### Abstract

We investigate regularity and well-posedness for a fluid evolution model in the presence of a three-phase contact point. We consider a fluid evolution governed by Darcy's Law. After linearization, we obtain a nonlocal third order operator which contains the Dirichlet-Neumann operator on the wedge with opening angle ∈ > 0. We show well-posedness and regularity for this linear evolution equation. In the limit of vanishing opening angle, we show the convergence of solutions to a fourth order degenerate parabolic operator, related to the thin-film equation. In the course of the analysis, we introduce and characterize a new type of sum of weighted Sobolev spaces which are suitable to capture the singular limit as ∈ → 0. In particular, the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem.

Original language | English (US) |
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Pages (from-to) | 395-424 |

Number of pages | 30 |

Journal | Communications in Mathematical Physics |

Volume | 320 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2013 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Well-Posedness and Uniform Bounds for a Nonlocal Third Order Evolution Operator on an Infinite Wedge.** / Knüpfer, Hans; Masmoudi, Nader.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 320, no. 2, pp. 395-424. https://doi.org/10.1007/s00220-013-1708-z

}

TY - JOUR

T1 - Well-Posedness and Uniform Bounds for a Nonlocal Third Order Evolution Operator on an Infinite Wedge

AU - Knüpfer, Hans

AU - Masmoudi, Nader

PY - 2013/6

Y1 - 2013/6

N2 - We investigate regularity and well-posedness for a fluid evolution model in the presence of a three-phase contact point. We consider a fluid evolution governed by Darcy's Law. After linearization, we obtain a nonlocal third order operator which contains the Dirichlet-Neumann operator on the wedge with opening angle ∈ > 0. We show well-posedness and regularity for this linear evolution equation. In the limit of vanishing opening angle, we show the convergence of solutions to a fourth order degenerate parabolic operator, related to the thin-film equation. In the course of the analysis, we introduce and characterize a new type of sum of weighted Sobolev spaces which are suitable to capture the singular limit as ∈ → 0. In particular, the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem.

AB - We investigate regularity and well-posedness for a fluid evolution model in the presence of a three-phase contact point. We consider a fluid evolution governed by Darcy's Law. After linearization, we obtain a nonlocal third order operator which contains the Dirichlet-Neumann operator on the wedge with opening angle ∈ > 0. We show well-posedness and regularity for this linear evolution equation. In the limit of vanishing opening angle, we show the convergence of solutions to a fourth order degenerate parabolic operator, related to the thin-film equation. In the course of the analysis, we introduce and characterize a new type of sum of weighted Sobolev spaces which are suitable to capture the singular limit as ∈ → 0. In particular, the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem.

UR - http://www.scopus.com/inward/record.url?scp=84877601136&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877601136&partnerID=8YFLogxK

U2 - 10.1007/s00220-013-1708-z

DO - 10.1007/s00220-013-1708-z

M3 - Article

VL - 320

SP - 395

EP - 424

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -