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
N1 - Funding Information:
N. M was partially supported by an NSF Grant DMS-1211806.
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
AN - SCOPUS:84877601136
SN - 0010-3616
VL - 320
SP - 395
EP - 424
JO - Communications In Mathematical Physics
JF - Communications In Mathematical Physics
IS - 2
ER -