L’equaci\u00f3 d’Euler (en forma de vorticitat) que descriu el moviment d’un fluid bidimensional \u00e9s<\/p>\n
$$\\partial_t \\omega + v\\cdot \\nabla \\omega = 0,$$<\/p>\n\n
on \\(\\omega\\) \u00e9s la vorticitat i \\(v\\) el camp de velocitats del fluid. Un cas d’especial rellev\u00e0ncia en la mec\u00e0nica de fluids \u00e9s quan la condici\u00f3 inicial per la vorticitat \u00e9s \\(\\omega(\\cdot,0)= \\chi_{D_0}\\) essent \\(D_0\\) un domini del pla. \u00c9s el conegut com a problema del\u00a0patch.<\/em><\/p>\n En aquest TFG estudiarem l’equaci\u00f3 d’Euler bidimensional entenent els conceptes de vorticitat i velocitat. Estudiarem les traject\u00f2ries que segueix una part\u00edcula dins del fluid. Finalment, ens traslladarem al cas del patch<\/em> on mirarem teoremes d’exist\u00e8ncia i unicitat aix\u00ed com t\u00e8cniques (stream function, funci\u00f3 definidora, problemes variacionals) aplicables a aquest cas.<\/p>\n <\/p>\n Conv\u00e9 que l’estudiant hagi cursat l’assignatura An\u00e0lisi Real i Funcional o estigui fent-ho, per tal d’aprofundir en q\u00fcestions relacionades amb l’exist\u00e8ncia, unicitat i regularitat de les solucions de la equaci\u00f3 en derivades parcials associada al problema. Tot i aix\u00ed, es podria arribar a fer un bon TFG evitant certs aspectes si l’estudiant no l’ha cursat per\u00f2 t\u00e9 prou inter\u00e8s.<\/p>\n","protected":false},"excerpt":{"rendered":" L’equaci\u00f3 d’Euler (en forma de vorticitat) que descriu el moviment d’un fluid bidimensional \u00e9s $$\\partial_t \\omega + v\\cdot \\nabla \\omega = 0,$$ on \\(\\omega\\) \u00e9s la vorticitat i \\(v\\) el camp de velocitats del fluid. Un cas d’especial rellev\u00e0ncia en la mec\u00e0nica de fluids \u00e9s quan la condici\u00f3 inicial per la vorticitat \u00e9s \\(\\omega(\\cdot,0)= \\chi_{D_0}\\) […]<\/p>\n","protected":false},"author":74,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29,30],"tags":[47],"class_list":["post-490","post","type-post","status-publish","format-standard","hentry","category-analisi-matematica","category-matematica-aplicada","tag-juan-carlos-cantero"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/490","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/users\/74"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=490"}],"version-history":[{"count":7,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/490\/revisions"}],"predecessor-version":[{"id":497,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/490\/revisions\/497"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=490"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=490"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=490"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}