Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves

Pages 909-948 from Volume 175 (2012), Issue 2 by Ángel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, María López-Fernández

Abstract

The Muskat problem models the evolution of the interface between two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition may hold initially but break down in finite time. As a consequence of the method used, we prove the existence of water waves turning.

  • [Am] Go to document D. M. Ambrose, "Well-posedness of two-phase Hele-Shaw flow without surface tension," European J. Appl. Math., vol. 15, iss. 5, pp. 597-607, 2004.
    @article {Am, MRKEY = {2128613},
      AUTHOR = {Ambrose, David M.},
      TITLE = {Well-posedness of two-phase {H}ele-{S}haw flow without surface tension},
      JOURNAL = {European J. Appl. Math.},
      FJOURNAL = {European Journal of Applied Mathematics},
      VOLUME = {15},
      YEAR = {2004},
      NUMBER = {5},
      PAGES = {597--607},
      ISSN = {0956-7925},
      MRCLASS = {76D27 (35Q35 35R35)},
      MRNUMBER = {2128613},
      MRREVIEWER = {Xuming Xie},
      DOI = {10.1017/S0956792504005662},
      ZBLNUMBER = {1076.76027},
      }
  • [Bardos] C. Bardos and S. Benachour, "Domaine d’analycité des solutions de l’équation d’Euler dans un ouvert de $R^{n}$," Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 4, iss. 4, pp. 647-687, 1977.
    @article {Bardos, MRKEY = {0454413},
      AUTHOR = {Bardos, C. and Benachour, S.},
      TITLE = {Domaine d'analycité des solutions de l'équation d'{E}uler dans un ouvert de {$R\sp{n}$}},
      JOURNAL = {Ann. Scuola Norm. Sup. Pisa Cl. Sci.},
      VOLUME = {4},
      YEAR = {1977},
      NUMBER = {4},
      PAGES = {647--687},
      MRCLASS = {35Q99},
      MRNUMBER = {0454413},
      MRREVIEWER = {W. F. Ames},
      ZBLNUMBER = {0366.35022},
      }
  • [BL] C. Bardos and D. Lannes, Mathematics for 2d interfaces, 2010.
    @misc{BL,
      author = {Bardos, C. and Lannes, D.},
      TITLE={Mathematics for 2d interfaces},
      ARXIV={1005.5329},
      NOTE={to appear in {\it Panorama et Syntheses}},
      YEAR={2010},
      }
  • [hou] Go to document T. J. Beale, T. Y. Hou, and J. Lowengrub, "Convergence of a boundary integral method for water waves," SIAM J. Numer. Anal., vol. 33, iss. 5, pp. 1797-1843, 1996.
    @article {hou, MRKEY = {1411850},
      AUTHOR = {Beale, J. Thomas and Hou, Thomas Y. and Lowengrub, John},
      TITLE = {Convergence of a boundary integral method for water waves},
      JOURNAL = {SIAM J. Numer. Anal.},
      FJOURNAL = {SIAM Journal on Numerical Analysis},
      VOLUME = {33},
      YEAR = {1996},
      NUMBER = {5},
      PAGES = {1797--1843},
      ISSN = {0036-1429},
      CODEN = {SJNAAM},
      MRCLASS = {76B15 (35Q35 65M12 76B45 76M25)},
      MRNUMBER = {1411850},
      DOI = {10.1137/S0036142993245750},
      ZBLNUMBER = {0858.76046},
      }
  • [ADCPM3] A. Castro, D. Córdoba, C. L. Fefferman, and F. Gancedo, Breakdown of smoothness for the Muskat problem, 2011.
    @misc{ADCPM3,
      author = {Castro, Angel and C{ó}rdoba, Diego and Fefferman, Charles L. and Gancedo, Francisco},
      TITLE={Breakdown of smoothness for the {M}uskat problem},
      YEAR={2011},
      NOTE={preprint},
     }
  • [turning] Go to document A. Castro, D. Córdoba, C. L. Fefferman, F. Gancedo, and M. López-Fernández, "Turning waves and breakdown for incompressible flows," Proc. Natl. Acad. Sci. USA, vol. 108, iss. 12, pp. 4754-4759, 2011.
    @article {turning, MRKEY = {2792311},
      AUTHOR = {Castro, Angel and C{ó}rdoba, Diego and Fefferman, Charles L. and Gancedo, Francisco and L{ó}pez-Fern{á}ndez, Mar{\'ı}a},
      TITLE = {Turning waves and breakdown for incompressible flows},
      JOURNAL = {Proc. Natl. Acad. Sci. USA},
      FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
