[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-157009-105":59,"doc-detail-157009-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","slow-manifolds-for-infinite-dimensional-evolution-equations","Slow manifolds for infinite-dimensional evolution equations","","The paper extends classical finite-dimensional Fenichel theory to infinite-dimensional fast-slow evolution systems. Under weak assumptions, solutions are well-approximated by the associated slow flow. A two-parameter family of slow manifolds is then constructed under stronger conditions on the linear part of the slow equation. The additional parameter captures a splitting of the slow variable space into a fast-decaying component and its complement, with the finite-dimensional case recovered as a special instance. The methods are applied to three PDE examples.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/slow-manifolds-for-infinite-dimensional-evolution-equations/157009/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/slow-manifolds-for-infinite-dimensional-evolution-equations/157009.png","ImageObject",300,407,{"name":92,"@type":93},"Quinn","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-01","2026-08-29",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does the paper extend from classical theory to infinite dimensions?","Question",{"text":113,"@type":114},"It extends finite-dimensional Fenichel theory to infinite-dimensional fast-slow evolution equations, proving that solutions are approximated by the corresponding slow flow under weak assumptions.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the slow manifold approximated and what assumptions are used?",{"text":118,"@type":114},"For a normally hyperbolic critical manifold in an attracting setting, the flow on the slow manifold is well-approximated by the slow subsystem flow, based on Fenichel-type theory extended to infinite-dimensional contexts.",{"name":120,"@type":111,"acceptedAnswer":121},"Why does the two-parameter family of slow manifolds matter?",{"text":122,"@type":114},"The second parameter, absent in the finite-dimensional setting, describes a decomposition of the slow variable space into a fast-decaying part and its complement.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},157009,1787971141,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},962075114765,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Comment. Math. Helv. 97 (2022), 61–132 DOI 10.4171/CMH/527  \nCommentarii Mathematici Helvetici  \n© 2022 Swiss Mathematical Society This work is licensed under a CC BY 4 .0 license  \nSlow manifolds for infinite-dimensional evolution equations  \nFelix Hummel and Christian Kuehn  \nAbstract. We extend classical finite-dimensional Fenichel theory in two directions to infinite dimensions. Under comparably weak assumptions we show that the solution of an infinitedimensional fast-slow system is approximated well by the corresponding slow flow. After that we construct a two-parameter family of slow manifolds S\";􀀐 under more restrictive assumptionson the linear part of the slow equation. The second parameter 􀀐 does not appear in the finitedimensional setting and describes a certain splitting of the slow variable space in a fast decaying part and its complement. The finite-dimensional setting is contained as a special case in which S\";􀀐 does not depend on 􀀐 . Finally, we apply our new techniques to three examples of fast-slow systems of partial differential equations.  \nMathematics Subject Classification (2020) . 35B25, 37D10, 37L25, 35A24 .  \nKeywords. Geometric singular perturbation theory, slow manifolds, infinite dimensions.  \n1. Introduction  \nIn this work, we study infinite-dimensional fast-slow evolution equations of the form  \n\"@tu\" D Au\" C f.u\" ; v\"/;@tv\" D B v\" C g.u\" ; v\"/;  \n(1.1)  \nwhere \" 􀀕 0 is a small parameter, A and B are linear operators on Banach spaces X and Y , respectively, f; g are sufficiently regular nonlinearities, and .u\" ; v\"/ D.u\".t/; v\".t// 2 X 􀀂 Y are the unknown functions, where the superscript indicates the dependence of the solution on \". In particular, the class of systems (1.1) are multiscale evolution equations, where the small parameter \" hints at a formal timescale separation between the variables u\" and v\".  \nThe motivation to study (1.1) is best explained via the finite-dimensional setting, where .u\" ; v\"/ 2 Rm 􀀂 Rn , A 2 Rm􀀂m , B 2 Rn􀀂n, and one often assumes that f; gare sufficiently smooth. Multiple time scale ordinary differential equations (ODEs) are employed across broad areas of mathematics [21] and form one of the few classes of higher-dimensional dynamical systems, where analytical results about nonlinear dynamics can be obtained due to the time scale separation structure. If we let \" ! 0  \n62 F. Hummel and C. Kuehn CMH in (1.1) we obtain the slow subsystem (or reduced system)  \n0 D Au0 C f.u0 ; v 0/;  \n(1.2)  \n@tv 0 D Bv 0 C g.u0 ; v 0/;  \nwhich is a differential-algebraic equation defined on the critical set  \nS0 WD ˚.u0 ; v 0/ 2 Rm 􀀂 Rn W 0 D Au0 C f.u0 ; v 0/ 􀀉 ;  \nwhich we shall assume to be a manifold referred to as the critical manifold. IfS0 􀀒 S0 is compact and normally hyperbolic submanifold, i.e., all eigenvalues of the matrix A CDuf.z/ 2 Rm􀀂m have nonzero real part for all z 2 S0 , then Fenichel–Tikhonov theory [14, 32] guarantees the existence of a locally invariant slow manifold S\". Of course, for practical applications, the case of a critical manifold, which is attracting in the fast directions, is the most frequently encountered. This case occurs when all eigenvalues of A C Du f.z/ have negative real part and we shall focus on the attracting setting here. For any normally hyperbolic critical manifold, the flow on S\"is approximated well by the slow subsystem flow of (1.1); see also [18, 21, 35] for detailed expositions of Fenichel theory. One reason to intuitively expect such an approximation result in finite dimensions is better visible on the fast time scale r W D t=\", which leads upon substitution in (1.1) to  \n@ru\" D Au\" C f.u\" ; v\"/;  \n@r v\" D \" 􀀀Bv\" C g.u\" ; v\"/ 􀀁 : (1.3)  \nIndeed, sending \" ! 0 in (1.3) yields the fast subsystem (or layer equations)  \n@ru0 D Au0 C f.u0 ; v 0/;  \n(1.4)  \n@rv 0 D 0:  \nThe full fast-slow system on Rm 􀀂 Rn can then be treated near S0 as a bounded perturbation of the fast subsystem since B and g satisfy local bounds due to the assumptions","cbCaieapLPnVcjNA","https://ap.wps.com/l/cbCaieapLPnVcjNA","pdf",541182,72,"English","# 1. Introduction\n## 1.1 Problem formulation and motivation\n## 1.2 Critical manifold and slow subsystem\n## 1.3 Attracting setting and approximation on slow manifold\n## 1.4 Challenges in infinite dimensions and prior related work","[{\"question\":\"What does the paper extend from classical theory to infinite dimensions?\",\"answer\":\"It extends finite-dimensional Fenichel theory to infinite-dimensional fast-slow evolution equations, proving that solutions are approximated by the corresponding slow flow under weak assumptions.\"},{\"question\":\"How is the slow manifold approximated and what assumptions are used?\",\"answer\":\"For a normally hyperbolic critical manifold in an attracting setting, the flow on the slow manifold is well-approximated by the slow subsystem flow, based on Fenichel-type theory extended to infinite-dimensional contexts.\"},{\"question\":\"Why does the two-parameter family of slow manifolds matter?\",\"answer\":\"The second parameter, absent in the finite-dimensional setting, describes a decomposition of the slow variable space into a fast-decaying part and its complement.\"}]","Slow manifolds for infinite-dimensional evolution equations | PDF",181]