### 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 language | English (US) |
---|---|

Pages (from-to) | 139-171 |

Number of pages | 33 |

Journal | Communications in Mathematical Physics |

Volume | 363 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 2018 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*363*(1), 139-171. https://doi.org/10.1007/s00220-018-3241-6

**On Singularity Formation in a Hele-Shaw Model.** / Constantin, Peter; Elgindi, Tarek; Nguyen, Huy; Vicol, Vlad.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 363, no. 1, pp. 139-171. https://doi.org/10.1007/s00220-018-3241-6

}

TY - JOUR

T1 - On Singularity Formation in a Hele-Shaw Model

AU - Constantin, Peter

AU - Elgindi, Tarek

AU - Nguyen, Huy

AU - Vicol, Vlad

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85052597330&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052597330&partnerID=8YFLogxK

U2 - 10.1007/s00220-018-3241-6

DO - 10.1007/s00220-018-3241-6

M3 - Article

AN - SCOPUS:85052597330

VL - 363

SP - 139

EP - 171

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -