Global existence for a thin film equation with subcritical mass

Abstract

In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case ht + ∂x(h n ∂xxxh) + ∂x(h n+2 ∂xh) = 0, where n ≥ 1. There exists a critical mass Mc = 2 √ 6π 3

A generalized Sz. Nagy inequality in higher dimensions and the critical thin film equation

“…Hence from (5.9) and the fact Furthermore, if the initial mass is less than M c , an entropy weak solution exists globally [25] for the one-dimensional case. For a multi-dimensional thin film equation with an unstable diffusion term, Taranets and King [32] showed short-time existence of solutions for the problem (1.1) and (5.1) with d = 2, 3 in a bounded domain with the boundary condition (1.13).…”

Existence and regularity of source-type self-similar solutions for stable thin-film equations

“…The second-order term in the equation, .h m / zz , arises as a cut off of van der Waals interactions [6,27]. In the case m D n C 3, the last equation enjoys a mass invariant scaling transformation [1,33]. If h is a solution of (1.1a), then h .t; z/ D h. nC4 t; z/; > 0 is a solution of (1.1a) on .…”

“…This justifies why we need part (ii) in the previous theorem. [33] for this restriction on !. See also [8,9,17,18,37,41,42,44] for the unstable thin-film equation.…”

Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation

“…On the critical line n = m+2 and when m < 3/2, there are a countable number of families of compactly supported self-similar blow-up solutions, where each solution is characterised by the region of compact support, and number of peaks [25,26,27,28]. These similarity solutions represent exact solutions of (1.1) that draw all the mass into the singularity.…”