Global Existence for the unstable Cahn-Hilliard equation in 2D with a Shear Flow
Bingyang Hu, Dinghua Xu, Yeyu Zhang
TL;DR
The authors study a 2D unstable Cahn–Hilliard equation with a horizontal shear advection on the torus and prove global existence and exponential decay of the $L^2$ energy under small cubic interaction ($|a|$ small) and small initial mean. They decompose the solution into mean and mean-zero components and leverage a dissipation-enhancing operator $H_\gamma$ associated with the shear to damp the oscillatory part, obtaining a decay rate $\lambda_\gamma$ that scales like $\gamma^{2/(2+m)}$. A bootstrap argument combines sharp nonlinear estimates with decay of the mixing operator to derive uniform bounds and close the bootstrap, yielding global mild solutions with $u \in L^\infty([0,\infty); L^2(\mathbb{T}^2)) \cap L^2([0,\infty); H^2(\mathbb{T}^2))$ and exponential decay of the $L^2$ norm. This work demonstrates how a carefully chosen shear can suppress finite-time blow-up in an unstable phase-field model by exploiting flow-induced dissipation.
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 Δ^2 u= Δ(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.