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arXiv · 2509.15773

Phase separation for the 2D Cahn-Hilliard equation with a background shear flow

Yu Feng·Yuanyuan Feng·Anna L. Mazzucato·Xiaoqian Xu
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Abstract

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.

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Yu Feng, Yuanyuan Feng, Anna L. Mazzucato, Xiaoqian Xu. 2025-09-19. Phase separation for the 2D Cahn-Hilliard equation with a background shear flow. https://arxiv.org/abs/2509.15773

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