On Singularity Formation in a Hele-Shaw Model

Peter Constantin, Tarek Elgindi, Huy Nguyen, Vlad Vicol

Research output: Contribution to journalArticle

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,∂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.

Original languageEnglish (US)
Pages (from-to)139-171
Number of pages33
JournalCommunications In Mathematical Physics
Volume363
Issue number1
DOIs
StatePublished - Oct 1 2018

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ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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