Geodesic
Growth in the Muskat problem
Rafael Granero-Belinchón1 ; Omar Lazar2
1 Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n, Santander, Spain.
2 Departamento de Análisis Matemático & IMUS, Universidad de Sevilla, C/ Tarifa s/n, Campus Reina Mercedes, 41012 Sevilla, Spain.
Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 7.

Voir la notice de l'article provenant de la source EDP Sciences

We review some recent results on the Muskat problem modelling multiphase flow in porous media. Furthermore, we prove a new regularity criteria in terms of some norms of the initial data in critical spaces (Ẇ1,∞ and Ḣ3∕2).
DOI : 10.1051/mmnp/2019021

Rafael Granero-Belinchón 1 ; Omar Lazar 2

1 Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n, Santander, Spain.
2 Departamento de Análisis Matemático & IMUS, Universidad de Sevilla, C/ Tarifa s/n, Campus Reina Mercedes, 41012 Sevilla, Spain. 
@article{MMNP_2020_15_a31,
     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 = {https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/2019021/}
}
TY  - JOUR
AU  - Rafael Granero-Belinchón
AU  - Omar Lazar
TI  - Growth in the Muskat problem
JO  - Mathematical modelling of natural phenomena
PY  - 2020
VL  - 15
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/2019021/
DO  - 10.1051/mmnp/2019021
LA  - en
ID  - MMNP_2020_15_a31
ER  - 
%0 Journal Article
%A Rafael Granero-Belinchón
%A Omar Lazar
%T Growth in the Muskat problem
%J Mathematical modelling of natural phenomena
%D 2020
%V 15
%I mathdoc
%U https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/2019021/
%R 10.1051/mmnp/2019021
%G en
%F MMNP_2020_15_a31
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. https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/2019021/

[1] D. Ambrose The zero surface tension limit of two-dimensional interfacial Darcy flow J. Math. Fluid Mech. 2014 105 143

[2] D.M. Ambrose Well-posedness of two-phase Hele-Shaw flow without surface tension Eur. J. Appl. Math. 2004 597 607

[3] C.-H. Arthur Cheng, R. Granero-Belinchón, S. Shkoller Well-posedness of the Muskat problem with H2 initial data Adv. Math 2016 32 104

[4] C.-H. Arthur Cheng, R. Granero-Belinchón, S. Shkoller, J. Wilkening Rigorous asymptotic models of water waves Water Waves 2019 1 60

[5] H. Bae, R. Granero-Belinchón 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] L.C Berselli Vanishing viscosity limit and long-time behavior for 2d quasi-geostrophic equations Indiana U. Math. J 2002 905 930

[9] L.C. Berselli, D. Córdoba, R. Granero-Belinchón Local solvability and turning for the inhomogeneous Muskat problem Interfaces Free Bound 2014 175 213

[10] O.V. Besov Investigation of a class of function spaces in connection with imbedding and extension theorems Trudy Matematicheskogo Instituta imeni VA Steklova 1961 42 81

[11] S.E. Buckley, M.C. Leverett Mechanism of fluid displacement in sands Trans. Aime 1941 107 116

[12] S. Cameron Global well-posedness for the 2d Muskat problem with slope less than 1 Anal. PDE 2019 997 1022

[13] A. Castro, D. Cordoba, C. Fefferman, F. Gancedo, M. Lopez-Fernandez Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves Ann. Math 2012 909 948

[14] A. Castro, D. Cordoba, C. Fefferman, F. Gancedo 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] A. Castro, D. Córdoba, C. Fefferman, F. Gancedo 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] M. Cerminara, A. Fasano Modelling the dynamics of a geothermal reservoir fed by gravity driven flow through overstanding saturated rocks J. Volcanol. Geotherm. Res 2012 37 54

[19] D. Chae, P. Constantin, D. Córdoba, F. Gancedo, J. Wu 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] X. Chen The hele-shaw problem and area-preserving curve-shortening motions Arch. Ration. Mech. Anal 1993 117 151

[22] P. Constantin, M. Pugh Global solutions for small data to the Hele-Shaw problem Nonlinearity 1993 393 415

[23] P. Constantin, A.J. Majda, E. Tabak Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar Nonlinearity 1994 1495

[24] P. Constantin, A.J. Majda, E.G. Tabak Singular front formation in a model for quasi-geostrophic flow Phys. Fluids 1994 9

[25] P. Constantin, D. Cordoba, F. Gancedo, R.M. Strain. On the global existence for the Muskat problem J. Eur. Math. Soc 2013 201 227

[26] P. Constantin, D. Cordoba, F. Gancedo, L. Rodríguez-Piazza, R.M. Strain On the Muskat problem: global in time results in2d and 3d Am. J. Math 2016 6

[27] P. Constantin, F. Gancedo, R. Shvydkoy, V. Vicol Global regularity for 2d Muskat equations with finite slope Ann. Inst. Henri Poincaré (C) Non Lin. Anal 2016 1041 1074

[28] A. Córdoba, D. Córdoba A maximum principle applied to quasi-geostrophic equations Commun. Math. Phys 2004 511 528

[29] D. Córdoba, F. Gancedo Contour dynamics of incompressible 3-D fluids in a porous medium with different densities Commun. Math. Phys 2007 445 471

[30] D. Córdoba, F. Gancedo A maximum principle for the Muskat problem for fluids with different densities Commun. Math. Phys 2009 681 696

[31] D. Córdoba, F. Gancedo Absence of squirt singularities for the multi-phase Muskat problem Commun. Math. Phys 2010 561 575

[32] D. Cordoba, T. Pernas-Castaño 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] A. Cordoba, D. Cordoba, F. Gancedo The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces Proc. Natl. Acad. Sci 2009 10955 10959