      VOLUME = {108},
      YEAR = {2011},
      NUMBER = {12},
      PAGES = {4754--4759},
      ISSN = {1091-6490},
      CODEN = {PNASFB},
      MRCLASS = {76B15 (35Q31 35Q35 76D27)},
      MRNUMBER = {2792311},
      DOI = {10.1073/pnas.1101518108},
      }
  • [CE] Go to document A. Constantin and J. Escher, "Wave breaking for nonlinear nonlocal shallow water equations," Acta Math., vol. 181, iss. 2, pp. 229-243, 1998.
    @article {CE, MRKEY = {1668586},
      AUTHOR = {Constantin, Adrian and Escher, Joachim},
      TITLE = {Wave breaking for nonlinear nonlocal shallow water equations},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {181},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {229--243},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {35Q35 (76B15)},
      MRNUMBER = {1668586},
      MRREVIEWER = {Alan Jeffrey},
      DOI = {10.1007/BF02392586},
      ZBLNUMBER = {0923.76025},
      }
  • [ccgs] P. Constantin, D. Córdoba, F. Gancedo, and R. M. Strain, On the global existence for the Muskat problem, 2012.
    @misc{ccgs,
      author={Constantin, P. and Córdoba, D. and Gancedo, F. and Strain, R.M. },
      TITLE={On the global existence for the {M}uskat problem},
      YEAR={2012},
      NOTE={to appear in {\it JEMS}},
     }
  • [Peter] Go to document P. Constantin and M. Pugh, "Global solutions for small data to the Hele-Shaw problem," Nonlinearity, vol. 6, iss. 3, pp. 393-415, 1993.
    @article {Peter, MRKEY = {1223740},
      AUTHOR = {Constantin, P. and Pugh, M.},
      TITLE = {Global solutions for small data to the {H}ele-{S}haw problem},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {6},
      YEAR = {1993},
      NUMBER = {3},
      PAGES = {393--415},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {35Q35 (30E99 76D99 76S05)},
      MRNUMBER = {1223740},
      MRREVIEWER = {M. Zerner},
      URL = {http://stacks.iop.org/0951-7715/6/393},
      ZBLNUMBER = {0808.35104},
      }
  • [AD] Go to document A. Córdoba and D. Córdoba, "A pointwise estimate for fractionary derivatives with applications to partial differential equations," Proc. Natl. Acad. Sci. USA, vol. 100, iss. 26, pp. 15316-15317, 2003.
    @article {AD, MRKEY = {2032097},
      AUTHOR = {C{ó}rdoba, Antonio and C{ó}rdoba, Diego},
      TITLE = {A pointwise estimate for fractionary derivatives with applications to partial differential equations},
      JOURNAL = {Proc. Natl. Acad. Sci. USA},
      FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
      VOLUME = {100},
      YEAR = {2003},
      NUMBER = {26},
      PAGES = {15316--15317},
      ISSN = {1091-6490},
      CODEN = {PNASFB},
      MRCLASS = {26A33 (35Q80)},
      MRNUMBER = {2032097},
      DOI = {10.1073/pnas.2036515100},
      ZBLNUMBER = {1111.26010},
      }
  • [ADY] Go to document A. Córdoba, D. Córdoba, and F. Gancedo, "Interface evolution: the Hele-Shaw and Muskat problems," Ann. of Math., vol. 173, iss. 1, pp. 477-542, 2011.
    @article {ADY, MRKEY = {2753607},
      AUTHOR = {C{ó}rdoba, Antonio and C{ó}rdoba, Diego and Gancedo, Francisco},
      TITLE = {Interface evolution: the {H}ele-{S}haw and {M}uskat problems},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {173},
      YEAR = {2011},
      NUMBER = {1},
      PAGES = {477--542},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {35R35 (35Q35 76D27 76S05 76T99)},
      MRNUMBER = {2753607},
      MRREVIEWER = {Alberto Valli},
      DOI = {10.4007/annals.2011.173.1.10},
      ZBLNUMBER = {05960663},
      }
  • [ADP2] Go to document A. Córdoba, D. Córdoba, and F. Gancedo, "Interface evolution: water waves in 2-D," Adv. Math., vol. 223, pp. 120-173, 2010.
    @article{ADP2,
      author = {C{ó}rdoba, Antonio and C{ó}rdoba, Diego and Gancedo, Francisco},
      TITLE = {Interface evolution: water waves in 2-{D}},
