades\/?p=1372"},"modified":"2025-09-15T20:21:58","modified_gmt":"2025-09-16T03:21:58","slug":"construction-of-nonunique-solutions-of-the-transport-and-continuity-equation-for-sobolev-vector-fields-in-diperna-lions-theory-2","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/hades\/2025\/09\/10\/construction-of-nonunique-solutions-of-the-transport-and-continuity-equation-for-sobolev-vector-fields-in-diperna-lions-theory-2\/","title":{"rendered":"Nonuniqueness of solutions to the Euler equations with integrable vorticity"},"content":{"rendered":"<div class=\"entry-content\">\n<p>The HADES seminar on Thursday, <strong>September<\/strong><strong>\u00a011th<\/strong>, will be at\u00a0<strong>3:30pm<\/strong>\u00a0in\u00a0<strong>Room 736<\/strong>.<\/p>\n<p><strong>Speaker:<\/strong> Anuj Kumar<\/p>\n<p><strong>Abstract: <\/strong>Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. DiPerna and Majda proved the existence of weak solutions with vorticity in $L^p (p &gt; 1)$.\u00a0 A celebrated open question is whether the uniqueness result can be generalized to solutions with $L^p$ vorticity. In this talk, we resolve this question in negative for some $p &gt; 1$. To prove nonuniqueness, we devise a new convex integration scheme that employs non-periodic, spatially-anisotropic perturbations, an idea that was inspired by our recent work on the transport equation. To construct the perturbation, we introduce a new family of building blocks based on the Lamb-Chaplygin dipole. This is a joint work with Elia Bru\u00e8 and Maria Colombo.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The HADES seminar on Thursday, September\u00a011th, will be at\u00a03:30pm\u00a0in\u00a0Room 736. Speaker: Anuj Kumar Abstract: Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. DiPerna and Majda proved the existence of weak solutions with vorticity in $L^p (p &gt; 1)$.\u00a0 A celebrated open question is whether the uniqueness result can [&hellip;]<\/p>\n","protected":false},"author":95,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25],"tags":[],"class_list":["post-1372","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\/1372","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\/95"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/comments?post=1372"}],"version-history":[{"count":3,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1372\/revisions"}],"predecessor-version":[{"id":1377,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1372\/revisions\/1377"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/media?parent=1372"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/categories?post=1372"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/tags?post=1372"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}