On Singularity Formation in a Hele-Shaw Model

Abstract

We discuss a lubrication approximation model of the interface between two immiscible fluids in a Hele-Shaw cell, derived in Constantin et al. (Phys Rev E 47(6):4169–4181, 1993) and widely studied since. The model consists of a single one dimensional evolution equation for the thickness 2h =  2h(x, t) of a thin neck of fluid,

$${\partial_{t} {h} + \partial_{x} (h \partial^{3}_{ x} h) = 0,}$$

for ${x \in (-1, 1)\, {\rm and}\, t \geq 0}$. The boundary conditions fix the neck height and the pressure jump:

$${h(\pm 1, t) = 1, \quad\quad \partial^{2}_{ x} h(\pm 1, t) = P > 0.}$$

We prove that starting from smooth and positive h, as long as h(x, t) >  0, for x ∈ [−1, 1], t ∈ [0, T ], no singularity can arise in the solution up to time T. As a consequence, we prove for any P > 2 and any smooth and positive initial datum that the solution pinches off in either finite or infinite time, i.e., ${{\rm inf}_{[-1,1]\times[0,T*)}h = 0}$, for some ${{T}_{*} \in (0,\infty]}$. These facts have been long anticipated on the basis of numerical and theoretical studies.