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

Hans Knüpfer, Nader Masmoudi

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)395-424
Number of pages30
JournalCommunications in Mathematical Physics
Volume320
Issue number2
DOIs
StatePublished - Jun 2013

Fingerprint

Uniform Bound
Evolution Operator
Wedge
Well-posedness
wedges
regularity
operators
linear evolution equations
Regularity
Sobolev space
Thin Film Equation
Angle
Parabolic Operator
Fluid
Singular Limit
Convergence of Solutions
Weighted Sobolev Spaces
fluids
linearization
Operator

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.

In: Communications in Mathematical Physics, Vol. 320, No. 2, 06.2013, p. 395-424.

Research output: Contribution to journalArticle

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