Global existence, singular solutions, and ill-posedness for the Muskat problem
The Muskat, or Muskat--Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell under applied pressure gradients or fluid injection/extraction. In contrast to the Hele-Shaw problem (the one-phase version of the Muskat problem), t...
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| Format: | Journal article |
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2004
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| author | Siegel, M Caflisch, R Howison, S |
| author_facet | Siegel, M Caflisch, R Howison, S |
| author_sort | Siegel, M |
| collection | OXFORD |
| description | The Muskat, or Muskat--Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell under applied pressure gradients or fluid injection/extraction. In contrast to the Hele-Shaw problem (the one-phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem. For the stable, forward Muskat problem, in which the higher viscosity fluid expands into the lower viscosity fluid, we show global in time existence for initial data that is a small perturbation of a flat interface. The initial data in this result may contain weak (e.g., curvature) singularities. For the unstable, backward problem, in which the higher viscosity fluid contracts, we construct singular solutions that start off with smooth initial data, but develop a point of infinite curvature at finite time. |
| first_indexed | 2024-03-06T21:58:29Z |
| format | Journal article |
| id | oxford-uuid:4dc6141e-80a6-4eb9-8342-7a8015d383bc |
| institution | University of Oxford |
| last_indexed | 2024-03-06T21:58:29Z |
| publishDate | 2004 |
| record_format | dspace |
| spelling | oxford-uuid:4dc6141e-80a6-4eb9-8342-7a8015d383bc2022-03-26T15:57:19ZGlobal existence, singular solutions, and ill-posedness for the Muskat problemJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4dc6141e-80a6-4eb9-8342-7a8015d383bcMathematical Institute - ePrints2004Siegel, MCaflisch, RHowison, SThe Muskat, or Muskat--Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell under applied pressure gradients or fluid injection/extraction. In contrast to the Hele-Shaw problem (the one-phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem. For the stable, forward Muskat problem, in which the higher viscosity fluid expands into the lower viscosity fluid, we show global in time existence for initial data that is a small perturbation of a flat interface. The initial data in this result may contain weak (e.g., curvature) singularities. For the unstable, backward problem, in which the higher viscosity fluid contracts, we construct singular solutions that start off with smooth initial data, but develop a point of infinite curvature at finite time. |
| spellingShingle | Siegel, M Caflisch, R Howison, S Global existence, singular solutions, and ill-posedness for the Muskat problem |
| title | Global existence, singular solutions, and ill-posedness for the Muskat problem |
| title_full | Global existence, singular solutions, and ill-posedness for the Muskat problem |
| title_fullStr | Global existence, singular solutions, and ill-posedness for the Muskat problem |
| title_full_unstemmed | Global existence, singular solutions, and ill-posedness for the Muskat problem |
| title_short | Global existence, singular solutions, and ill-posedness for the Muskat problem |
| title_sort | global existence singular solutions and ill posedness for the muskat problem |
| work_keys_str_mv | AT siegelm globalexistencesingularsolutionsandillposednessforthemuskatproblem AT caflischr globalexistencesingularsolutionsandillposednessforthemuskatproblem AT howisons globalexistencesingularsolutionsandillposednessforthemuskatproblem |