Voir la notice de l'article provenant de la source EDP Sciences
Rafael Granero-Belinchón 1 ; Omar Lazar 2
@article{10_1051_mmnp_2019021,
author = {Rafael Granero-Belinch\'on and Omar Lazar},
title = {Growth in the {Muskat} problem},
journal = {Mathematical modelling of natural phenomena},
eid = {7},
publisher = {mathdoc},
volume = {15},
year = {2020},
doi = {10.1051/mmnp/2019021},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2019021/}
}
Rafael Granero-Belinchón; Omar Lazar. Growth in the Muskat problem. Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 7. doi: 10.1051/mmnp/2019021
[1] The zero surface tension limit of two-dimensional interfacial Darcy flow J. Math. Fluid Mech. 2014 105 143
[2] Well-posedness of two-phase Hele-Shaw flow without surface tension Eur. J. Appl. Math. 2004 597 607
[3] , , Well-posedness of the Muskat problem with H2 initial data Adv. Math 2016 32 104
[4] , , , Rigorous asymptotic models of water waves Water Waves 2019 1 60
[5] , Global existence for some transport equations with nonlocal velocity Adv. Math 2015 197 219
[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).
[7] J. Bear, Dynamics of Fluids in Porous Media. Dover Publications, USA (1988).
[8] Vanishing viscosity limit and long-time behavior for 2d quasi-geostrophic equations Indiana U. Math. J 2002 905 930
[9] , , Local solvability and turning for the inhomogeneous Muskat problem Interfaces Free Bound 2014 175 213
[10] Investigation of a class of function spaces in connection with imbedding and extension theorems Trudy Matematicheskogo Instituta imeni VA Steklova 1961 42 81
[11] , Mechanism of fluid displacement in sands Trans. Aime 1941 107 116
[12] Global well-posedness for the 2d Muskat problem with slope less than 1 Anal. PDE 2019 997 1022
[13] , , , , Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves Ann. Math 2012 909 948
[14] , , , Breakdown of smoothness for the Muskat problem Arch. Ration. Mech. Anal. 2013 805 909
[15] A. Castro, D. Córdoba and D. Faraco, Mixing solutions for the Muskat problem. Preprint arXiv:1605.04822 (2016).
[16] , , , Splash singularities for the one-phase Muskat problem in stable regimes Arch. Ration. Mech. Anal 2016 213 243
[17] Á. Castro, D. Faraco and F. Mengual, Degraded mixing solutions for the Muskat problem. Preprint arXiv:1805.12050 (2018).
[18] , Modelling the dynamics of a geothermal reservoir fed by gravity driven flow through overstanding saturated rocks J. Volcanol. Geotherm. Res 2012 37 54
[19] , , , , Generalized surface quasi-geostrophic equations with singular velocities Commun. Pure Appl. Math 2012 1037 1066
[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).
[21] The hele-shaw problem and area-preserving curve-shortening motions Arch. Ration. Mech. Anal 1993 117 151
[22] , Global solutions for small data to the Hele-Shaw problem Nonlinearity 1993 393 415
[23] , , Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar Nonlinearity 1994 1495
[24] , , Singular front formation in a model for quasi-geostrophic flow Phys. Fluids 1994 9
[25] , , , On the global existence for the Muskat problem J. Eur. Math. Soc 2013 201 227
[26] , , , , On the Muskat problem: global in time results in2d and 3d Am. J. Math 2016 6
[27] , , , Global regularity for 2d Muskat equations with finite slope Ann. Inst. Henri Poincaré (C) Non Lin. Anal 2016 1041 1074
[28] , A maximum principle applied to quasi-geostrophic equations Commun. Math. Phys 2004 511 528
[29] , Contour dynamics of incompressible 3-D fluids in a porous medium with different densities Commun. Math. Phys 2007 445 471
[30] , A maximum principle for the Muskat problem for fluids with different densities Commun. Math. Phys 2009 681 696
[31] , Absence of squirt singularities for the multi-phase Muskat problem Commun. Math. Phys 2010 561 575
[32] , Non-splat singularity for the one-phase Muskat problem Trans. Am. Math. Soc 2017 711 754
[33] D. Cordoba and O. Lazar, Global well-posedness for the 2d stable Muskat problem in H3∕2. Preprint arXiv:1803.07528 (2018).
[34] , , The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces Proc. Natl. Acad. Sci 2009 10955 10959
[35] , , Interface evolution: the Hele-Shaw and Muskat problems Ann. Math 2011 477 542
[36] , , Porous media: the Muskat problem in three dimensions Anal. PDE 2013 447 497
[37] , , On the confined Muskat problem: differences with the deep water regime Commun. Math. Sci 2014 423 455
[38] , , A note on stability shifting for the Muskat problem Phil. Trans. R. Soc. A 2015 20140278
[39] , , A note on stability shifting for the Muskat problem ii: from stable to unstable and back to stable Anal. PDE 2017 367 378
[40] , Well-posedness of the free-surface incompressible Euler equations with or without surface tension J. Am. Math. Soc 2006 829 930
[41] , On the impossibility of finite-time splash singularities for vortex sheets Arch. Ration. Mech. Anal 2016 987 1033
[42] H. Darcy, Les fontaines publiques de la ville de Dijon: exposition et application. Victor Dalmont, Paris (1856).
