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

TL;DR

The paper analyzes the 2D Cahn–Hilliard equation on the torus with a background horizontal shear, focusing on the large‑shear (small $\nu$) regime. By projecting the solution onto the kernel of the advection operator and its orthogonal complement, the authors prove that the orthogonal component experiences enhanced dissipation, while the parallel component converges to the one‑dimensional Cahn–Hilliard dynamics in $x_2$, under well‑prepared data. A bootstrap argument, together with energy methods and enhanced dissipation estimates for the operator $H_\nu = \mu\nu \Delta^2 + v(x_2)\partial_{x_1}$, establishes uniform bounds and exponential decay for the perpendicular part, leading to a rigorous 1Dization of the long‑time behavior. The results provide a rigorous justification for the striated, banded patterns observed in simulations and connect mixing and enhanced dissipation mechanisms to phase separation under shear.

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.

Theorems & Definitions (23)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Remark 2.2
• Lemma 3.1
• Remark 3.2
• proof
• Lemma 4.1
• proof
• Lemma 4.2