Growth phenomena
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Review
Issue
Math. Model. Nat. Phenom.
Volume 15, 2020
Growth phenomena
Article Number 7
Number of page(s) 23
DOI https://doi.org/10.1051/mmnp/2019021
Published online 17 February 2020
  1. D. Ambrose, The zero surface tension limit of two-dimensional interfacial Darcy flow. J. Math. Fluid Mech. 16 (2014) 105–143. [CrossRef] [Google Scholar]
  2. D.M. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension. Eur. J. Appl. Math. 15 (2004) 597–607. [Google Scholar]
  3. C.-H. Arthur Cheng, R. Granero-Belinchón and S. Shkoller, Well-posedness of the Muskat problem with H2 initial data. Adv. Math. 286 (2016) 32–104. [CrossRef] [Google Scholar]
  4. C.-H. Arthur Cheng, R. Granero-Belinchón, S. Shkoller and J. Wilkening, Rigorous asymptotic models of water waves. Water Waves (2019) 1–60. [Google Scholar]
  5. H. Bae and R. Granero-Belinchón, Global existence for some transport equations with nonlocal velocity. Adv. Math. 269 (2015) 197–219. [CrossRef] [Google Scholar]
  6. H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations. In Vol. 343. Springer Science & Business Media, Switzerland (2011). [Google Scholar]
  7. J. Bear, Dynamics of Fluids in Porous Media. Dover Publications, USA (1988). [Google Scholar]
  8. L.C Berselli, Vanishing viscosity limit and long-time behavior for 2d quasi-geostrophic equations. Indiana U. Math. J. 51 (2002) 905–930. [CrossRef] [Google Scholar]
  9. L.C. Berselli, D. Córdoba and R. Granero-Belinchón, Local solvability and turning for the inhomogeneous Muskat problem. Interfaces Free Bound. 16 (2014) 175–213. [CrossRef] [Google Scholar]
  10. O.V. Besov, Investigation of a class of function spaces in connection with imbedding and extension theorems. Trudy Matematicheskogo Instituta imeni VA Steklova 60 (1961) 42–81. [Google Scholar]
  11. S.E. Buckley and M.C. Leverett, Mechanism of fluid displacement in sands. Trans. Aime 146 (1941) 107–116. [Google Scholar]
  12. S. Cameron, Global well-posedness for the 2d Muskat problem with slope less than 1. Anal. PDE 12 (2019) 997–1022. [CrossRef] [Google Scholar]
  13. A. Castro, D. Cordoba, C. Fefferman, F. Gancedo and M. Lopez-Fernandez, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. 175 (2012) 909–948. [Google Scholar]
  14. A. Castro, D. Cordoba, C. Fefferman and F. Gancedo, Breakdown of smoothness for the Muskat problem. Arch. Ration. Mech. Anal. 208 (2013) 805–909. [Google Scholar]
  15. A. Castro, D. Córdoba and D. Faraco, Mixing solutions for the Muskat problem. Preprint arXiv:1605.04822 (2016). [Google Scholar]
  16. A. Castro, D. Córdoba, C. Fefferman and F. Gancedo, Splash singularities for the one-phase Muskat problem in stable regimes. Arch. Ration. Mech. Anal. 222 (2016) 213–243. [Google Scholar]
  17. Á. Castro, D. Faraco and F. Mengual, Degraded mixing solutions for the Muskat problem. Preprint arXiv:1805.12050 (2018). [Google Scholar]
  18. M. Cerminara and A. Fasano, Modelling the dynamics of a geothermal reservoir fed by gravity driven flow through overstanding saturated rocks. J. Volcanol. Geotherm. Res. 233 (2012) 37–54. [CrossRef] [Google Scholar]
  19. D. Chae, P. Constantin, D. Córdoba, F. Gancedo and J. Wu, Generalized surface quasi-geostrophic equations with singular velocities. Commun. Pure Appl. Math. 65 (2012) 1037–1066. [Google Scholar]
  20. H.A. Chang-Lara and N. Guillen, From the free boundary condition for hele-shaw to a fractional parabolic equation. Preprint arXiv:1605.07591 (2016). [Google Scholar]
  21. X. Chen, The hele-shaw problem and area-preserving curve-shortening motions. Arch. Ration. Mech. Anal. 123 (1993) 117–151. [Google Scholar]
  22. P. Constantin and M. Pugh, Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6 (1993) 393–415. [Google Scholar]
  23. P. Constantin, A.J. Majda and E. Tabak, Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7 (1994) 1495. [Google Scholar]
