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

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.