Abstract
We perform energy estimates for a sharp-interface model of two-dimensional, two-phase Darcy flow with surface tension. A proof of well-posedness of the initial value problem follows from these estimates. In general, the time of existence of these solutions will go to zero as the surface tension parameter vanishes. We then make two additional estimates, in the case that a stability condition is satisfied by the initial data: we make an additional energy estimate which is uniform in the surface tension parameter, and we make an estimate for the difference of two solutions with different values of the surface tension parameter. These additional estimates allow the zero surface tension limit to be taken, showing that solutions of the initial value problem in the absence of surface tension are the limit of solutions of the initial value problem with surface tension as surface tension vanishes.
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Ambrose, D.M.: Well-posedness of vortex sheets with surface tension. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)–Duke University (2002)
Ambrose, D.M.: Well-posedness of vortex sheets with surface tension. SIAM J. Math. Anal. 35(1), 211–244 (2003) (electronic)
Ambrose D.M.: Well-posedness of two-phase Hele-Shaw flow without surface tension. Eur. J. Appl. Math. 15(5), 597–607 (2004)
Ambrose D.M.: Well-posedness of two-phase Darcy flow in 3D. Quart. Appl. Math. 65(1), 189–203 (2007)
Ambrose D.M., Bona J.L., Nicholls D.P.: Well-posedness of a model for water waves with viscosity. Discret. Cont. Dyn. Syst. Ser. B 17(4), 1113–1137 (2012)
Ambrose D.M., Masmoudi N.: The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58(10), 1287–1315 (2005)
Ambrose D.M., Masmoudi N.: Well-posedness of 3D vortex sheets with surface tension. Commun. Math. Sci. 5(2), 391–430 (2007)
Ambrose D.M., Masmoudi N.: The zero surface tension limit of three-dimensional water waves. Indiana Univ. Math. J. 58(2), 479–521 (2009)
Ambrose D.M., Siegel M.: A non-stiff boundary integral method for 3d porous media flow with surface tension. Math. Comput. Simul. 82(6), 968–983 (2012)
Ambrose, D.M., Siegel, M., Tlupova, S.: A small-scale decomposition for 3D boundary integral computations with surface tension. J. Comput. Phys. 247, 168–191 (2013)
Ambrose, D.M., Wright, J.D.: Dispersion vs. anti-diffusion: well-posedness in variable coefficient and quasilinear equations of KdV-type. Indiana Univ. Math. J. (2012, in press)
Bailly J.-H.: Local existence of classical solutions to first-order parabolic equations describing free boundaries. Nonlinear Anal. 32(5), 583–599 (1998)
Baker G.R., Meiron D.I., Orszag S.A.: Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501 (1982)
Batchelor, G.K.: An Introduction to Fluid Dynamics, paperback edition. Cambridge Mathematical Librarym Cambridge University Press, Cambridge (1999)
Beale J.T., Hou T.Y., Lowengrub J.S.: Growth rates for the linearized motion of fluid interfaces away from equilibrium. Commun. Pure Appl. Math. 46(9), 1269–1301 (1993)
Bear, J.: Dynamics of Fluids in Porous Media, Dover Books on Physics and Chemistry. Dover Publications, New York (1988)
Ceniceros H.D., Hou T.Y.: The singular perturbation of surface tension in Hele-Shaw flows. J. Fluid Mech. 409, 251–272 (2000)
Ceniceros, H.D., Hou, T.Y.: Numerical study of interfacial problems with small surface tension. In: First International Congress of Chinese Mathematicians (Beijing, 1998), vol. 20 of AMS/IP Studies in Advanced Mathemetics, pp 63–92. American Mathemetical Society, Providence, RI (2001)
Christianson H., Hur V.M., Staffilani G.: Strichartz estimates for the water-wave problem with surface tension. Commun. Partial Diff. Equ. 35(12), 2195–2252 (2010)
Constantin P., Córdoba D., Gancedo F., Strain R.: On the global existence for the Muskat problem. J. Eur. Math. Soc. 15, 201–227 (2013)
Constantin P., Pugh M.: Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6(3), 393–415 (1993)
Córdoba A., Córdoba D., Gancedo F.: The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces. Proc. Natl. Acad. Sci. USA 106(27), 10955–10959 (2009)
Córdoba A., Córdoba D., Gancedo F.: Interface evolution: water waves in 2-D. Adv. Math. 223(1), 120–173 (2010)
Córdoba, A., Córdoba, D., Gancedo, F.: Porous media: the Muskat problem in 3D. Anal. PDE. 6(2), 447–497 (2013)
Córdoba A., Córdoba D., Gancedo F.: Interface evolution: the Hele-Shaw and Muskat problems. Ann. Math. (2) 1(173), 477–542 (2011)
