Well-posedness and self-similar asymptotics for a thin-film equation

Journal article (2015)

Authors

M.V. Gnann University of Michigan
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Uniqueness Stability Lubrication theory Capillarity Free boundary problems Classical solutions Degenerate parabolic equations Smoothness and regularity of solutions Nonlinear parabolic equations Asymptotic behavior of solutions Fourth-order equations Hele-Shaw flows Self-similar solutions Thin fluid films
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Published Date

2015

Language

English

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Abstract

We investigate compactly supported solutions for a thin-film equation with linear mobility in the regime of perfect wetting. This problem has already been addressed by Carrillo and Toscani, proving that the source-type self-similar profile is a global attractor of entropy solutions with compactly supported initial data. Here we study small perturbations of source-type self-similar solutions for the corresponding classical free boundary problem and set up a global existence and uniqueness theory within weighted L2-spaces under minimal assumptions. Furthermore, we derive asymptotics for the evolution of the solution, the free boundary, and the center of mass. As spatial translations are scaled out in our reference frame, the rate of convergence is higher than the one obtained by Carrillo and Toscani.