On Singularity Formation in a Hele-Shaw Model

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,∂th+∂x(h∂x3h)=0, for x∈(-1,1)andt≥0. The boundary conditions fix the neck height and the pressure jump: h(±1,t)=1,∂x2h(±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., inf _{[ - 1 , 1 ] × [ 0 , T ∗ )}h= 0 , for some T_∗∈ (0 , ∞]. These facts have been long anticipated on the basis of numerical and theoretical studies.