[35] A. Cordoba, D. Córdoba, F. Gancedo Interface evolution: the Hele-Shaw and Muskat problems Ann. Math 2011 477 542

[36] A. Córdoba, D. Córdoba, F. Gancedo Porous media: the Muskat problem in three dimensions Anal. PDE 2013 447 497

[37] D. Córdoba, R. Granero-Belinchón, R. Orive On the confined Muskat problem: differences with the deep water regime Commun. Math. Sci 2014 423 455

[38] D. Córdoba, J. Gómez-Serrano, A. Zlatoš A note on stability shifting for the Muskat problem Phil. Trans. R. Soc. A 2015 20140278

[39] D. Córdoba, J. Gómez-Serrano, A. Zlatoš A note on stability shifting for the Muskat problem ii: from stable to unstable and back to stable Anal. PDE 2017 367 378

[40] D. Coutand, S. Shkoller Well-posedness of the free-surface incompressible Euler equations with or without surface tension J. Am. Math. Soc 2006 829 930

[41] D. Coutand, S. Shkoller 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] L. Dawson, H. Mcgahagan, G. Ponce 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] J. Escher, B.-V. Matioc On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results Z. Anal. Anwend 2011 193 218

[46] J. Escher, G. Simonett Classical solutions for Hele–Shaw models with surface tension Adv. Differ. Equ 1997 619 642

[47] J. Escher, G. Simonett A center manifold analysis for the Mullins–Sekerka model J. Differ. Equ 1998 267 292

[48] J. Escher, A.-V. Matioc, B.-V. Matioc A generalized Rayleigh-Taylor condition for the Muskat problem Nonlinearity 2012 73 92

[49] J. Escher, B.-V. Matioc, C. Walker The domain of parabolicity for the Muskat problem Indiana Univ. Math. J 2018 679 737

[50] C. Fefferman, A.D. Ionescu, V. Lie On the absence of splash singularities in the case of two-fluid interfaces Duke Math. J 2016 417 462

[51] C. Förster, L. Székelyhidi Piecewise constant subsolutions for the Muskat problem Commun. Math. Phys 2018 1051 1080

[52] S. Friedlander, V. Vicol 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] S. Friedlander, V. Vicol On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations Nonlinearity 2011 3019

[54] S. Friedlander, W. Rusin, V. Vicol 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] A. Friedman Free boundary problems arising in tumor models Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 2004

[57] F. Gancedo Existence for the α-patch model and the QG sharp front in Sobolev spaces Adv. Math 2008 2569 2598

[58] F. Gancedo, R.M. Strain Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem Proc. Natl. Acad. Sci 2014 635 639

[59] F. Gancedo, E. Garcia-Juarez, N. Patel, R.M. Strain On the muskat problem with viscosity jump: Global in time results Adv. Math 2019 552 597

[60] J. Gómez-Serrano, R. Granero-Belinchón On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof Nonlinearity 2014 1471 1498

[61] R. Granero-Belinchón Global existence for the confined Muskat problem SIAM J. Math. Anal 2014 1651 1680

[62] R. Granero-Belinchón, S. Shkoller Well-posedness and decay to equilibrium for the muskat problem with discontinuouspermeability (2016) Trans. Amer. Math. Soc 2019 2255 2286

[63] R. Granero-Belinchón, S. Scrobogna 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] S.M. Hassanizadeh, W.G. Gray Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries Adv. Water Resour 1990 169 186

[66] H.S. Hele-Shaw The flow of water Nature 1898 34 36

[67] H.S. Hele-Shaw 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] A. Kiselev, F. Nazarov, A. Volberg Global well-posedness for the critical 2D dissipative quasi-geostrophic equation Invent. Math 2007 445 453

[70] O. Lazar 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.J. Majda, E.G. Tabak 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] B.-V. Matioc The muskat problem in 2d: equivalence of formulations, well-posedness, and regularity results Anal. PDE 2018 281 332

[75] B.-V. Matioc 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] A.-V. Matioc, B.-V. Matioc Well-posedness and stability results for a quasilinear periodic muskat problem J. Differ. Equ 2019 5500 5531

[78] H.K. Moffatt, D.E. Loper The magnetostrophic rise of a buoyant parcel in the earth’s core Geophys. J. Int 1994 394 402

[79] M. Muskat Two fluid systems in porous media. the encroachment of water into an oil sand Physics 1934 250 264

[80] M. Muskat The flow of fluids through porous media J. Appl. Phys 1937 274 282

[81] M. Muskat 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] F. Otto 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] N. Patel, R.M. Strain Large time decay estimates for the Muskat equation Commun. Part. Differ. Equ 2017 977 999

[86] T. Pernas-Castaño Local-existence for the inhomogeneous Muskat problem Nonlinearity 2017 2063

[87] J. Pruess, G. Simonett On the Muskat flow Evol. Equ. Control Theory 2016 631 645

[88] L. Rayleigh On the instability of jets Proc. London Math. Soc 1878 4 13

[89] J. Rodrigo 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] P.G. Saffman, 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 1958 312 329

[92] M. Siegel, R.E. Caflisch, S. Howison Global existence, singular solutions, and ill-posedness for the Muskat problem Commun. Pure Appl. Math. 2004 1374 1411

[93] L. Székelyhidi 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] A.R. Thornton, A.J. Van Der Horn, E. Gagarina, W. Zweers, D. Van Der Meer, O. Bokhove Hele-shaw beach creation by breaking waves: a mathematics-inspired experiment Environ. Fluid Mech 2014 1123 1145

[96] S. Tofts On the existence of solutions to the Muskat problem with surface tension J. Math. Fluid Mech 2017 581 611

Cité par Sources :