Global Existence for the unstable Cahn-Hilliard equation in 2D with a Shear Flow

TL;DR

This paper proves global existence and exponential decay of solutions for a 2D unstable Cahn-Hilliard equation with shear flow, under specific conditions on the shear and initial data.

Contribution

It establishes the global existence and decay rates for the unstable Cahn-Hilliard equation with shear flow in 2D, extending understanding of stability in such advective systems.

Findings

Solutions' L^2-energy decays exponentially over time.

Global existence of solutions is guaranteed under small initial data.

Decay rate depends on shear flow properties and initial data size.

Abstract

In this paper, we consider the advective unstable Cahn-Hilliard equation in 2D with shear flow: \begin{equation*} \begin{cases} u_t+Av_1(y) \partial_x u+\varepsilon \Delta^2 u= \Delta(a u^3+ b u^2) \quad & \quad \textrm{on} \quad \mathbb T^2; \\ \\ u \ \textrm{periodic} \quad & \quad \textrm{on} \quad \partial \mathbb T^2, \end{cases} \end{equation*} with an initial data $u_0 \in H_0^2(\mathbb T^2)$, where $\mathbb T^2$ is the two-dimensional torus, $A, \varepsilon>0$, $a<0$, $b \in \mathbb R$. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we show the $L^2$-energy of the solutions to such problems converges expotentially to zero, if in addition, both $|a|$ and $\left\| \int_{\mathbb T} u_0(x, \cdot ) dx \right\|_{L_y^2}$ are sufficiently small.