Global existence, singular solutions, and ill-posedness for the Muskat problem

Communications on Pure and Applied Mathematics

Volumn 57, Issue 10, 2004, Pages 1374-1411

  Siegel, Michael ^a   Caflisch, Russel E ^b   Howison, Sam ^c  

^a NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4544228729     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.20040     Document Type: Article

References (17)

1. Caflisch, R. E. Vortex dynamics. Unpublished manuscript, 1989.
2. Caflisch, R. E. A simplified version of the abstract Cauchy-Kowalewski theorem with weak singularities. Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 495-500.
3. Caflisch, R. E.; Orellana, O. F. Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20 (1989), no. 2, 293-307.
4. Caflisch, R. E.; Orellana, O.; Siegel, M. A localized approximation method for vortical flows. SIAM J. Appl. Math. 50 (1990), no. 6, 1517-1532.
5. Ceniceros, H. D.; Hou, T. Y.; Si, H. Numerical study of Hele-Shaw flow with suction. Phys. Fluids 11 (1999), no. 9, 2471-2486.
6. Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. 5th ed. Academic, Boston, 1994.
7. Howison, S. D. Complex variable methods in Hele-Shaw moving boundary problems. European J. Appl. Math. 3 (1992), no. 3, 209-224.
8. Howison, S. D. A note on the two-phase Hele-Shaw problem. J. Fluid Mech. 409 (2000), 243-249.
9. Jiang, L.; Chen, Y. 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, J. R.; Lacey, A. A.; Vázquez, J.-L. Persistence of comers in free boundaries in Hele-Shaw flow. European J. Appl. Math. 6 (1995), no. 5, 455-490.
11. Muskat, M. Two-fluid systems in porous media. The encrpachment of water into an oil sand. Physics 5 (1934), 250-264.
12. Otto, F. Viscous fingering: an optimal bound on the growth rate of the mixing zone. SIAM J. Appl. Math. 57 (1997), no. 4, 982-990.
13. Otto, F. Evolution of microstructure in unstable porous media flow: a relaxational approach. Comm. Pure Appl. Math. 52 (1999), no. 7, 873-915.
14. Saffman, P. G.; Taylor, G. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. Roy. Soc. London. Ser. A 245 (1958), 312-329.
15. Schroll, H. J.; Tveito, A. Local existence and stability for a hyperbolic-elliptic system modeling two-phase reservoir flow. Electron. J. Differential Equations 2000, no. 4.
16. Tryggvason, G.; Aref, H. Numerical experiments on Hele-Shaw flow with a sharp interface. J. Fluid Mech. 136 (1983), 1-30.
17. Zygmund, A. Trigonometrical series. 2nd ed. Chelsea, New York, 1952.