Suppression of epitaxial thin film growth by mixing

TL;DR

This paper analyzes a fourth-order parabolic PDE modeling epitaxial thin film growth, proving local existence of solutions and demonstrating that strong mixing advection prevents blow-up, leading to global existence and exponential convergence.

Contribution

It establishes conditions under which advection ensures global existence and convergence, contrasting with finite-time blow-up without advection.

Findings

Advection with sufficient mixing guarantees global solutions.

Without advection, solutions can blow up in finite time.

Exponential convergence to a homogeneous state under strong mixing.

Abstract

We consider following fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus with and without advection of an incompressible vector field in the case $2<p<3$: \begin{equation*} \partial_t u + (-\Delta)^2 u = -\nabla\cdot(|\nabla u|^{p-2}\nabla u). \end{equation*} The study of this form of equations arises from mathematical models that simulate the epitaxial growth of the thin film. We prove the local existence of mild solutions for any initial data lies in $L^2$ in both cases. Our main result is: in the advective case, if the imposed advection is sufficiently mixing, then the global existence of solution can be proved, and the solution will converge exponentially to a homogeneous mixed state. While in the absence of advection, there exist initial data in $H^2\cap W^{1,\infty}$ such that the solution will blow up in finite time.