Towards optimal regularity for the fourth-order thin film equation in RN: Graveleau-type focusing self-similarity

Abstract

An approach to some "optimal" (more precisely, non-improvable) regularity of solutions of the thin film equationwhere n ∈ (0, 2) is a fixed exponent, with smooth compactly supported initial data u 0 (x), in dimensions N ≥ 2 is discussed. Namely, a precise exponent for the Hölder continuity with respect to the spatial radial variable |x| is obtained by construction of a Graveleau-type focusing self-similar solution. As a consequence, optimal regularity of the gradient ∇u in certain L p spaces, as well as a Hölder continuity property of solutions with respect to x and t, are derived, which cannot be obtained by classic standard methods of integral identities-inequalities. Several profiles for the solutions in the cases n = 0 and n > 0 are also plotted. In general, we claim that, even for arbitrarily small n > 0 and positive analytic initial data u 0 (x), the solutions u(x, t) cannot be better than C 2−ε x -smooth, where ε(n) = O(n) as n → 0.

Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting

“…Regularity and smoothness of the solution and it's free boundary in the one dimensional setting was obtained in the papers [1] - [5]. As for the case of more than one spatial variable (the multidimensional setting), the author is aware only of the paper [15] (see also the paper [17] in this connection). It is well known that multidimensional setting is fundamentally differ from the one-dimensional one.…”