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

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 $L^2$-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.

[1]  G. Burton.  Sobolev Spaces , 2013 .

[2]  Christian Seis.  Long-time asymptotics for the porous medium equation: The spectrum of the linearized operator , 2013, 1304.0834.

[3]  Francisco Bernis,et al.  Source type solutions of a fourth order nonlinear degenerate parabolic equation , 1992 .

[4]  C. Bennett,et al.  Interpolation of operators , 1987 .

[5]  D. John.  On Uniqueness of Weak Solutions for the Thin-Film Equation , 2013, 1310.6222.

[6]  G. Hardy,et al.  Note on a theorem of Hilbert , 1920 .

[7]  Thorsten Gerber,et al.  Semigroups Of Linear Operators And Applications To Partial Differential Equations , 2016 .

[8]  E. Carlen,et al.  Asymptotic equipartition and long time behavior of solutions of a thin-film equation☆ , 2007 .

[9]  R. McCann,et al.  The Spectrum of a Family of Fourth-Order Nonlinear Diffusions Near the Global Attractor , 2015 .

[10]  Lorenzo Giacomelli,et al.  Rigorous lubrication approximation , 2003 .

[11]  A. Mellet.  The Thin Film Equation with Non-Zero Contact Angle: A Singular Perturbation Approach , 2015 .

[12]  J. A. Carrillo,et al.  Asymptotic L1-decay of solutions of the porous medium equation to self-similarity , 2000 .

[13]  Noel F. Smyth,et al.  High-Order Nonlinear Diffusion , 1988 .

[14]  Andrew J. Bernoff,et al.  Linear stability of source-type similarity solutions of the thin film equation , 2002, Appl. Math. Lett..

[15]  R. McCann,et al.  A Family of Nonlinear Fourth Order Equations of Gradient Flow Type , 2009, 0901.0540.

[16]  S. Herminghaus,et al.  Wetting: Statics and dynamics , 1997 .

[17]  Alexander Mielke.  Über maximaleLp-Regularität für Differentialgleichungen in Banach- und Hilbert-Räumen , 1987 .

[18]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[19]  Francisco Bernis,et al.  Finite speed of propagation for thin viscous flows when 2 ≤ n ≤ 3 , 1996 .

[20]  J. Goldstein.  Semigroups of Linear Operators and Applications , 1985 .

[21]  Felix Otto,et al.  Lubrication approximation with prescribed nonzero contact anggle , 1998 .

[22]  Thin-film equations with “partial wetting” energy: Existence of weak solutions , 2005 .

[23]  P. Gennes.  Wetting: statics and dynamics , 1985 .

[24]  Well-posedness for the Navier-slip thin-film equation in the case of complete wetting , 2014 .

[25]  Lorenzo Giacomelli,et al.  Smooth zero-contact-angle solutions to a thin-film equation around the steady state☆ , 2008 .

[26]  Giuseppe Toscani,et al.  Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation , 2002 .

[27]  James Stuart Tanton,et al.  Encyclopedia of Mathematics , 2005 .

[28]  Francisco Bernis,et al.  Finite speed of propagation and continuity of the interface for thin viscous flows , 1996, Advances in Differential Equations.

[29]  Well-Posedness and Uniform Bounds for a Nonlocal Third Order Evolution Operator on an Infinite Wedge , 2013 .

[30]  D. Bonn,et al.  Wetting and Spreading , 2009 .

[31]  S. Bankoff,et al.  Long-scale evolution of thin liquid films , 1997 .

[32]  E. Weinan,et al.  Boundary Conditions for the Moving Contact Line Problem , 2007 .

[33]  F. Otto,et al.  Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3 , 2013, European Journal of Applied Mathematics.