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| We are interested in the thin-film equation with quadratic mobility, modeling the spreading of a thin liquid film with a Navier-slip condition at the solid substrate. This degenerate fourth-order parabolic equation has the contact line (where liquid, solid, and vapor meet) as a free boundary. There, a zero-contact angle condition is imposed, modeling the so-called ``complete wetting'' regime.
We first argue that the self-similar source-type solution, once its leading order profile is factored-off, is analytic as a function of two variables $(x,x^\beta)$ with $\beta$ irrational, where $x$ denotes the distance from the contact line. Motivated by this preliminary, we then argue that the full free-boundary problem is well-posed in weighted $L^2$-spaces which capture the leading order terms of such $(x,x^\beta)$-expansion. This is part of a joint project with Manuel V. Gnann, Hans Kn\"upfer, and Felix Otto. |
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