The two-phase hele-shaw problem with a nonregular initial interface and without surface tension

Journal of Mathematical Physics, Analysis, Geometry

Volumn 10, Issue 1, 2014, Pages 3-43

  Bazaliy, B V ^a   Vasylyeva, N ^a  

^a INSTITUTE OF APPLIED MATHEMATICS AND MECHANICS   (Ukraine)

Author keywords

Free boundary problems; Laplace equation; Muskat problem; Waiting time property; Weighted H lder spaces

Indexed keywords

EID: 84892764232     PISSN: 18129471     EISSN: 18175805     Source Type: Journal    
DOI: 10.15407/mag10.01.003     Document Type: Article

References (31)

1. Ambrose, Well-Posedness of Two-Phase Hele-Shaw Flow without Surface Ten- sion. European J. Appl. Math. 15 (2004), 597-607.
2. H. Bateman and A. Erdélyi, Tables of Integral Transforms. Vol. 1. Book Company, INC, New York, Toronto, London, 1954.
3. B.V. Bazaliy, Stefan Problem for the Laplace Equation with Regard for the Curvature of the Free Boundary. Ukr. Math. J. 40 (1997), 1465-1484.
4. B.V. Bazaliy, Classical Solvability of the Free Boundary Hele-Shaw Problem. Ukr. Math. J. 50 (1998), 1452-1462.
5. B.V. Bazaliy and A. Friedman, The Hele-Shaw Problem with Surface Tension in a Half-Plane. J. Diff. Eqs. 216 (2005), 439-469.
6. E. Di Benedetto and A. Friedman, The Ill-Posed Hele-Shaw and Stefan Problems for Supercoold Water. Trans. Amer. Math. Soc. 282 (1984), 183-203.
7. B.V. Bazaliy and N. Vasylyeva, The Muskat Problem with Surface Tension and a Nonregular Initial Interface. Nonlinear Analysis: Theory, Methods and Applications 74 (2011), 6074-6096.
8. B.V. Bazaliy and N. Vasylyeva, The Transmission Problem in Domains with a Corner Point for the Laplace Operator in Weighted Hölder Spaces. J. Diff. Eqs. 249 (2010), 2476-2499.
9. B.V. Bazaliy and N. Vasylyeva, On the Solvability of a Transmission Problem for the Laplace Operator with a Dynamic Boundary Condition on a Nonregular Interface. J. Math. Anal. Appl. 393 (2012), 651-670.
10. J.A. Cima, A.L. Matheson, and W.T. Ross, The Cauchy Transform. Mathematical Surveys and Monographs 125, AMS, 2006.
11. I.I. Daniliuk, Nonregular Boundary Problems on a Plane. Nauka, Moscow, 2006. (Russian)
12. S.P. Degtyarev, The Existence of a Smooth Interface in the Nonstationary Elliptic Muskat-Verigin Problem with a Nonlinear Source. Ukr. Math. Bull. 7 (2010), 301-330.
13. C. Elliott and J.R. Ockendon, Weak and Variational Methods for Moving Boundary Problem. Pitman, London, 1982.
14. J. Esher and B.V. Matioc, On the Parabolicity of the Muskat Problem: Well-Posedness, Fingering and Stability Results. Z. Anal. Anwend. 30 (2011), No. 2, 193-218.
15. J. Esher and G. Simonett, Classical Solutions of Multidimensional Hele-Shaw Models. SIAM J. Math. Anal. 28 (1997), 1028-1047.
16. E.I. Hanzawa, Classical Solution of the Stefan Problem. Tohoku Math. J. 33 (1981), 297-335.
17. Y.E. Hohlov and S. Howison, The Classification of Solutions in the Free Boundary Hele-Shaw Problem. Dokl. Acad. Nauk USSR 325 (1992), 1161-1166.
18. S. Howison, A Note on the Two-Phase Hele-Shaw Problem. J. Fluid Mech. 409 (2000), 243-249.
19. L. Jiang and Y. Chen, Weak Formulation of a Multidimensional Muskat Problem. Free Boundary Problems: Theory and Applications, Vol. II (Irsee,1987), 509-513. Pitman Research Notes in Mathematics Seris, 186. Longman, Harlow, 1990.
20. J.R. King, A.A. Lacey, and J.L. Vazquez, Persestence of Corners in Free Boundaries in Hele-Shaw Flow. European J. Appl. Math. 6 (1995), 455-490.
21. M.V. Krasnoschok, On an Initial-Boundary Value Problem for a Stationary System of the Theory of Elasticity with Additional Dynamic Condition on a Boundary of a Domain. Transactions of IAMM 21 (2010), 137-150.
22. O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural'tseva, Linear and Quasilinear Parabolic Equations. Transl. Math. Monogr. 23 AMS, Providence, RI, 1968.
23. A. Lundardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in NoDEA. 16 BirkhaÄuser, Verlag, Basel, 1995.
24. P.B. Mucha, On the Stefan Problem with Surface Tension in the Lp Framework. Adv. Diff. Eqs. 10 (2005), No. 8, 861-900.
25. M. Muskat, Two Fluid Systems in Porous Media. The Encroachment of Water into an Oil Sand. Physics 5 (1934), 250-264.
26. F. Otto, Viscous Fingering: an Optimal Bound on the Growth Rate of the Mixing Zone. SIAM J. Appl. Math. 57 (1997), No. 4, 982-990.
27. Ja. A. Roitberg and Z.G. Sheftel, General Boundary Value Problems for Elliptic Equations with Discontinuous Coeffcients. Soviet. Math. Dokl. 4 (1963), 231-234. (Russian)
28. M. Siegel, R. Caflisch, and S. Howison, Global Existence, Singular Solutions, and Ill-Posedness for the Muskat Problem. Comm. Pure and Appl. Math. 57 (2004), 1374-1411.
29. N. Vasylyeva, On the Solvability of the Hele-Shaw Problem in the Case of Nonsmooth Initial Data in Weighted Hölder Classes. Ukr. Math. Bull. 2 (2005), No. 3, 323-349.
30. F. Yi, Local Classical Solution of Muskat Free Boundary Problem. J. Partial Diff. Eqs. 9 (1996), 84-96.
31. F. Yi, Global Classical Solution of Muskat Free Boundary Problem. J. Math. Anal. Appl. 288 (2003), 442-461.