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Global existence, singular solutions, and ill-posedness for the Muskat problem. (English) Zbl 1062.35089
Author’s summary: 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 constrast 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 the 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.

MSC:
35Q35 PDEs in connection with fluid mechanics
76D27 Other free boundary flows; Hele-Shaw flows
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[1] Vortex dynamics. Unpublished manuscript, 1989.
[2] Caflisch, Bull Amer Math Soc (NS) 23 pp 495– (1990)
[3] Caflisch, SIAM J Math Anal 20 pp 293– (1989)
[4] Caflisch, SIAM J Appl Math 50 pp 1517– (1990)
[5] Ceniceros, Phys Fluids 11 pp 2471– (1999)
[6] ; Table of integrals, series, and products. 5th ed. Academic, Boston, 1994.
[7] Howison, European J Appl Math 3 pp 209– (1992)
[8] Howison, J Fluid Mech 409 pp 243– (2000)
[9] ; Weak formulation of a multidimensional Muskat problem. Free boundary problems: theory and applications, vol. II (Irsee, 1987), 509-513. Pitman Research Notes in Mathematics Series, 186. Longman, Harlow, 1990.
[10] King, European J Appl Math 6 pp 455– (1995)
[11] Muskat, Physics 5 pp 250– (1934)
[12] Otto, SIAM J Appl Math 57 pp 982– (1997)
[13] Otto, Comm Pure Appl Math 52 pp 873– (1999)
[14] Saffman, Proc Roy Soc London Ser A 245 pp 312– (1958)
[15] Schroll, Electron J Differential Equations 2000
[16] Tryggvason, J Fluid Mech 136 pp 1– (1983)
[17] Trigonometrical series. 2nd ed. Chelsea, New York, 1952.
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