On the energy-minimizing steady states of a thin film equation

TL;DR

This paper analyzes steady states of a thin film equation, proving existence, uniqueness, symmetry, and characterizing the conditions under which solutions touch zero or remain positive, with implications for stability and convergence.

Contribution

It establishes the existence and uniqueness of energy-minimizing steady states for each mass, characterizes their symmetry and positivity properties, and links these to stability and convergence behaviors.

Findings

Unique minimal energy steady states exist for each mass.

Steady states are symmetric and decreasing about the center.

Touchdown zero occurs at a critical mass threshold.

Abstract

Steady states of the thin film equation $u_t+[u^3 (u_xxx + \alpha^2 u_x -\sin(x) )]_x=0$ are considered on the periodic domain $\Omega = (-\pi,\pi)$. The equation defines a generalized gradient flow for an energy functional that controls the $H^1$-norm. The main result establishes that there exists for each given mass a unique nonnegative function of minimal energy. This minimizer is symmetric decreasing about $x=0$. For $\alpha<1$ there is a critical value for the mass at which the minimizer has a touchdown zero. If the mass exceeds this value, the minimizer is strictly positive. Otherwise, it is supported on a proper subinterval of the domain and meets the dry region at zero contact angle. A second result explores the relation between strict positivity and exponential convergence for steady states. It is shown that positive minimizers are locally exponentially attractive, while the distance from a steady state with a dry region cannot decay faster than a power law.