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A survey for the Muskat problem and a new estimate

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Abstract

This paper shows a summary of mathematical results about the Muskat problem. The main concern is well-posed scenarios which include the possible formation of singularities in finite time or existence of solutions for all time. These questions are important in mathematical physics but also have a strong mathematical interest. Stressing some recent results of the author, we also give a new estimate for the problem in the last section. Initial data with \(L^2\) decay and slope less than one provide weak solutions which satisfy a parabolic inequality as in the linear regime.

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References

  1. Ambrose, D.M.: Well-posedness of two-phase Hele-Shaw flow without surface tension. Eur. J. Appl. Math. 15(5), 597–607 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrose, D.M.: Well-posedness of two-phase Darcy flow in 3D. Q. Appl. Math. 65(1), 189–203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrose, D.M.: The zero surface tension limit of two-dimensional interfacial Darcy flow. J. Math. Fluid Mech. 16(1), 105143 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bardos, C., Lannes, D.: Mathematics for 2d interfaces. Singularities in mechanics: formation, propagation and microscopic description, 37–67, Panor. Synthèses, 38, Soc. Math. France, Paris (2012)

  5. Berselli, L.C., Córdoba, D., Granero-Belinchón, R.: Local solvability and turning for the inhomogeneous Muskat problem. Interface Free Bound. 16(2), 175–213 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York (1988)

    MATH  Google Scholar 

  7. Beck, T., Sosoe, P., Wong, P.: Duchon-Robert solutions for the Rayleigh-Taylor and Muskat problems. J. Differ. Equ. 256(1), 206–222 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bertozzi, A.L., Constantin, P.: Global regularity for vortex patches. Commun. Math. Phys. 152(1), 1993 (1928)

    MathSciNet  MATH  Google Scholar 

  9. Castro, A., Córdoba, D., Gancedo, F.: Some recent results on the Muskat problem. Journees Equations aux Derivees Partielles 5, 14 (2010)

    Google Scholar 

  10. Castro, A., Córdoba, D., Fefferman, C., Gancedo, F.: Breakdown of smoothness for the Muskat problem. Arch. Ration. Mech. Anal. 208(3), 805–909 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castro, A., Córdoba, D., Fefferman, C., Gancedo, F.: Splash singularities for the one-phase Muskat problem in stable regimes. arXiv:1311.7653 (2013)

  12. Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., López-Fernández, M.: Turning waves and breakdown for incompressible flows. Proc. Natl. Acad. Sci. USA 108(12), 4754–4759 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., López-Fernández, M.: Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. 175(2), 909–948 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chemin, J.Y.: Two-Dimensional Euler System and the Vortex Patches Problem. Handbook of Mathematical Fluid Dynamics, vol. 3. North-Holland, Amsterdam (2004)

    MATH  Google Scholar 

  15. Constantin, P., Córdoba, D., Gancedo, F., Rodriguez-Piazza, L., Strain, R.M.: On the Muskat problem: global in time results in 2D and 3D. arXiv:1310.0953 (2014)

  16. Constantin, P., Córdoba, D., Gancedo, F., Strain, R.M.: On the global existence for the Muskat problem. J. Eur. Math. Soc. 15, 201–227 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Constantin, P., Gancedo, F., Shvydkoy, R., Vicol, V.: Global regularity for 2D Muskat equations with finite slope. arXiv:1507.0138 (2015)

  18. Constantin, P., Pugh, M.: Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6(3), 393–415 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Córdoba, A., Córdoba, D., Gancedo, F.: Interface evolution: the Hele-Shaw and Muskat problems. Ann. Math. 173(1), 477–542 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Córdoba, A., Córdoba, D., Gancedo, F.: Porous media: the Muskat problem in three dimensions. Anal. PDE 6(2), 447–497 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Córdoba, D., Gancedo, F.: Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Commun. Math. Phys. 273(2), 445–471 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Córdoba, D., Gancedo, F.: A maximum principle for the Muskat problem for fluids with different densities. Commun. Math. Phys. 286(2), 681–696 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Córdoba, D., Gancedo, F.: Contour dynamics for 2D active scalars. Newsletter of the European Mathematical Society, vol. 71 (2009)

  24. Córdoba, D., Gancedo, F.: Absence of squirt singularities for the multi-phase Muskat problem. Commun. Math. Phys. 299(2), 561–575 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Córdoba, D., Gancedo, F., Orive, R.: A note on the interface dynamics for convection in porous media. Phys. D 237, 1488–1497 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Córdoba, D., Gómez-Serrano, J., Zlatos, A.: A note in stability shifting for the Muskat problem. Phil. Trans. R. Soc. A 373, 2050 (2015)

