# Localization, Smoothness, and Convergence to Equilibrium for a Thin Film Equation - Mathematics > Analysis of PDEs

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Abstract: We investigate the long-time behavior of weak solutions to the thin-film typeequation $$v t =xv - vv {xxx} x\ ,$$ which arises in the Hele-Shaw problem.We estimate the rate of convergence of solutions to the Smyth-Hill equilibriumsolution, which has the form $\frac{1}{24}C^2-x^2^2 +$, in the norm $$|\!|\!|f |\!|\!| {m,1}^2 = \int {\R}1+ |x|^{2m}|fx|^2\dd x +\int {\R}|f xx|^2\dd x\ .$$ We obtain exponential convergence in the $|\!|\!|\cdot |\!|\!| {m,1}$ norm for all $m$ with $1\leq m< 2$, thus obtaining ratesof convergence in norms measuring both smoothness and localization. Thelocalization is the main novelty, and in fact, we show that there is a closeconnection between the localization bounds and the smoothness bounds:Convergence of second moments implies convergence in the $H^1$ Sobolev norm. Wethen use methods of optimal mass transportation to obtain the convergence ofthe required moments. We also use such methods to construct an appropriateclass of weak solutions for which all of the estimates on which our convergenceanalysis depends may be rigorously derived. Though our main results onconvergence can be stated without reference to optimal mass transportation,essential use of this theory is made throughout our analysis.

Autor: Eric A. Carlen, Suleyman Ulusoy

Fuente: https://arxiv.org/