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Symmetric singularity formation in lubrication-type equations for interface motion. (English) Zbl 0856.35002

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. 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 suggests 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 of 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 (MLE) is a transition from a finite time singularity to an infinite time singularity 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 blow up in derivatives of order greater than 4-2n.

MSC:
35A20Analytic methods, singularities (PDE)
35C20Asymptotic expansions of solutions of PDE
35K55Nonlinear parabolic equations
76B45Capillarity (surface tension)
76M20Finite difference methods (fluid mechanics)