  24. P. Constantin, A.J. Majda and E.G. Tabak, Singular front formation in a model for quasi-geostrophic flow. Phys. Fluids 6 (1994) 9. [CrossRef] [Google Scholar]
  25. P. Constantin, D. Cordoba, F. Gancedo and R.M. Strain. On the global existence for the Muskat problem. J. Eur. Math. Soc. 15 (2013) 201–227. [CrossRef] [Google Scholar]
  26. P. Constantin, D. Cordoba, F. Gancedo, L. Rodríguez-Piazza and R.M. Strain, On the Muskat problem: global in time results in2d and 3d. Am. J. Math. 138 (2016) 6. [CrossRef] [Google Scholar]
  27. P. Constantin, F. Gancedo, R. Shvydkoy and V. Vicol, Global regularity for 2d Muskat equations with finite slope. Ann. Inst. Henri Poincaré (C) Non Lin. Anal. 34 (2016) 1041–1074. [CrossRef] [Google Scholar]
  28. A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249 (2004) 511–528. [CrossRef] [Google Scholar]
  29. D. Córdoba and F. Gancedo, Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Commun. Math. Phys. 273 (2007) 445–471. [CrossRef] [Google Scholar]
  30. D. Córdoba and F. Gancedo, A maximum principle for the Muskat problem for fluids with different densities. Commun. Math. Phys. 286 (2009) 681–696. [CrossRef] [Google Scholar]
  31. D. Córdoba and F. Gancedo, Absence of squirt singularities for the multi-phase Muskat problem. Commun. Math. Phys. 299 (2010) 561–575. [CrossRef] [Google Scholar]
  32. D. Cordoba and T. Pernas-Castaño, Non-splat singularity for the one-phase Muskat problem. Trans. Am. Math. Soc. 369 (2017) 711–754. [Google Scholar]
  33. D. Cordoba and O. Lazar, Global well-posedness for the 2d stable Muskat problem in H3∕2. Preprint arXiv:1803.07528 (2018). [Google Scholar]
  34. A. Cordoba, D. Cordoba and F. Gancedo, The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces. Proc. Natl. Acad. Sci. 106 (2009) 10955–10959. [CrossRef] [Google Scholar]
  35. A. Cordoba, D. Córdoba and F. Gancedo, Interface evolution: the Hele-Shaw and Muskat problems. Ann. Math. 173 (2011) 477–542. [Google Scholar]
  36. A. Córdoba, D. Córdoba and F. Gancedo, Porous media: the Muskat problem in three dimensions. Anal. PDE 6 (2013) 447–497. [CrossRef] [Google Scholar]
  37. D. Córdoba, R. Granero-Belinchón and R. Orive, On the confined Muskat problem: differences with the deep water regime. Commun. Math. Sci. 12 (2014) 423–455. [Google Scholar]
  38. D. Córdoba, J. Gómez-Serrano and A. Zlatoš, A note on stability shifting for the Muskat problem. Phil. Trans. R. Soc. A 373 (2015) 20140278. [Google Scholar]
  39. D. Córdoba, J. Gómez-Serrano and A. Zlatoš, A note on stability shifting for the Muskat problem ii: from stable to unstable and back to stable. Anal. PDE 10 (2017) 367–378. [CrossRef] [Google Scholar]
  40. D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20 (2006) 829–930. [CrossRef] [Google Scholar]
  41. D. Coutand and S. Shkoller, On the impossibility of finite-time splash singularities for vortex sheets. Arch. Ration. Mech. Anal. 221 (2016) 987–1033. [Google Scholar]
  42. H. Darcy, Les fontaines publiques de la ville de Dijon: exposition et application. Victor Dalmont, Paris (1856). [Google Scholar]
  43. L. Dawson, H. McGahagan and G. Ponce, On the decay properties of solutions to a class of Schrödinger equations. Proc. Am. Math. Soc. 136 (2008) 2081–2090. [Google Scholar]
  44. C.M. Elliott and J.R. Ockendon, Vol. 59 of Weak and Variational Methods for Moving Boundary Problems. Pitman Publishing, London (1982). [Google Scholar]
  45. J. Escherand B.-V. Matioc, On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results. Z. Anal. Anwend. 30 (2011) 193–218. [Google Scholar]
  46. J. Escher and G. Simonett, Classical solutions for Hele–Shaw models with surface tension. Adv. Differ. Equ. 2 (1997) 619–642. [Google Scholar]
