[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-202026-105":59,"doc-detail-202026-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","solutions-to-a-model-with-nonuniformly-parabolic-terms-for-phase-evolution-driven-by-configurational-forces","Solutions to a Model with Nonuniformly Parabolic Terms for Phase Evolution Driven by Configurational Forces","","Existence of solutions global in time is established for an initial-boundary value problem combining linear elasticity with a second-order nonlinear, non-uniformly parabolic phase-field equation. The system models temporal evolution of material phases governed by configurational forces. The phase-field model is obtained by adding a parabolic regularization term to an underlying hyperbolic sharp-interface formulation for a strain-discontinuity surface. The analytical result verifying the model’s consistency holds in one spatial dimension.",{"@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/solutions-to-a-model-with-nonuniformly-parabolic-terms-for-phase-evolution-driven-by-configurational-forces/202026/",{"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/solutions-to-a-model-with-nonuniformly-parabolic-terms-for-phase-evolution-driven-by-configurational-forces/202026.png","ImageObject",300,407,{"name":92,"@type":93},"Jordan Avery","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-04",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",14,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What type of mathematical system is studied in the paper?","Question",{"text":113,"@type":114},"The paper studies an initial-boundary value problem coupling linear elasticity with a second-order nonlinear, non-uniformly parabolic phase-field equation that governs phase evolution.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the phase-field model derived from the sharp-interface model?",{"text":118,"@type":114},"The model is obtained by inserting a parabolic regularizing term into an original hyperbolic character model, which itself is derived from a sharp-interface formulation for a strain-discontinuity surface.",{"name":120,"@type":111,"acceptedAnswer":121},"In what spatial setting does the global-in-time existence result hold?",{"text":122,"@type":114},"The existence proof establishing global-in-time solutions is valid only in one space dimension.","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},202026,1788541138,{"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},1099523882367,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Solutions to a Model with Nonuniformly Parabolic Terms for Phase Evolution Driven by Con􀀌gurational Forces  \nHans-Dieter Alber􀀃 and Peicheng Zhuy  \nDepartment of Mathematics  \nDarmstadt University of Technology  \nSchlossgartenstr. 7, 64289 Darmstadt  \nGermany  \nAbstract  \nWe prove existence of solutions global in time to an initial-boundary value problem for a system of partial di􀀋erential equations, which consists of the equations of linear elasticity and a nonlinear, non-uniformly parabolic equation of second order. This problem models the behavior of material phases, whose evolution in time is driven by con􀀌gurational forces. The model is obtained by inserting a parabolic, regularizing term into an original model with hyperbolic character, which is derived in [2, 3] by transforming a well known sharp interface model for the evolution of a surface of strain discontinuity. Our existence proof, which contributes to the veri􀀌cation of the model, is only valid in one space dimension.  \n1 Introduction  \nIn [3] a model has been derived for the behavior of materials with phase transistions, for which the evolution of the interfacial regions is driven by con􀀌gurational forces. The model consists of the partial di􀀋erential equations of linear elasticity coupled to a quasilinear, non-uniformly parabolic equation of second order. To verify the validity of a new model investigations are necessary, in which not only simulations must be carried out but also the analytical properties of the model must be determined. Here we contribute to the veri􀀌cation of the new model by showing that in the case of one space dimension an initial-boundary value problem to this model has solutions global in time. We 􀀌rst formulate this initial-boundary value problem in the three-dimensional case, give a short sketch of the derivation of the model  \n􀀃 E-mail: [alber@mathematik.tu-darmstadt.de](alber@mathematik.tu-darmstadt.de)yE-mail: [zhu@mathematik.tu-darmstadt.de](zhu@mathematik.tu-darmstadt.de)  \nand conclude this introduction by reducing the model to the one-dimensional case and by stating our main result.  \nLet 􀀊 􀀚 R3 be an open set. It represents the material points of a solid body. The di􀀋erent phases are characterized by the order parameter S(t; x) 2 R. A value of S(t; x) near to zero indicates that the material is in the matrix phase at the point x 2 􀀊 at time t, a value near to one indicates that the material is in the second phase. The other unknowns are the displacement u(t; x) 2 R3 of the material point x at time t and the Cauchy stress tensor T(t; x) 2 S 3 , where S 3 denotes the set of symmetric 3 􀀂 3-matrices. The unknowns must satisfy the quasi-static equations  \n􀀀divx T (t; x) = b (t; x) ; (1.1)  \nT (t; x) = D 􀀀\"(rx u (t; x)) 􀀀 S (t; x)􀀁 ; (1.2)  \nSt (t; x) = 􀀀c 􀀐 S (\"(rxu (t; x)); S(t; x)) 􀀀 􀀗􀀁xS (t; x)􀀑 jrxS (t; x)j (1.3) for (t; x) 2 (0 ; 1 ) 􀀂 􀀊 . The boundary and initial conditions are  \nu (t; x) = 􀀍(t; x); S (t; x) = 0; (t; x) 2 [0 ; 1 ) 􀀂 @􀀊 ; (1.4)  \nS(0; x) = S0 (x); x 2 􀀊 : (1.5)  \nHere rxu denotes the 3 􀀂 3-matrix of 􀀌rst order derivatives of u, the deformation gradient, (\"(rxu))T denotes the transposed matrix and  \n1 T\" (rxu) = 2 􀀀 rxu + (rxu) 􀀁  \nis the strain tensor.  2 S 3 is a given matrix, the mis􀀌t strain, and D : S 3 ! S 3 is a linear, symmetric, positive de􀀌nite mapping, the elasticity tensor. In the free energy  \n(\"; S) = 12 􀀀 D (\" 􀀀 S)􀀁 􀀁 (\" 􀀀 S) + ^ (S) (1.6)  \nwe choose for ^ 2 C2 (R ; [0 ; 1 )) a double well potential with minima at S = 0 and S = 1 . Also, c is a positive constant and 􀀗 is a small nonnegative constant. Given are the volume force b : [0; 1 ) 􀀂 􀀊 ! R3 and the boundary and initial data 􀀍 : [0 ; 1 ) 􀀂 @􀀊 ! R3 , S0 : 􀀊 ! R.  \nThis completes the formulation of the initial-boundary value problem. The equations (1.1) and (1.2) di􀀋er from the system of linear elasticity only by the term S, which couples this system to equation (1.3) . The evolution equation (1.3) for the order parameter S is non-uniformly ","cbCaidGpkY50lsHl","https://ap.wps.com/l/cbCaidGpkY50lsHl","pdf",215285,22,"English","# Introduction\n## Model formulation in three dimensions\n## Quasi-static linear elasticity coupled to phase-field evolution\n## Derivation from sharp interface to phase-field model\n## Reduction to the one-dimensional case and main result","[{\"question\":\"What type of mathematical system is studied in the paper?\",\"answer\":\"The paper studies an initial-boundary value problem coupling linear elasticity with a second-order nonlinear, non-uniformly parabolic phase-field equation that governs phase evolution.\"},{\"question\":\"How is the phase-field model derived from the sharp-interface model?\",\"answer\":\"The model is obtained by inserting a parabolic regularizing term into an original hyperbolic character model, which itself is derived from a sharp-interface formulation for a strain-discontinuity surface.\"},{\"question\":\"In what spatial setting does the global-in-time existence result hold?\",\"answer\":\"The existence proof establishing global-in-time solutions is valid only in one space dimension.\"}]","Solutions to a Model with Nonuniformly Parabolic Terms for Phase Evolution Driven by Configurational Forces | PDF",55]