Local smoothing effect and existence for the one-phase Hele-Shaw problem with zero surface tension

Complex Variables and Elliptic Equations

Volumn 57, Issue 2-4, 2012, Pages 351-368

  Xie, Xuming ^a  

^a MORGAN STATE UNIVERSITY   (United States)

Author keywords

Sobolev space, initial value problem; unsteady Hele Shaw problem; well posedness

Indexed keywords

EID: 84857159071     PISSN: 17476933     EISSN: 17476941     Source Type: Journal    
DOI: 10.1080/17476933.2011.575463     Document Type: Article

References (53)

1. Howison, SD. Bibliography of free and moving boundary problems in Hele-Shaw and Stokes flow, 1998. Available athttp://people.maths.ox.ac.uk/howison/Hele-Shaw/
2. Polubarinova-Kochina, PYa. On the motion of the oil contour. Dokl. Akad. Nauk. SSSR, 47254-257.
3. Galin, LA. 1945. Unsteady filtration with a free surface. Dokl. Akad. Nauk USSR, 47: 246-249.
4. Saffman, PG and Taylor, GI. 1958. The penetration of a fluid into a porous medium of Hele-Shaw cell containing a more viscous fluid. Proc. R. Soc. Lond. Ser. A., 145: 312
5. Cummings, LJ, Howison, SD and King, JR. 1999. Two-dimensional Stokes and Hele-Shaw flows with free surfaces. Euro. J. Appl. Math., 10: 635-680.
6. Tanveer, S. 2000. Surprises in Viscous fingering. J. Fluid Mech., 409: 273-308.
7. Gustafsson, B and Vasilev, A. 2006. Conformal and Potential Analysis in Hele-Shaw Cell, Basel, Boston, Berlin: Birkhauser Verlag.
8. Richardson, S. 1972. Hele-Shaw flows with a free boundary produced by injection of fluid into a narrow channel. J. Fluid Mech., 56: 609
9. Howison, SD. 1985. Bubble growth in porous media and Hele-Shaw cells. Proc. R. Soc. Edinb. A, 102: 141-148.
10. Bensimon, D, Kadanoff, LP, Liang, S, Shraiman, BI and Tang, C. 1986. Viscous flow in two dimensions. Rev. Mod. Phys., 58: 977
11. Gustafsson, B. 1985. Application of variational inequalities to a moving boundary problem for Hele-Shaw flow. SIAM J. Math. Anal., 9: 279-300.
12. DiBenedetto, E and Friedman, A. 1984. The ill-posed Hele-Shaw model and the Stefan problem for supercooled water. Trans. AMS, 282: 183-204.
13. Andreucci, D, Caruso, G and DiBenedetto, E. 2004. Ill-posed Hele-Shaw flows. Int. Ser. Numer. Math., 147: 27-51.
14. Gustafsson, B. 1984. On a differential equation arising in a Hele-Shaw flow moving boundary problem. Ark. Mat., 22: 251-268.
15. Reissig, M. 1994. The existence and uniqueness of analytic solutions for a moving boundary problem for Hele-Shaw flows in the plane. Nonlinear Anal. TMA, 23: 565-574.
16. Reissig, M and Wolfersdorf, LV. 1992. A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane. Ark. Mat., 31: 101-116.
17. Kim, I. 2003. Uniqueness and existence results on the Hele-Shaw and the Stefan Problems. Arch. Ration. Mech. Anal., 168: 299-328.
18. Kim, I. 2005. A free boundary problem with curvature. Commun. PDE, 30: 121-138.
19. Choi, S, Jerison, DS and Kim, IC. 2009. Local regularization of the one-phase Hele-Shaw flow. Indiana Univ. Math. J., 58: 2765-2804.
20. Choi, S, Jerison, DS and Kim, IC. 2007. Regularity for the one-phase Hele-Shaw problem from a Lipschitz surface. Am. J. Math., 129: 527-582.
21. Duchon, J and Robert, R. 1984. Evolution d'une interface par capillarite' t difusion de volume. Ann. Inst. H. Poincare' Anal. Non Linéaire, 1: 361-378.
22. Constantin, P and Pugh, M. 1993. Global solutions for small data to Hele-Shaw problem. Nonlinearity, 6: 393
23. Tanveer, S. 1993. Evolution of Hele-Shaw interface for small surface tension. Proc. R. Soc. Lond. A, 343: 155-204.
24. Escher, J and Simonett, G. 1997. Classical solutions for Hele-Shaw models with surface tension. Adv. Differ. Eqns, 2: 619-642.
