Details
| Original language | English |
|---|---|
| Pages (from-to) | 679-737 |
| Number of pages | 59 |
| Journal | Indiana University Mathematics Journal |
| Volume | 67 |
| Issue number | 2 |
| Publication status | Published - 2018 |
Abstract
We address the well-posedness of the Muskat problem in a periodic geometry and in a setting which allows us to consider general initial and boundary data, gravity effects, as well as surface tension effects. In the absence of surface tension, we prove that the Rayleigh-Taylor condition identifies a domain of parabolicity for the Muskat problem. This property is used to establish the well-posedness of the problem. In the presence of surface tension effects, the Muskat problem is of parabolic type for general initial and boundary data. As a biproduct of our analysis, we obtain that Dirichlet-Neumann type operators associated with certain diffraction problems are negative generators of strongly continuous and analytic semigroups in the scale of small Hölder spaces.
Keywords
- Diffraction problem, Dirichlet-neumann operator, Muskat problem, Rayleigh-taylor condition
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Indiana University Mathematics Journal, Vol. 67, No. 2, 2018, p. 679-737.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The Domain of Parabolicity for the Muskat Problem
AU - Escher, Joachim
AU - Matioc, Bogdan-Vasile
AU - Walker, Christoph
PY - 2018
Y1 - 2018
N2 - We address the well-posedness of the Muskat problem in a periodic geometry and in a setting which allows us to consider general initial and boundary data, gravity effects, as well as surface tension effects. In the absence of surface tension, we prove that the Rayleigh-Taylor condition identifies a domain of parabolicity for the Muskat problem. This property is used to establish the well-posedness of the problem. In the presence of surface tension effects, the Muskat problem is of parabolic type for general initial and boundary data. As a biproduct of our analysis, we obtain that Dirichlet-Neumann type operators associated with certain diffraction problems are negative generators of strongly continuous and analytic semigroups in the scale of small Hölder spaces.
AB - We address the well-posedness of the Muskat problem in a periodic geometry and in a setting which allows us to consider general initial and boundary data, gravity effects, as well as surface tension effects. In the absence of surface tension, we prove that the Rayleigh-Taylor condition identifies a domain of parabolicity for the Muskat problem. This property is used to establish the well-posedness of the problem. In the presence of surface tension effects, the Muskat problem is of parabolic type for general initial and boundary data. As a biproduct of our analysis, we obtain that Dirichlet-Neumann type operators associated with certain diffraction problems are negative generators of strongly continuous and analytic semigroups in the scale of small Hölder spaces.
KW - Diffraction problem
KW - Dirichlet-neumann operator
KW - Muskat problem
KW - Rayleigh-taylor condition
UR - http://www.scopus.com/inward/record.url?scp=85051763940&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1507.02601
DO - 10.48550/arXiv.1507.02601
M3 - Article
AN - SCOPUS:85051763940
VL - 67
SP - 679
EP - 737
JO - Indiana University Mathematics Journal
JF - Indiana University Mathematics Journal
SN - 0022-2518
IS - 2
ER -