      JOURNAL={Adv. Math.},
      VOLUME={223},
      YEAR={2010},
      PAGES={120--173},
      MRNUMBER = {2563213},
      ZBLNUMBER = {1183.35276},
      DOI = {10.1016/j.aim.2009.07.016},
     }
  • [DY] Go to document D. Córdoba and F. Gancedo, "Contour dynamics of incompressible 3-D fluids in a porous medium with different densities," Comm. Math. Phys., vol. 273, iss. 2, pp. 445-471, 2007.
    @article {DY, MRKEY = {2318314},
      AUTHOR = {C{ó}rdoba, Diego and Gancedo, Francisco},
      TITLE = {Contour dynamics of incompressible 3-{D} fluids in a porous medium with different densities},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {273},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {445--471},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {76D27 (35B30 35Q30 76S05)},
      MRNUMBER = {2318314},
      MRREVIEWER = {Xuming Xie},
      DOI = {10.1007/s00220-007-0246-y},
      ZBLNUMBER = {1120.76064},
      }
  • [DP2] Go to document D. Córdoba and F. Gancedo, "A maximum principle for the Muskat problem for fluids with different densities," Comm. Math. Phys., vol. 286, iss. 2, pp. 681-696, 2009.
    @article {DP2, MRKEY = {2472040},
      AUTHOR = {C{ó}rdoba, Diego and Gancedo, Francisco},
      TITLE = {A maximum principle for the {M}uskat problem for fluids with different densities},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {286},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {681--696},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {35Q35 (35B50 35R35 76D27)},
      MRNUMBER = {2472040},
      MRREVIEWER = {Xiaoming Wang},
      DOI = {10.1007/s00220-008-0587-1},
      ZBLNUMBER = {1173.35637},
      }
  • [DP3] Go to document D. Córdoba and F. Gancedo, "Absence of squirt singularities for the multi-phase Muskat problem," Comm. Math. Phys., vol. 299, iss. 2, pp. 561-575, 2010.
    @article {DP3, MRKEY = {2679821},
      AUTHOR = {C{ó}rdoba, Diego and Gancedo, Francisco},
      TITLE = {Absence of squirt singularities for the multi-phase {M}uskat problem},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {299},
      YEAR = {2010},
      NUMBER = {2},
      PAGES = {561--575},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {35R37 (35A20 35B30 35Q35 35R35 76S05 76T30)},
      MRNUMBER = {2679821},
      MRREVIEWER = {Jinhae Park},
      DOI = {10.1007/s00220-010-1084-x},
      ZBLNUMBER = {1198.35176},
      }
  • [DPR] Go to document D. Córdoba, F. Gancedo, and R. Orive, "A note on interface dynamics for convection in porous media," Phys. D, vol. 237, iss. 10-12, pp. 1488-1497, 2008.
    @article {DPR, MRKEY = {2454601},
      AUTHOR = {C{ó}rdoba, Diego and Gancedo, Francisco and Orive, Rafael},
      TITLE = {A note on interface dynamics for convection in porous media},
      JOURNAL = {Phys. D},
      FJOURNAL = {Physica D. Nonlinear Phenomena},
      VOLUME = {237},
      YEAR = {2008},
      NUMBER = {10-12},
      PAGES = {1488--1497},
      ISSN = {0167-2789},
      CODEN = {PDNPDT},
      MRCLASS = {76S05 (35Q35 35R35)},
      MRNUMBER = {2454601},
      DOI = {10.1016/j.physd.2008.03.042},
      ZBLNUMBER = {1143.76574},
      }
  • [Drit] Go to document D. G. Dritschel, "Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows," Comput. Phys. Rep., vol. 10, pp. 77-146, 1989.
    @article{Drit,
      author={Dritschel, D. G.},
      TITLE={Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows},
      JOURNAL={Comput. Phys. Rep.},
      VOLUME={10},
      YEAR={1989},
      PAGES={77--146},
      DOI={10.1016/0167-7977(89)90004-X},
     }
  • [Esch1] J. Escher and G. Simonett, "Classical solutions for Hele-Shaw models with surface tension," Adv. Differential Equations, vol. 2, iss. 4, pp. 619-642, 1997.
    @article {Esch1, MRKEY = {1441859},
      AUTHOR = {Escher, Joachim and Simonett, Gieri},
