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

Abstract

Steady states of the thin film equation $u_t+[u^3(u_{x}xx+{\alpha }^2{u}_x-\sin (x)){]}_x=0$ 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.

arXiv comment14 pages, 5 figures

References in corpus (1)

• Technical report on a long-wave unstable thin film equation with convection