  47. J. Escherand G. Simonett, A center manifold analysis for the Mullins–Sekerka model. J. Differ. Equ. 143 (1998) 267–292. [Google Scholar]
  48. J. Escher, A.-V. Matioc and B.-V. Matioc, A generalized Rayleigh-Taylor condition for the Muskat problem. Nonlinearity 25 (2012) 73–92. [Google Scholar]
  49. J. Escher, B.-V. Matioc and C. Walker, The domain of parabolicity for the Muskat problem. Indiana Univ. Math. J. 67 (2018) 679–737. [CrossRef] [Google Scholar]
  50. C. Fefferman, A.D. Ionescu, V. Lie, On the absence of splash singularities in the case of two-fluid interfaces. Duke Math. J. 165 (2016) 417–462. [CrossRef] [Google Scholar]
  51. C. Förster and L. Székelyhidi Jr. Piecewise constant subsolutions for the Muskat problem. Commun. Math. Phys. 363 (2018) 1051–1080. [CrossRef] [Google Scholar]
  52. S. Friedlander and V. Vicol, Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics. Ann. Inst. Henri Poincaré (C) Non Lin. Anal. 28 (2011) 283–301. [CrossRef] [Google Scholar]
  53. S. Friedlander and V. Vicol, On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations. Nonlinearity 24 (2011) 3019. [Google Scholar]
  54. S. Friedlander, W. Rusin and V. Vicol, On the supercritically diffusive magnetogeostrophic equations. Nonlinearity 25 (2012) 3071. [Google Scholar]
  55. S. Friedlander, W. Rusin, V. Vicol and A.I. Nazarov, The magneto-geostrophic equations: a survey. Proceedings of the St. Petersburg Mathematical Society, Volume XV: Advances in Mathematical Analysis of Partial Differential Equations. American Mathematical Society, Providence, USA (2014). [Google Scholar]
  56. A. Friedman, Free boundary problems arising in tumor models. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 (2004). [Google Scholar]
  57. F. Gancedo, Existence for the α-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217 (2008) 2569–2598. [CrossRef] [Google Scholar]
  58. F. Gancedo and R.M. Strain, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem. Proc. Natl. Acad. Sci. 111 (2014) 635–639. [CrossRef] [Google Scholar]
  59. F. Gancedo, E. Garcia-Juarez, N. Patel and R.M. Strain, On the muskat problem with viscosity jump: Global in time results. Adv. Math. 345 (2019) 552–597. [CrossRef] [Google Scholar]
  60. J. Gómez-Serrano and R. Granero-Belinchón, On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof. Nonlinearity 27 (2014) 1471–1498. [Google Scholar]
  61. R. Granero-Belinchón. Global existence for the confined Muskat problem. SIAM J. Math. Anal. 46 (2014) 1651–1680. [CrossRef] [Google Scholar]
  62. R. Granero-Belinchón and S. Shkoller, Well-posedness and decay to equilibrium for the muskat problem with discontinuouspermeability (2016). Trans. Amer. Math. Soc. 372 (2019) 2255–2286. [CrossRef] [Google Scholar]
  63. R. Granero-Belinchón and S. Scrobogna, Asymptotic models for free boundary flow in porous media, Phys. D: Nonlinear Phenom. 392 (2019) 1–16. [CrossRef] [Google Scholar]
  64. R. Granero-Belinchón, The inhomogeneous Muskat problem. Ph.D thesis, University of Cantabria, Spain (2013). [Google Scholar]
  65. S.M. Hassanizadeh and W.G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Resour. 13 (1990) 169–186. [Google Scholar]
  66. H.S. Hele-Shaw, The flow of water. Nature 58 (1898) 34–36. [Google Scholar]
  67. H.S. Hele-Shaw, On the motion of a viscous fluid between two parallel plates. Trans. Roy. Inst. Nav. Archit. 40 (1898) 218. [Google Scholar]
  68. U. Hornung, Vol. 6 of Homogenization and Porous Media. Springer Verlag, New York (1997). [CrossRef] [Google Scholar]
  69. A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167 (2007) 445–453. [Google Scholar]
  70. O. Lazar, Global existence for the critical dissipative surface quasi-geostrophic equation. Commun. Math. Phys. 322 (2013) 73–93. [CrossRef] [Google Scholar]
  71. P.G. Lemarié-Rieusset, The Navier–Stokes problem in the 21st century. Chapman and Hall/, Boca Raton (2016). [CrossRef] [Google Scholar]