      TITLE = {Classical solutions for {H}ele-{S}haw models with surface tension},
      JOURNAL = {Adv. Differential Equations},
      FJOURNAL = {Advances in Differential Equations},
      VOLUME = {2},
      YEAR = {1997},
      NUMBER = {4},
      PAGES = {619--642},
      ISSN = {1079-9389},
      MRCLASS = {35Q99 (35B40 35K55 76D45)},
      MRNUMBER = {1441859},
      MRREVIEWER = {Jing Xue Yin},
      ZBLNUMBER = {1023.35527},
      }
  • [Esch2] Go to document J. Escher and B. Matioc, "On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results," Z. Anal. Anwend., vol. 30, iss. 2, pp. 193-218, 2011.
    @article {Esch2, MRKEY = {2793001},
      AUTHOR = {Escher, Joachim and Matioc, Bogdan-Vasile},
      TITLE = {On the parabolicity of the {M}uskat problem: well-posedness, fingering, and stability results},
      JOURNAL = {Z. Anal. Anwend.},
      FJOURNAL = {Zeitschrift für Analysis und ihre Anwendungen. Journal of Analysis and its Applications},
      VOLUME = {30},
      YEAR = {2011},
      NUMBER = {2},
      PAGES = {193--218},
      ISSN = {0232-2064},
      MRCLASS = {35K55 (35A08 35B35 35R35 47J15 76D27)},
      MRNUMBER = {2793001},
      DOI = {10.4171/ZAA/1431},
      ZBLNUMBER = {1223.35199},
      }
  • [Esch3] J. Escher, A. V. Matioc, and B. Matioc, A generalised Rayleigh-Taylor condition for the Muskat problem, 2010.
    @misc{Esch3,
      author = {Escher, Joachim and Matioc, A. V. and Matioc, Bogdan-Vasile},
      TITLE={A generalised {R}ayleigh-{T}aylor condition for the {M}uskat problem},
      ARXIV={1005.2511},
      NOTE={preprint},
      YEAR={2010},
     }
  • [Hai] E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff problems, New York: Springer-Verlag, 1987, vol. 8.
    @book {Hai, MRKEY = {0868663},
      AUTHOR = {Hairer, E. and Nørsett, S. P. and Wanner, G.},
      TITLE = {Solving Ordinary Differential Equations. {\rm I}. {N}onstiff problems},
      SERIES = {Springer Ser. Comput. Math.},
      VOLUME = {8},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      PAGES = {xiv+480},
      ISBN = {3-540-17145-2},
      MRCLASS = {65-02 (34A40 65L99)},
      MRNUMBER = {0868663},
      MRREVIEWER = {W. F. Ames},
      ZBLNUMBER = {1185.65115},
      }
  • [H-S] Hele-Shaw, "On the motion of a viscous fluid between two parallel plates," Trans. Royal Inst. Nav. Archit., vol. 40,21, 1989.
    @article{H-S,
      author={Hele-Shaw},
      TITLE={On the motion of a viscous fluid between two parallel plates},
      JOURNAL={Trans. Royal Inst. Nav. Archit.},
      VOLUME={40,21},
      YEAR={1989},
     }
  • [Howison] Go to document S. D. Howison, "A note on the two-phase Hele-Shaw problem," J. Fluid Mech., vol. 409, pp. 243-249, 2000.
    @article {Howison, MRKEY = {1756390},
      AUTHOR = {Howison, Sam D.},
      TITLE = {A note on the two-phase {H}ele-{S}haw problem},
      JOURNAL = {J. Fluid Mech.},
      FJOURNAL = {Journal of Fluid Mechanics},
      VOLUME = {409},
      YEAR = {2000},
      PAGES = {243--249},
      ISSN = {0022-1120},
      CODEN = {JFLSA7},
      MRCLASS = {76D27 (76T99)},
      MRNUMBER = {1756390},
      MRREVIEWER = {Peter A. McCoy},
      DOI = {10.1017/S0022112099007740},
      ZBLNUMBER = {0962.76028},
      }
  • [HLS] Go to document T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, "Removing the stiffness from interfacial flows with surface tension," J. Comput. Phys., vol. 114, iss. 2, pp. 312-338, 1994.
    @article {HLS, MRKEY = {1294935},
      AUTHOR = {Hou, Thomas Y. and Lowengrub, John S. and Shelley, Michael J.},
      TITLE = {Removing the stiffness from interfacial flows with surface tension},
      JOURNAL = {J. Comput. Phys.},
      FJOURNAL = {Journal of Computational Physics},
      VOLUME = {114},
      YEAR = {1994},
      NUMBER = {2},
      PAGES = {312--338},
