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

2010/09/21 by Burchard, Almut, Chugunova, Marina, Stephens, Benjamin K.
#35K25 #35K35 #35Q35 #37L05 #76A20 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1009.4092

Abstract

Steady states of the thin film equation u_t+[u³ (u_xxx + α² u_x -sin(x) )]_x=0 are considered on the periodic domain Ω= (-π,π). The equation defines a generalized gradient flow for an energy functional that controls the H¹-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 α<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.

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