Coutand, D., Hole, J., Shkoller, S.: Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit (2012)
Duchon J., Robert R.: Évolution d’une interface par capillarité et diffusion de volume. I. Existence locale en temps. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(5), 361–378 (1984)
Düll W.-P.: Validity of the Korteweg-de Vries approximation for the two-dimensional water wave problem in the arc length formulation. Comm. Pure Appl. Math. 65(3), 381–429 (2012)
Escher J., Simonett G.: On Hele-Shaw models with surface tension. Math. Res. Lett. 3(4), 467–474 (1996)
Escher J., Simonett G.: Classical solutions for Hele-Shaw models with surface tension. Adv. Diff. Equ. 2(4), 619–642 (1997)
Escher J., Simonett G.: Classical solutions of multidimensional Hele-Shaw models. SIAM J. Math. Anal. 28(5), 1028–1047 (1997)
Gillow, K.A., Howison, S.D.: A bibliography of free and moving boundary problems for Hele-Shaw and Stokes flow. http://people.maths.ox.ac.uk/howison/Hele-Shaw/ (2012). Accessed 20 August 2012
Guo Y., Hallstrom C., Spirn D.: Dynamics near unstable, interfacial fluids. Commun. Math. Phys. 270(3), 635–689 (2007)
Hadzic, M., Shkoller, S.: Well-posedness for the classical Stefan problem and the zero surface tension limit (2011)
Hartman, P.: Ordinary Differential Equations, volume 38 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e:34002)], With a foreword by Peter Bates (2002)
Helson, H.: Harmonic Analysis. Addison-Wesley Publishing Company Advanced Book Program, Reading (1983)
Hou T.Y., Lowengrub J.S., Shelley M.J.: The long-time motion of vortex sheets with surface tension. Phys. Fluids 9(7), 1933–1954 (1997)
Hou T.Y., Lowengrub J.S., Shelley M.J.: Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114(2), 312–338 (1994)
Hou T.Y., Zhang P.: Convergence of a boundary integral method for 3-D water waves. Discret. Cont. Dyn. Syst. Ser. B 2(1), 1–34 (2002)
Howison S.D.: Cusp development in Hele-Shaw flow with a free surface. SIAM J. Appl. Math. 46(1), 20–26 (1986)
Kim I.C.: Uniqueness and existence results on the Hele-Shaw and the Stefan problems. Arch. Ration. Mech. Anal. 168(4), 299–328 (2003)
Majda A.J., Bertozzi A.L.: Vorticity and Incompressible Flow, volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
Nie Q.: The nonlinear evolution of vortex sheets with surface tension in axisymmetric flows. J. Comput. Phys. 174(1), 438–459 (2001)
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, 312–329. (2 plates) (1958)
Saffman P.G.: Vortex Dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York (1992)
Siegel M., Caflisch R.E., Howison S.: Global existence, singular solutions, and ill-posedness for the Muskat problem. Commun. Pure Appl. Math. 57(10), 1374–1411 (2004)
Siegel M., Tanveer S.: Singular perturbation of smoothly evolving hele-shaw solutions. Phys. Rev. Lett. 76, 419–422 (1996)
Siegel M., Tanveer S., Dai W.-S.: Singular effects of surface tension in evolving Hele-Shaw flows. J. Fluid Mech. 323, 201–236 (1996)
Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130(1), 39–72 (1997)
Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12(2), 445–495 (1999)
Xie X.: Local smoothing effect and existence for the one-phase Hele-Shaw problem with zero surface tension. Complex Var. Elliptic Equ. 57(2-4), 351–368 (2012)
Ye J., Tanveer S.: Global existence for a translating near-circular Hele-Shaw bubble with surface tension. SIAM J. Math. Anal. 43(1), 457–506 (2011)
Ye J., Tanveer S.: Global solutions for a two-phase Hele-Shaw bubble for a near-circular initial shape. Complex Var. Elliptic Equ. 57(1), 23–61 (2012)
Yi F.: Local classical solution of Muskat free boundary problem. J. Partial Differ. Equ. 9(1), 84–96 (1996)
Yi F.: Global classical solution of Muskat free boundary problem. J. Math. Anal. Appl. 288(2), 442–461 (2003)
Zeidler, E.: Nonlinear functional analysis and its applications. I. Springer-Verlag, New York, Fixed-point theorems, Translated from the German by Peter R. Wadsack (1986)
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Communicated by A. Constantin
The author gratefully acknowledges support from the National Science Foundation through grants DMS-1008387 and DMS-1016267.
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Ambrose, D.M. The Zero Surface Tension Limit of Two-Dimensional Interfacial Darcy Flow. J. Math. Fluid Mech. 16, 105–143 (2014). https://doi.org/10.1007/s00021-013-0146-1
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DOI: https://doi.org/10.1007/s00021-013-0146-1