    Article  MATH  Google Scholar 

  27. Córdoba, D., Granero, R., Orive, R.: The confined Muskat problem: differences with the deep water regime. Commun. Math. Sci. 12(3), 423–455 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Córdoba, D., Pernas-Castaño, T.: Non-splat singularity for the one-phase Muskat problem. To appear in Trans. Amer. Math, Soc (2015)

  29. Chen, X.: The hele-shaw problem and area-preserving curve-shortening motions. Arch. Ration. Mech. Anal. 123(2), 117–151 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Cheng, C.H., Granero-Belinchón, R., Shkoller, S.: Well-posedness of the Muskat problem with \(H^{2}\) initial data. Adv. Math. 286, 32–104 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Darcy, H.: Les Fontaines Publiques de la Ville de Dijon. Dalmont, Paris (1856)

    Google Scholar 

  32. Dombre, T., Pumir, A., Siggia, E.: On the interface dynamics for convection in porous media. Phys. D 57, 311–329 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. Duchon, J., Robert, R.: Evolution dune interface par capillarité et diffusion de volume. I. Existence locale en temps. Ann. Inst. H. Poincaré Anal. Non Linéaire 5(1), 361–378 (1984)

    MATH  Google Scholar 

  34. Escher, J., Matioc, B.V.: On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results. Z. Anal. Anwend. 30(2), 193–218 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Escher, J., Simonett, G.: Classical solutions for Hele-Shaw models with surface tension. Adv. Differ. Equ. 2, 619–642 (1997)

    MathSciNet  MATH  Google Scholar 

  36. Gancedo, F., Strain, R.M.: Absence of splash singularities for SQG sharp fronts and the Muskat problem. Proc. Natl. Acad. Sci. 111(2), 635–639 (2014)

    Article  MathSciNet  Google Scholar 

  37. 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/. Accessed August 19, 2015

  38. Gómez-Serrano, J., Granero-Belinchón, R.: On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof. Nonlinearity 27(6), 1471–1498 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Granero-Belinchón, R.: Global existence for the confined Muskat problem. SIAM J. Math. Anal. 46(2), 1651–1680 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Guo, Y., Hallstrom, C., Spirn, D.: Dynamics near unstable, interfacial fluids. Commun. Math. Phys. 270(3), 635–689 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Guo, Y., Tice, I.: Decay of viscous surface waves without surface tension. arXiv:1011.5179 (2011)

  42. Hou, T.Y., Lowengrub, J.S., Shelley, M.J.: Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114, 312–338 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  43. Knüpfer, H., Masmoudi, N.: Darcys flow with prescribed contact angle: well-posedness and lubrication approximation. Arch. Ration. Mech. Anal. 218, 1–58 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Lannes, D.: The Water Waves Problem: Mathematical Analysis and Asymptotics. Mathematical Surveys and Monographs, vol. 188. American Mathematical Society (2013)

  45. Muskat, M.: Two Fluid systems in porous media. The encroachment of water into an oil sand. Physics 5, 250–264 (1934)

    Article  MATH  Google Scholar 

  46. Otto, F.: Evolution of microstructure in unstable porous media flow: a relaxational approach. Commun. Pure Appl. Math. 52, 873–915 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  47. Saffman, P.G., Taylor, G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245, 312–329 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  48. Siegel, M., Caflisch, R., Howison, S.: Global existence, singular solutions, and Ill-posedness for the Muskat problem. Commun. Pure Appl. Math. 57, 1374–1411 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  49. Székelyhidi Jr., L.: Relaxation of the incompressible porous media equation. Ann. Sci. Éc. Norm. Supér. 45(3), 491–509 (2012)

    MathSciNet  MATH  Google Scholar 

  50. Wu, S.: Recent progress in mathematical analysis of vortex sheets. In: Proceedings of the International Congress of Mathematicians, vol. III, pp. 233–242. Higher Ed. Press, Beijing (2002)

  51. Yi, F.: Local classical solution of Muskat free boundary problem. J. Partial Differ. Equ. 9, 84–96 (1996)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

FG were partially supported by the Grant MTM2014-59488-P, the Ramón y Cajal program RyC-2010-07094 and by the Grant P12-FQM-2466 from Junta de Andalucía (Spain).

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  1. Departamento de Análisis Matemático and IMUS, Universidad de Sevilla, C/ Tarfia s/n, Campus Reina Mercedes, 41012, Seville, Spain

    Francisco Gancedo

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  1. Francisco Gancedo
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Gancedo, F. A survey for the Muskat problem and a new estimate. SeMA 74, 21–35 (2017). https://doi.org/10.1007/s40324-016-0078-9

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