      ISSN = {0021-9991},
      CODEN = {JCTPAH},
      MRCLASS = {76M25 (76C05 76D45 76D99 76M99)},
      MRNUMBER = {1294935},
      DOI = {10.1006/jcph.1994.1170},
      ZBLNUMBER = {0810.76095},
      }
  • [Lannes] D. Lannes, A stability criterion for two-fluid interfaces and applications, 2010.
    @misc{Lannes,
      author={Lannes, D.},
      TITLE={A stability criterion for two-fluid interfaces and applications},
      NOTE={preprint},
      YEAR={2010},
      ARXIV={1005.4565},
      }
  • [Muskat] M. Muskat, "Two fluid systems in porous media. The encroachment of water into an oil sand," Physics, vol. 5, pp. 250-264, 1934.
    @article{Muskat,
      author={Muskat, M.},
      TITLE={Two fluid systems in porous media. {T}he encroachment of water into an oil sand},
      JOURNAL={Physics},
      VOLUME={5},
      YEAR={1934},
      PAGES={250--264},
      JFMNUMBER = {60.1388.01},
      }
  • [Nirenberg] L. Nirenberg, "An abstract form of the nonlinear Cauchy-Kowalewski theorem," J. Differential Geometry, vol. 6, pp. 561-576, 1972.
    @article {Nirenberg, MRKEY = {0322321},
      AUTHOR = {Nirenberg, L.},
      TITLE = {An abstract form of the nonlinear {C}auchy-{K}owalewski theorem},
      JOURNAL = {J. Differential Geometry},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {6},
      YEAR = {1972},
      PAGES = {561--576},
      ISSN = {0022-040X},
      MRCLASS = {35A10 (35R20)},
      MRNUMBER = {0322321},
      MRREVIEWER = {W. Watzlawek},
      ZBLNUMBER = {0257.35001},
      }
  • [Nishida] Go to document T. Nishida, "A note on a theorem of Nirenberg," J. Differential Geom., vol. 12, iss. 4, pp. 629-633 (1978), 1977.
    @article {Nishida, MRKEY = {0512931},
      AUTHOR = {Nishida, Takaaki},
      TITLE = {A note on a theorem of {N}irenberg},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {12},
      YEAR = {1977},
      NUMBER = {4},
      PAGES = {629--633 (1978)},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {58E05 (35A10)},
      MRNUMBER = {0512931},
      MRREVIEWER = {J. Chrastina},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214434231},
      ZBLNUMBER = {0368.35007},
      }
  • [Otto] Go to document F. Otto, "Viscous fingering: an optimal bound on the growth rate of the mixing zone," SIAM J. Appl. Math., vol. 57, iss. 4, pp. 982-990, 1997.
    @article {Otto, MRKEY = {1462048},
      AUTHOR = {Otto, Felix},
      TITLE = {Viscous fingering: an optimal bound on the growth rate of the mixing zone},
      JOURNAL = {SIAM J. Appl. Math.},
      FJOURNAL = {SIAM Journal on Applied Mathematics},
      VOLUME = {57},
      YEAR = {1997},
      NUMBER = {4},
      PAGES = {982--990},
      ISSN = {0036-1399},
      CODEN = {SMJMAP},
      MRCLASS = {76D99 (76E99 76S05 76T05)},
      MRNUMBER = {1462048},
      MRREVIEWER = {Sam D. Howison},
      DOI = {10.1137/S003613999529438X},
      ZBLNUMBER = {0901.76082},
      }
  • [Ray] Go to document L. Rayleigh, "On the instability of jets," Proc. Lond. Math. Soc., vol. 10, 1879.
    @article{Ray,
      author={Rayleigh, Lord},
      TITLE={On the instability of jets},
      JOURNAL={Proc. Lond. Math. Soc.},
      VOLUME={10},
      PAGE={413},
      YEAR={1879},
      DOI = {10.1112/plms/s1-10.1.4},
     }
  • [S-T] P. G. Saffman and G. Taylor, "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid," Proc. Roy. Soc. London. Ser. A, vol. 245, pp. 312-329. (2 plates), 1958.
    @article {S-T, MRKEY = {0097227},
      AUTHOR = {Saffman, P. G. and Taylor, Geoffrey},
      TITLE = {The penetration of a fluid into a porous medium or {H}ele-{S}haw cell containing a more viscous liquid},
      JOURNAL = {Proc. Roy. Soc. London. Ser. A},
      FJOURNAL = {Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences},
      VOLUME = {245},
      YEAR = {1958},