[43] , , On the decay properties of solutions to a class of Schrödinger equations Proc. Am. Math. Soc 2008 2081 2090
[44] C.M. Elliott and J.R. Ockendon, Vol. 59 of Weak and Variational Methods for Moving Boundary Problems. Pitman Publishing, London (1982).
[45] , On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results Z. Anal. Anwend 2011 193 218
[46] , Classical solutions for Hele–Shaw models with surface tension Adv. Differ. Equ 1997 619 642
[47] , A center manifold analysis for the Mullins–Sekerka model J. Differ. Equ 1998 267 292
[48] , , A generalized Rayleigh-Taylor condition for the Muskat problem Nonlinearity 2012 73 92
[49] , , The domain of parabolicity for the Muskat problem Indiana Univ. Math. J 2018 679 737
[50] , , On the absence of splash singularities in the case of two-fluid interfaces Duke Math. J 2016 417 462
[51] , Piecewise constant subsolutions for the Muskat problem Commun. Math. Phys 2018 1051 1080
[52] , Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics Ann. Inst. Henri Poincaré (C) Non Lin. Anal 2011 283 301
[53] , On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations Nonlinearity 2011 3019
[54] , , On the supercritically diffusive magnetogeostrophic equations Nonlinearity 2012 3071
[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).
[56] Free boundary problems arising in tumor models Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 2004
[57] Existence for the α-patch model and the QG sharp front in Sobolev spaces Adv. Math 2008 2569 2598
[58] , Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem Proc. Natl. Acad. Sci 2014 635 639
[59] , , , On the muskat problem with viscosity jump: Global in time results Adv. Math 2019 552 597
[60] , On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof Nonlinearity 2014 1471 1498
[61] Global existence for the confined Muskat problem SIAM J. Math. Anal 2014 1651 1680
[62] , Well-posedness and decay to equilibrium for the muskat problem with discontinuouspermeability (2016) Trans. Amer. Math. Soc 2019 2255 2286
[63] , Asymptotic models for free boundary flow in porous media Phys. D: Nonlinear Phenom 2019 1 16
[64] R. Granero-Belinchón, The inhomogeneous Muskat problem. Ph.D thesis, University of Cantabria, Spain (2013).
[65] , Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries Adv. Water Resour 1990 169 186
[66] The flow of water Nature 1898 34 36
[67] On the motion of a viscous fluid between two parallel plates Trans. Roy. Inst. Nav. Archit 1898 218
[68] U. Hornung, Vol. 6 of Homogenization and Porous Media. Springer Verlag, New York (1997).
[69] , , Global well-posedness for the critical 2D dissipative quasi-geostrophic equation Invent. Math 2007 445 453
[70] Global existence for the critical dissipative surface quasi-geostrophic equation Commun. Math. Phys 2013 73 93
[71] P.G. Lemarié-Rieusset, The Navier–Stokes problem in the 21st century. Chapman and Hall/, Boca Raton (2016).
[72] , A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow Phys. D: Nonlin. Phenom 1996 515 522
[73] A.J. Majda and A.L. Bertozzi, Vorticity and incompressible flow. In Vol. 27. Cambridge University Press, Cambridge (2002).
[74] The muskat problem in 2d: equivalence of formulations, well-posedness, and regularity results Anal. PDE 2018 281 332
[75] Viscous displacement in porous media: the Muskat problem in 2D Trans. Am. Math. Soc 2018 7511 7556
[76] B.-V. Matioc, Well-posedness and stability results for some periodic Muskat problems. Preprint arXiv:1804.10403 (2018).
[77] , Well-posedness and stability results for a quasilinear periodic muskat problem J. Differ. Equ 2019 5500 5531
[78] , The magnetostrophic rise of a buoyant parcel in the earth’s core Geophys. J. Int 1994 394 402
[79] Two fluid systems in porous media. the encroachment of water into an oil sand Physics 1934 250 264
[80] The flow of fluids through porous media J. Appl. Phys 1937 274 282
[81] The flow of homogeneous fluids through porous media Soil Sci 1938 169
[82] D.A. Nield and A. Bejan, Convection in Porous Media. Springer Verlag, New York (2006).
[83] Evolution of microstructure in unstable porous media flow: a relaxational approach Commun. Pure Appl. Math 1999 873 915
[84] F. Otto, Evolution of microstructure: an example, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin (2001) 501–522.
[85] , Large time decay estimates for the Muskat equation Commun. Part. Differ. Equ 2017 977 999
[86] Local-existence for the inhomogeneous Muskat problem Nonlinearity 2017 2063
[87] , On the Muskat flow Evol. Equ. Control Theory 2016 631 645
[88] On the instability of jets Proc. London Math. Soc 1878 4 13
[89] On the evolution of sharp fronts for the quasi-geostrophic equation Comm. Pure Appl. Math 2005 821 866
[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).
[91] , The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid Proc. Roy. Soc. London Ser. A 1958 312 329
[92] , , Global existence, singular solutions, and ill-posedness for the Muskat problem Commun. Pure Appl. Math. 2004 1374 1411
[93] Relaxation of the incompressible porous media equation Ann. Sci. Éc. Norm. Supér. 2012 491 509
[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).
[95] , , , , , Hele-shaw beach creation by breaking waves: a mathematics-inspired experiment Environ. Fluid Mech 2014 1123 1145
[96] On the existence of solutions to the Muskat problem with surface tension J. Math. Fluid Mech 2017 581 611
Cité par Sources :