Vol. 11, No. 5, 2018

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The thin-film equation close to self-similarity

Christian Seis

Vol. 11 (2018), No. 5, 1303–1342

We study well-posedness and regularity of the multidimensional thin-film equation with linear mobility in a neighborhood of the self-similar Smyth–Hill solutions. To be more specific, we perform a von Mises change of dependent and independent variables that transforms the thin-film free boundary problem into a parabolic equation on the unit ball. We show that the transformed equation is well-posed and that solutions are smooth and even analytic in time and angular direction. The latter gives the analyticity of level sets of the original equation, and thus, in particular, of the free boundary.

thin-film equation, fourth-order equation, self-similar solution, well-posedness
Mathematical Subject Classification 2010
Primary: 35K30
Secondary: 76A20
Received: 5 September 2017
Accepted: 2 January 2018
Published: 11 April 2018
Christian Seis
Institut für Analysis und Numerik
Westfälische Wilhelms-Universität Münster