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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 $h(x(t), t)\to 0$ 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 $n\in \bbfR$. 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 $4- 2n$.
Reviewer: A.Bertozzi (Durham/NC)

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