      PAGES = {312--329. (2 plates)},
      ISSN = {0962-8444},
      MRCLASS = {76.00},
      MRNUMBER = {0097227},
      MRREVIEWER = {K. Bhagwandin},
      ZBLNUMBER = {0086.41603},
      }
  • [SCH] Go to document M. Siegel, R. E. Caflisch, and S. Howison, "Global existence, singular solutions, and ill-posedness for the Muskat problem," Comm. Pure Appl. Math., vol. 57, iss. 10, pp. 1374-1411, 2004.
    @article {SCH, MRKEY = {2070208},
      AUTHOR = {Siegel, Michael and Caflisch, Russel E. and Howison, Sam},
      TITLE = {Global existence, singular solutions, and ill-posedness for the {M}uskat problem},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {57},
      YEAR = {2004},
      NUMBER = {10},
      PAGES = {1374--1411},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {35Q35 (35R35 76D03 76D27 76S05)},
      MRNUMBER = {2070208},
      MRREVIEWER = {Alberto Valli},
      DOI = {10.1002/cpa.20040},
      ZBLNUMBER = {1062.35089},
      }
  • [SSBF] Go to document C. Sulem, P. -L. Sulem, C. Bardos, and U. Frisch, "Finite time analyticity for the two- and three-dimensional Kelvin-Helmholtz instability," Comm. Math. Phys., vol. 80, iss. 4, pp. 485-516, 1981.
    @article {SSBF, MRKEY = {0628507},
      AUTHOR = {Sulem, C. and Sulem, P.-L. and Bardos, C. and Frisch, U.},
      TITLE = {Finite time analyticity for the two- and three-dimensional {K}elvin-{H}elmholtz instability},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {80},
      YEAR = {1981},
      NUMBER = {4},
      PAGES = {485--516},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {76C05 (35Q20 58E07)},
      MRNUMBER = {0628507},
      MRREVIEWER = {Charles J. Amick},
      ZBLNUMBER = {0476.76032},
      DOI = {10.1007/BF01941659},
      }
  • [Yi] F. Yi, "Local classical solution of Muskat free boundary problem," J. Partial Differential Equations, vol. 9, iss. 1, pp. 84-96, 1996.
    @article {Yi, MRKEY = {1384001},
      AUTHOR = {Yi, Fahuai},
      TITLE = {Local classical solution of {M}uskat free boundary problem},
      JOURNAL = {J. Partial Differential Equations},
      FJOURNAL = {Journal of Partial Differential Equations},
      VOLUME = {9},
      YEAR = {1996},
      NUMBER = {1},
      PAGES = {84--96},
      ISSN = {1000-940X},
      CODEN = {JPDEEX},
      MRCLASS = {35R35 (76S05)},
      MRNUMBER = {1384001},
      ZBLNUMBER = {0847.35152},
      }
  • [Yi2] Go to document F. Yi, "Global classical solution of Muskat free boundary problem," J. Math. Anal. Appl., vol. 288, iss. 2, pp. 442-461, 2003.
    @article {Yi2, MRKEY = {2019452},
      AUTHOR = {Yi, Fahuai},
      TITLE = {Global classical solution of {M}uskat free boundary problem},
      JOURNAL = {J. Math. Anal. Appl.},
      FJOURNAL = {Journal of Mathematical Analysis and Applications},
      VOLUME = {288},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {442--461},
      ISSN = {0022-247X},
      CODEN = {JMANAK},
      MRCLASS = {76S05 (35R35 76D03)},
      MRNUMBER = {2019452},
      MRREVIEWER = {Li Shang Jiang},
      DOI = {10.1016/j.jmaa.2003.09.003},
      ZBLNUMBER = {1038.35083},
      }

Keywords
, , , , ,
DOI
https://doi.org/10.4007/annals.2012.175.2.9
MR
2993754
Zbl
06025004
Milestones
Received: 8 February 2011
Revised: 30 May 2011
Accepted: 7 May 2011
Published online: 1 March 2012

Authors

Ángel Castro

Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain

Diego Córdoba

Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Cientíificas, 28049 Madrid, Spain

Charles Fefferman

Department of Mathematics, Princeton University, Fine Hall - Washington Rd., Princeton, NJ 08544

Francisco Gancedo

Department of Mathematical Analysis, University of Seville, C/ Tarfia s/n, Campus Reina Mercedes, 41012 Sevilla, Spain

María López-Fernández

Institut für Mathematik, Universität Zürich, Zürich, Switzerland