Symmetric singularity formation in lubrication-type equations for interface motion

Fourth-order degenerate diffusion equations arise in a ''lubrication approximation'' of a thin film or neck driven by surface tension. Numerical studies of the lubrication equation (LE) h(t) + (h(n)h(xxx))(x) = 0 with various boundary conditions indicate that singularity formation in which h(x(t),t) --> 0 occurs for small enough n with ''anomalous'' or ''second type'' scaling inconsistent with usual dimensional analysis. This paper considers locally symmetric or even singularities in the (LE) and in the modified lubrication equation (MLE) h(t) + h(n)h(xxxx) = 0. Both equations have the property that entropy bounds forbid finite-time singularities when n is sufficiently large. Power series expansions for local symmetric similarity solutions are proposed for equation (LE) with n < 1 and (MLE) for all n is an element of R. In the latter case, special boundary conditions that force singularity formation are required to produce singularities when n is large. Matching conditions at higher-order terms in the expansion suggest a simple functional form for the time dependence of the solution. Computer simulations presented here resolve the self-similarity in the onset of the singularity for approximately 30 decades in min(x)(h(x,t)). Measurements of the similarity shape and time dependences show excellent agreement with the theoretical prediction. One striking feature of the solution to (RILE) is a transition from a finite-time singularity to an infinite-time singularity at n = 3/2. Also, both equations (LE) and (MLE) exhibit symmetric singularities for n < 0 with derivatives of order k vanishing at the singular point for all 2 < k < 4-2n. They also exhibit a blowup in derivatives of order greater than 4 - 2n.