Phase separation for the 2D Cahn-Hilliard equation with a background shear flow
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918491481702400 |
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| author | Feng, Yu Feng, Yuanyuan Mazzucato, Anna L. Xu, Xiaoqian |
| author_facet | Feng, Yu Feng, Yuanyuan Mazzucato, Anna L. Xu, Xiaoqian |
| contents | We consider the Cahn-Hilliard equation, which models phase separation in binary fluids, on the two-dimen\-sional torus in the presence of advection by a given background shear flow, satisfying certain conditions and of sufficiently large amplitude. By exploiting the resulting enhanced dissipation for the linearized operator, we prove that, with well-prepared data, the solution converges asymptotically at large times to the solution of a one-dimensional Cahn-Hilliard equation, obtained by projecting the full equation in the direction orthogonal to the shear in a suitable sense. This result rigorously justified the observed phenomenon of striation in the concentration field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_15773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase separation for the 2D Cahn-Hilliard equation with a background shear flow Feng, Yu Feng, Yuanyuan Mazzucato, Anna L. Xu, Xiaoqian Analysis of PDEs 35K91, 76F25, 76E05 We consider the Cahn-Hilliard equation, which models phase separation in binary fluids, on the two-dimen\-sional torus in the presence of advection by a given background shear flow, satisfying certain conditions and of sufficiently large amplitude. By exploiting the resulting enhanced dissipation for the linearized operator, we prove that, with well-prepared data, the solution converges asymptotically at large times to the solution of a one-dimensional Cahn-Hilliard equation, obtained by projecting the full equation in the direction orthogonal to the shear in a suitable sense. This result rigorously justified the observed phenomenon of striation in the concentration field. |
| title | Phase separation for the 2D Cahn-Hilliard equation with a background shear flow |
| topic | Analysis of PDEs 35K91, 76F25, 76E05 |
| url | https://arxiv.org/abs/2509.15773 |