[{"_id":"10005","year":"2021","acknowledgement":"M. Bulíček and J. Málek acknowledge the support of the project No. 18-12719S financed by the Czech\r\nScience foundation (GAČR). E. Maringová acknowledges support from Charles University Research program \r\nUNCE/SCI/023, the grant SVV-2020-260583 by the Ministry of Education, Youth and Sports, Czech Republic\r\nand from the Austrian Science Fund (FWF), grants P30000, W1245, and F65. M. Bulíček and J. Málek are\r\nmembers of the Nečas Center for Mathematical Modelling.\r\n","date_created":"2021-09-12T22:01:25Z","volume":31,"type":"journal_article","oa_version":"Preprint","month":"08","date_updated":"2023-09-04T11:43:45Z","abstract":[{"lang":"eng","text":"We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone p-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations."}],"citation":{"ista":"Bulíček M, Maringová E, Málek J. 2021. On nonlinear problems of parabolic type with implicit constitutive equations involving flux. Mathematical Models and Methods in Applied Sciences. 31(09).","mla":"Bulíček, Miroslav, et al. “On Nonlinear Problems of Parabolic Type with Implicit Constitutive Equations Involving Flux.” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 31, no. 09, World Scientific, 2021, doi:<a href=\"https://doi.org/10.1142/S0218202521500457\">10.1142/S0218202521500457</a>.","apa":"Bulíček, M., Maringová, E., &#38; Málek, J. (2021). On nonlinear problems of parabolic type with implicit constitutive equations involving flux. <i>Mathematical Models and Methods in Applied Sciences</i>. World Scientific. <a href=\"https://doi.org/10.1142/S0218202521500457\">https://doi.org/10.1142/S0218202521500457</a>","ama":"Bulíček M, Maringová E, Málek J. On nonlinear problems of parabolic type with implicit constitutive equations involving flux. <i>Mathematical Models and Methods in Applied Sciences</i>. 2021;31(09). doi:<a href=\"https://doi.org/10.1142/S0218202521500457\">10.1142/S0218202521500457</a>","short":"M. Bulíček, E. Maringová, J. Málek, Mathematical Models and Methods in Applied Sciences 31 (2021).","ieee":"M. Bulíček, E. Maringová, and J. Málek, “On nonlinear problems of parabolic type with implicit constitutive equations involving flux,” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 31, no. 09. World Scientific, 2021.","chicago":"Bulíček, Miroslav, Erika Maringová, and Josef Málek. “On Nonlinear Problems of Parabolic Type with Implicit Constitutive Equations Involving Flux.” <i>Mathematical Models and Methods in Applied Sciences</i>. World Scientific, 2021. <a href=\"https://doi.org/10.1142/S0218202521500457\">https://doi.org/10.1142/S0218202521500457</a>."},"intvolume":"        31","status":"public","external_id":{"arxiv":["2009.06917"],"isi":["000722222900004"]},"main_file_link":[{"url":"https://arxiv.org/abs/2009.06917","open_access":"1"}],"date_published":"2021-08-25T00:00:00Z","oa":1,"publication_status":"published","article_type":"original","article_processing_charge":"No","scopus_import":"1","publication":"Mathematical Models and Methods in Applied Sciences","department":[{"_id":"JuFi"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"World Scientific","arxiv":1,"title":"On nonlinear problems of parabolic type with implicit constitutive equations involving flux","day":"25","author":[{"first_name":"Miroslav","last_name":"Bulíček","full_name":"Bulíček, Miroslav"},{"id":"dbabca31-66eb-11eb-963a-fb9c22c880b4","full_name":"Maringová, Erika","first_name":"Erika","last_name":"Maringová"},{"full_name":"Málek, Josef","first_name":"Josef","last_name":"Málek"}],"issue":"09","language":[{"iso":"eng"}],"keyword":["Nonlinear parabolic systems","implicit constitutive theory","weak solutions","existence","uniqueness"],"isi":1,"project":[{"grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"quality_controlled":"1","doi":"10.1142/S0218202521500457","publication_identifier":{"issn":["0218-2025"],"eissn":["1793-6314"]}},{"keyword":["Mean curvature flow","gradient flows","varifolds","weak solutions","weak-strong uniqueness","calibrated geometry","gradient-flow calibrations"],"citation":{"ama":"Hensel S, Laux T. A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2109.04233\">10.48550/arXiv.2109.04233</a>","ista":"Hensel S, Laux T. A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. arXiv, 2109.04233.","mla":"Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.” <i>ArXiv</i>, 2109.04233, doi:<a href=\"https://doi.org/10.48550/arXiv.2109.04233\">10.48550/arXiv.2109.04233</a>.","apa":"Hensel, S., &#38; Laux, T. (n.d.). A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2109.04233\">https://doi.org/10.48550/arXiv.2109.04233</a>","ieee":"S. Hensel and T. Laux, “A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness,” <i>arXiv</i>. .","chicago":"Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2109.04233\">https://doi.org/10.48550/arXiv.2109.04233</a>.","short":"S. Hensel, T. Laux, ArXiv (n.d.)."},"language":[{"iso":"eng"}],"project":[{"_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","call_identifier":"H2020","name":"Bridging Scales in Random Materials","grant_number":"948819"}],"status":"public","external_id":{"arxiv":["2109.04233"]},"date_published":"2021-09-09T00:00:00Z","doi":"10.48550/arXiv.2109.04233","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2109.04233"}],"publication_status":"submitted","oa":1,"_id":"10011","publication":"arXiv","article_processing_charge":"No","ec_funded":1,"acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 948819), and from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. The content of this paper was developed and parts of it were written during a visit of the first author to the Hausdorff Center of Mathematics (HCM), University of Bonn. The hospitality and the support of HCM are gratefully acknowledged.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"JuFi"}],"year":"2021","article_number":"2109.04233","title":"A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness","arxiv":1,"date_created":"2021-09-13T12:17:10Z","author":[{"orcid":"0000-0001-7252-8072","id":"4D23B7DA-F248-11E8-B48F-1D18A9856A87","full_name":"Hensel, Sebastian","first_name":"Sebastian","last_name":"Hensel"},{"first_name":"Tim","last_name":"Laux","full_name":"Laux, Tim"}],"month":"09","oa_version":"Preprint","type":"preprint","abstract":[{"text":"We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.","lang":"eng"}],"date_updated":"2023-05-03T10:34:38Z","day":"09"}]