Convergence of a one-dimensional Cahn-Hilliard equation with degenerate mobility
We consider a one-dimensional periodic forward-backward parabolic equation, regularized by a degenerate nonlinear fourth-order term of order $\varepsilon^2\ll 1$. This equation is known in the literature as Cahn--Hilliard equation with degenerate mobility. Under the hypothesis of the initial data be...
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| Format: | Journal article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2018
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| _version_ | 1826276280416337920 |
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| author | Delgadino, MG |
| author_facet | Delgadino, MG |
| author_sort | Delgadino, MG |
| collection | OXFORD |
| description | We consider a one-dimensional periodic forward-backward parabolic equation, regularized by a degenerate nonlinear fourth-order term of order $\varepsilon^2\ll 1$. This equation is known in the literature as Cahn--Hilliard equation with degenerate mobility. Under the hypothesis of the initial data being well prepared, we prove that as $\varepsilon\to0$, the solution converges to the solution of a well-posed degenerate parabolic equation. The proof exploits the gradient flow nature of the equation in $\mathcal{W}^2(\Bbb{T})$ and utilizes the framework of convergence of gradient flows developed by Sandier and Serfaty (Comm. Pure Appl. Math., 57 (2004), pp. 1627--1672; Discrete Contin. Dyn. Syst., 31 (2011), pp. 1427--1451). As an incidental, we study fine energetic properties of solutions to the thin-film equation $\partial_t\nu=-(\nu\nu_{xxx})_x$. |
| first_indexed | 2024-03-06T23:11:35Z |
| format | Journal article |
| id | oxford-uuid:65a9c5d2-6c49-4fb2-9b0d-b71253c6d06a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T23:11:35Z |
| publishDate | 2018 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:65a9c5d2-6c49-4fb2-9b0d-b71253c6d06a2022-03-26T18:26:52ZConvergence of a one-dimensional Cahn-Hilliard equation with degenerate mobilityJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:65a9c5d2-6c49-4fb2-9b0d-b71253c6d06aEnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2018Delgadino, MGWe consider a one-dimensional periodic forward-backward parabolic equation, regularized by a degenerate nonlinear fourth-order term of order $\varepsilon^2\ll 1$. This equation is known in the literature as Cahn--Hilliard equation with degenerate mobility. Under the hypothesis of the initial data being well prepared, we prove that as $\varepsilon\to0$, the solution converges to the solution of a well-posed degenerate parabolic equation. The proof exploits the gradient flow nature of the equation in $\mathcal{W}^2(\Bbb{T})$ and utilizes the framework of convergence of gradient flows developed by Sandier and Serfaty (Comm. Pure Appl. Math., 57 (2004), pp. 1627--1672; Discrete Contin. Dyn. Syst., 31 (2011), pp. 1427--1451). As an incidental, we study fine energetic properties of solutions to the thin-film equation $\partial_t\nu=-(\nu\nu_{xxx})_x$. |
| spellingShingle | Delgadino, MG Convergence of a one-dimensional Cahn-Hilliard equation with degenerate mobility |
| title | Convergence of a one-dimensional Cahn-Hilliard equation with degenerate mobility |
| title_full | Convergence of a one-dimensional Cahn-Hilliard equation with degenerate mobility |
| title_fullStr | Convergence of a one-dimensional Cahn-Hilliard equation with degenerate mobility |
| title_full_unstemmed | Convergence of a one-dimensional Cahn-Hilliard equation with degenerate mobility |
| title_short | Convergence of a one-dimensional Cahn-Hilliard equation with degenerate mobility |
| title_sort | convergence of a one dimensional cahn hilliard equation with degenerate mobility |
| work_keys_str_mv | AT delgadinomg convergenceofaonedimensionalcahnhilliardequationwithdegeneratemobility |