25. Prokert, G. 1998. Existence result for Hele-Shaw flow driven by surface tension. Euro. J. Appl. Math., 9: 195-221.
26. Gunther, M and Prokert, G. 2005. On a Hele-Shaw-Type domain evolution with convected surface energy density. SIAM J. Math. Anal., 37: 372-410.
27. Gunther, M and Prokert, G. 2005. On a Hele-Shaw-Type domain evolution with convected surface energy density: The third order problem. SIAM J. Math. Anal., 38: 1154-1185.
28. Bazaliy, BV and Friedman, A. 2005. The Hele-Shaw problem with surface tension in a half-plane. J. Differ. Eqns,: 439-469.
29. Chapman, SJ. 1999. On the role of Stokes lines in the selection of Saffman-Taylor fingers with small surface tension. Eur. J. Appl. Math., 10: 513-534.
30. Xie, X and Tanveer, S. 2003. Rigorous results in steady finger selection in viscous fingering. Arch. Ration. Mech. Anal., 166: 219-286.
31. Tanveer, S and Xie, X. 2003. Analyticity and nonexistence of classical steady Hele-Shaw Fingers. Commun. Pure Applied Math., LVI: 353-402.
32. Xie, X. 2008. Existence and selection of steady bubbles in a Hele-Shaw cell. Nonlinear Anal. Theory, Method Appl., 69: 2009-2048.
33. Howison, SD. 2000. A note on the two-phase Hele-Shaw problem. J. Fluid Mech., 409: 243-249.
34. Chen, X. 1993. The Hele-Shaw problem and area-preserving curve-shortening motions. Arch. Ration. Mech. Anal., 123: 117-151.
35. Siegel, M, Caflisch, RE and Howison, S. 2004. Global existence, singular solutions and Ill-posedness for the Muskat problem. Commun. Pure Appl. Math., LVII: 1374-1411.
36. Ambrose, D. 2004. Well-posedness of two-phase Hele-Shaw flow without surface tension. Euro. J. Appl. Math., 15: 597-607.
37. Entov, VM and Etingof, P. 2007. On a generalizes two-fluid Hele-Shaw flow. Euro. J. Appl. Math., 18: 103-128.
38. Ye, J and Tanveer, S. 2011. Global existence for a translating near-circular Hele-Shaw bubble with surface tension. SIAM. J. Math. Anal., 43: 457-506.
39. Howison, SD. 1988. Hele-Shaw free-boundary problems with suction. Q. J. Mech. Appl. Math., 41
40. Tian, F. 1995. On the breakdown of Hele-Shaw solutions with nonzero surface tension. J. Nonlinear Sci., 5: 479
41. Nie, Q and Tian, F. 2001. Singularity in Hele-Shaw flows driven by a multipole. SIAM J. Appl. Math., 62(2): 385
42. Nie, Q and Tian, F. 1998. Singularity in Hele-Shaw flows. SIAM J. Appl. Math., 58: 34
43. Cummings, LJ and King, JR. 2004. Hele-Shaw flow with a point sink: Generic solution break down. Euro. J. Appl. Math., 15: 1-37.
44. King, JR, Lacey, AA and Vazquez, JL. 1995. Persistence of corners in free boundaries in Hele-Shaw flow. Euro. J. Appl. Math., 6: 455-490.
45. Xie, X. Local smoothing effect and existence for the one-phase Hele-Shaw problem with surface tension, (submitted)
46. Constantin, P and Kadanoff, LP. 1991. Dynamics of a complex interface. Phys. D, 47: 450-460.
47. Kato, T. 1983. On the Cauchy problem for the (generalized) KDV equation. Stud. Appl. Math., 8: 93-128.
48. Constantin, P and Saut, JC. 1988. Local smoothing properties of dispersive equations. J. Am. Math. Soc., 1(2): 413-439.
49. Kenig, CE, Ponce, G and Vega, L. 1998. Smoothing effect and local existence theory for generalized nonlinear Schrodinger equations. Invent. Math., 134: 489-545.
50. Christianson, H, Hur, VM and Staffilani, G. 2009. Local smoothing effects for the water-wave problem with surface tension. C. R. Math. Acad. Sci. Paris, 347(3-4): 159-162.
51. Wu, S. 2006. Mathematical analysis of Vortex Sheets. Commun. Pure Appl. Math., LIX: 1-142.
52. Taylor, ME. 1996. Partial Differential Equations I,II,III, New York: Springer-Verlag.
53. Carrier, GF, Krook, M and Pearson, CE. 1966. Functions of a Complex Variable, New York: McGraw-Hill.