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 \cite{CDGKSZ93} 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, \[ \partial_t h + \partial_x( h \, \partial_x^3 h) = 0\, , \] for $x\in (-1,1)$ and $t\ge 0$. The boundary conditions fix the neck height and the pressure jump: \[ h(\pm 1,t) = 1, \qquad \partial_{x}^2 h(\pm 1,t) = P>0. \] We prove that starting from smooth and positive $h$, as long as $h(x,t)>0$, for $x\in [-1,1],t\in [0,T]$, 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., ${\inf }_{[-1,1]\times [0,T_{\ast })}h=0$, for some $T_{\ast }\in (0,\infty ]$. These facts have been long anticipated on the basis of numerical and theoretical studies.

References in corpus (1)

• Stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem

Cited by in corpus (2)

• Compressible fluids and active potentials
• Classical solutions to the thin-film equation with general mobility in the perfect-wetting regime