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MiS Preprint

Well-Posedness and Self-Similar Asymptotics for a Thin-Film Equation

Manuel Gnann


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.
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MSC Codes:
35A02, 35A09, 35B35, 35B40, 35B65, 35C06, 35K25, 35K55, 35K65, 35K35, 76A20, 76D08, 76D27, 76D45
self-similar solutions, degenerate parabolic equations, fourth-order equations, Nonlinear parabolic equations, free boundary problems, stability, Asymptotic behavior of solutions, Smoothness and regularity of solutions, uniqueness, classical solutions, thin fluid films, lubrication theory, hele-shaw flow, capillarity

Related publications

2015 Repository Open Access
Manuel V. Gnann

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

In: SIAM journal on mathematical analysis, 47 (2015) 4, pp. 2868-2902