|
Well-posedness of two-phase Darcy flow in 3D
Author(s):
David
M.
Ambrose
Journal:
Quart. Appl. Math.
65
(2007),
189-203.
MSC (2000):
Primary 35Q35
Posted:
February 12, 2007
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
We prove the well-posedness, locally in time, of the motion of two fluids flowing according to Darcy's law, separated by a sharp interface in the absence of surface tension. We first reformulate the problem using favorable variables and coordinates. This results in a quasilinear parabolic system. Energy estimates are performed, and these estimates imply that the motion is well-posed for a short time with data in a Sobolev space, as long as a condition is satisfied. This condition essentially says that the more viscous fluid must displace the less viscous fluid. It should be true that small solutions exist for all time; however, this question is not addressed in the present work.
References:
-
- 1.
- Ambrose, D.M. Well-posedness of vortex sheets with surface tension. SIAM J. Math. Anal. 35 (2003), 211-244. MR 2001473 (2005g:76006)
- 2.
- Ambrose, D.M. Well-posedness of two-phase Hele-Shaw flow without surface tension. European J. Appl. Math. 15 (2004) 597-607. MR 2128613 (2005m:76065)
- 3.
- Ambrose, D.M.; Masmoudi, N. Well-posedness of 3D vortex sheets with surface tension. (2006) Submitted.
- 4.
- Ambrose, D.M.; Masmoudi, N. The zero surface tension limit of three-dimensional water waves. (2006) In preparation.
- 5.
- Baker, G.; Meiron, D.; Orszag, S. Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123 (1982), 477-501. MR 0687014 (84a:76002)
- 6.
- Caflisch, R.E.; Li, X.-F. Lagrangian theory for 3D vortex sheets with axial or helical symmetry. Transport Theory Statist. Phys. 21 (1992), 559-578. MR 1194461 (93h:76014)
- 7.
- Chen, X.; Friedman, A. A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth. SIAM J. Math. Anal. 35 (2003) 974-986. MR 2049029 (2005f:35333)
- 8.
- Cordoba, D.; Gancedo, F. Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. (2006) Preprint.
- 9.
- Escher, J.; Simonett, G. Classical solutions of multidimensional Hele-Shaw models. SIAM J. Math. Anal. 28 (1997) 1028-1047. MR 1466667 (98i:35213)
- 10.
- Escher, J.; Simonett, G. A center manifold analysis for the Mullins-Sekerka model. J. Differential Equations 143 (1998), 267-292. MR 1607952 (98m:35228)
- 11.
- Friedman, A. Time dependent free boundary problems. SIAM Rev. 21 (1979) 213-221. MR 0524512 (81g:76097)
- 12.
- Friedman, A. Free boundary problems with surface tension conditions. Nonlinear Analysis 63 (2005) 666-671. MR 2188139 (2006f:35301)
- 13.
- Friedman, A.; Reitich, F. Nonlinear stability of a quasi-static Stefan problem with surface tension: A continuation approach. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 341-403. MR 1895715 (2003e:35326)
- 14.
- Hou, T.; Lowengrub, J.; Shelley, M. Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114 (1994), 312-338. MR 1294935 (95e:76069)
- 15.
- Hou, T.; Lowengrub, J.; Shelley, M. The long-time motion of vortex sheets with surface tension. Phys. Fluids 9 (1997), 1933-1954. MR 1455083 (98d:76033)
- 16.
- Majda, A.; Bertozzi, A. Vorticity and Incompressible Flow. Cambridge University Press, Cambridge, UK, 2002. MR 1867882 (2003a:76002)
- 17.
- Saffman, P.G. Vortex dynamics. Cambridge University Press, Cambridge, UK, 1992. MR 1217252 (94c:76015)
- 18.
- Siegel, M.; Caflisch, R.; Howison, S. Global existence, singular solutions, and ill-posedness for the Muskat problem. Comm. Pure Appl. Math. 57 (2004), 1374-1411. MR 2070208
- 19.
- Wu, S. Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc. 12 (1999), 445-495. MR 1641609 (2001m:76019)
Similar Articles:
Retrieve articles in Quarterly of Applied Mathematics
with MSC
(2000):
35Q35
Retrieve articles in all Journals with MSC
(2000):
35Q35
Additional Information:
David
M.
Ambrose
Affiliation:
Department of Mathematical Sciences, Clemson University, Martin Hall, Clemson, South Carolina 29634
PII:
S0033-569X-07-01055-3
Received by editor(s):
November 16, 2006
Posted:
February 12, 2007
Additional Notes:
The author was supported by NSF grant DMS-0610898
Copyright of article:
Copyright
2007,
Brown University
The copyright for this article reverts to public domain after 28 years from publication.
|