  72. A.J. Majda and E.G. Tabak, A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Phys. D: Nonlin. Phenom. 98 (1996) 515–522. [CrossRef] [Google Scholar]
  73. A.J. Majda and A.L. Bertozzi, Vorticity and incompressible flow. In Vol. 27. Cambridge University Press, Cambridge (2002). [Google Scholar]
  74. B.-V. Matioc, The muskat problem in 2d: equivalence of formulations, well-posedness, and regularity results. Anal. PDE 12 (2018) 281–332. [CrossRef] [Google Scholar]
  75. B.-V. Matioc, Viscous displacement in porous media: the Muskat problem in 2D. Trans. Am. Math. Soc. 370 (2018) 7511–7556. [Google Scholar]
  76. B.-V. Matioc, Well-posedness and stability results for some periodic Muskat problems. Preprint arXiv:1804.10403 (2018). [Google Scholar]
  77. A.-V. Matioc and B.-V. Matioc, Well-posedness and stability results for a quasilinear periodic muskat problem. J. Differ. Equ. 266 (2019) 5500–5531. [Google Scholar]
  78. H.K. Moffatt and D.E. Loper, The magnetostrophic rise of a buoyant parcel in the earth’s core. Geophys. J. Int. 117 (1994) 394–402. [Google Scholar]
  79. M. Muskat, Two fluid systems in porous media. the encroachment of water into an oil sand. Physics 5 (1934) 250–264. [Google Scholar]
  80. M. Muskat, The flow of fluids through porous media. J. Appl. Phys. 8 (1937) 274–282. [Google Scholar]
  81. M. Muskat, The flow of homogeneous fluids through porous media. Soil Sci. 46 (1938) 169. [Google Scholar]
  82. D.A. Nield and A. Bejan, Convection in Porous Media. Springer Verlag, New York (2006). [Google Scholar]
  83. F. Otto, Evolution of microstructure in unstable porous media flow: a relaxational approach. Commun. Pure Appl. Math. 52 (1999) 873–915. [Google Scholar]
  84. F. Otto, Evolution of microstructure: an example, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin (2001) 501–522. [CrossRef] [Google Scholar]
  85. N. Patel and R.M. Strain, Large time decay estimates for the Muskat equation. Commun. Part. Differ. Equ. 42 (2017) 977–999. [CrossRef] [Google Scholar]
  86. T. Pernas-Castaño, Local-existence for the inhomogeneous Muskat problem. Nonlinearity 30 (2017) 2063. [Google Scholar]
  87. J. Pruessand G. Simonett, On the Muskat flow. Evol. Equ. Control Theory 5 (2016) 631–645. [CrossRef] [Google Scholar]
  88. L. Rayleigh, On the instability of jets. Proc. London Math. Soc. s1-10 (1878) 4–13. [CrossRef] [MathSciNet] [Google Scholar]
  89. J. Rodrigo, On the evolution of sharp fronts for the quasi-geostrophic equation. Comm. Pure Appl. Math. 58 (2005) 821–866. [CrossRef] [Google Scholar]
  90. T. Runst and W. Sickel, Vol. 3 of Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Walter de Gruyter, Germany (1996). [CrossRef] [Google Scholar]
  91. 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 245 (1958) 312–329. [CrossRef] [MathSciNet] [Google Scholar]
  92. M. Siegel, R.E. Caflisch and S. Howison, Global existence, singular solutions, and ill-posedness for the Muskat problem. Commun. Pure Appl. Math. 57 (2004) 1374–1411. [Google Scholar]
  93. L. Székelyhidi Jr. Relaxation of the incompressible porous media equation. Ann. Sci. Éc. Norm. Supér. 45 (2012) 491–509. [CrossRef] [Google Scholar]
  94. L. Tartar, Incompressible fluid flow in a porous medium-convergence of the homogenization process, in Nonhomogeneous media and vibration theory, edited by E. Sánchez-Palencia. Springer-Verlag Berlin (1980). [Google Scholar]
  95. A.R. Thornton, A.J. van der Horn, E. Gagarina, W. Zweers, D. van der Meer and O. Bokhove, Hele-shaw beach creation by breaking waves: a mathematics-inspired experiment. Environ. Fluid Mech. 14 (2014) 1123–1145. [CrossRef] [Google Scholar]
  96. S. Tofts, On the existence of solutions to the Muskat problem with surface tension. J. Math. Fluid Mech. 19 (2017) 581–611. [CrossRef] [Google Scholar]

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