On an asymptotic model for free boundary darcy flow in porous media

SIAM Journal on Mathematical Analysis

Volumn 52, Issue 5, 2020, Pages 4937-4970

  Granero Belinchon, Rafael ^a   Scrobogna, Stefano ^b  

^a UNIVERSITY OF CANTABRIA   (Spain)

^b UNIVERSITY OF SEVILLE   (Spain)

Author keywords

Free boundary problems; Moving interfaces; Muskat problem

Indexed keywords

POROUS MATERIALS;

ASYMPTOTIC MODEL; CRITICAL SPACES; DARCY FLOWS; FREE BOUNDARY; SMALL AMPLITUDE; WAVELENGTH APPROXIMATION; WELLPOSEDNESS;

FLOW OF FLUIDS;

EID: 85096354441     PISSN: 00361410     EISSN: 10957154     Source Type: Journal    
DOI: 10.1137/19M126623X     Document Type: Article

References (47)

1. T. Alazard, Convexity and the Hele-Shaw Equation, Water Waves, 2020, in press.
2. T. Alazard and D. Bresch, Functional Inequalities and Strong Lyapunov Functionals for Free Surface Flows in Fluid Dynamics, preprint, arXiv:2004.03440.
3. T. Alazard and J.-M. Delort, Sobolev estimates for two dimensional gravity water waves, Astérisque, 374 (2015), pp. 1-241
4. T. Alazard and O. Lazar, Paralinearization of the Muskat equation and application to the Cauchy problem, Arch. Ration. Mech. Anal., 237 (2020), pp. 545-583.
5. T. Alazard, N. Meunier, and D. Smets, Lyapunov functions, identities and the Cauchy problem for the Hele-Shaw equation, Comm. Math. Phys., 377 (2020), pp. 1421-1459.
6. H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss. 343, Springer, Heidelberg, 2011.
7. L. C. Berselli, D. Córdoba, and R. Granero-Belinchón, Local solvability and turning for the inhomogeneous Muskat problem, Interfaces Free Bound., 16 (2014), pp. 175-213.
8. J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. 17 (1872), pp. 55-108.
9. S. Cameron, Global well-posedness for the 2D Muskat problem with slope less than 1, Anal. PDE, 12 (2017), pp. 997-1022.
10. A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, and M. Lopez-Fernandez, Rayleigh- Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2), 175 (2012), pp. 909-948.
11. 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), pp. 213-243.
12. J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), pp. 314-328.
13. C.-H. A. Cheng, R. Granero-Belinchón, and S. Shkoller, Well-posedness of the Muskat problem with H2 initial data, Adv. Math., 286 (2016), pp. 32-104.
14. C.-H. A. Cheng, R. Granero-Belinchón, S. Shkoller, and J. Wilkening, Rigorous Asymptotic models of water waves, Water Waves, 1 (2019), pp. 71-130.
15. P. Constantin, D. Cordoba, F. Gancedo, and R. M. Strain, On the global existence for the Muskat problem, J. Eur. Math. Soc. (JEMS), 15 (2013), pp. 201-227.
16. P. Constantin, F. Gancedo, R. Shvydkoy, and V. Vicol, Global regularity for 2D Muskat equations with finite slope, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2016), pp. 1041-1074.
17. A. Cordoba, D. Córdoba, and F. Gancedo, Interface evolution: The Hele-Shaw and Muskat problems, Ann. of Math. (2), 173 (2011), pp. 477-542.
18. D. Córdoba and F. Gancedo, Contour dynamics of incompressible 3-D fluids in a porous medium with different densities, Comm. Math. Phys., 273 (2007), pp. 445-471.
19. D. Córdoba and F. Gancedo, A maximum principle for the Muskat problem for fluids with different densities, Comm. Math. Phys., 286 (2009), pp. 681-696.
20. D. Córdoba, F. Gancedo, and R. Orive, A note on interface dynamics for convection in porous media, Phys. D, 237 (2008), pp. 1488-1497.
21. D. Córdoba, J. Gómez-Serrano, and A. Zlatoš, A note on stability shifting for the Muskat problem, Philos. Trans. Roy. Soc. A, 373 (2015), 20140278.
22. 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), pp. 367-378.
23. D. Córdoba, R. Granero-Belinchón, and R. Orive, On the confined Muskat problem: Differences with the deep water regime, Comm. Math. Sci., 12 (2014), pp. 423-455.
24. D. Cordoba and T. Pernas-Castaño, Non-splat singularity for the one-phase Muskat problem, Trans. Amer. Math. Soc., 369 (2017), pp. 711-754.
25. D. Cordoba and O. Lazar, Global Well-posedness for the 2D Stable Muskat Problem in H3/2, preprint, arXiv:1803.07528, 2018.
26. D. Coutand and S. Shkoller, On the impossibility of finite-time splash singularities for vortex sheets, Arch. Ration. Mech. Anal., 221 (2016), pp. 987-1033.
27. W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-deVries scaling limits, Comm. Partial Differential Equations, 10 (1985), pp. 787-1003.
28. W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), pp. 73-83.
29. W. Craig, U. Schanz, and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), pp. 615-667.
30. C. Fefferman, A. D. Ionescu, and V. Lie, On the absence of splash singularities in the case of two-fluid interfaces, Duke Math. J., 165 (2016), pp. 417-462.
31. F. Gancedo and O. Lazar, Global Well-posedness for the 3D Muskat Problem in the Critical Sobolev Space, preprint, arXiv:2006.01787, 2020.
32. F. Gancedo and R. M Strain, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA, 111 (2014), pp. 635-639.
33. J. Gómez-Serrano and R. Granero-Belinchón, On turning waves for the inhomogeneous Muskat problem: A computer-assisted proof, Nonlinearity, 27 (2014), pp. 1471-1498.
34. R. Granero-Belinchón, Global existence for the confined Muskat problem, SIAM J. Math. Anal., 46 (2014), pp. 1651-1680.
35. F. Gancedo, R. Granero-Belinchón, and S. Scrobogna, Surface tension stabilization of the Rayleigh-Taylor instability for a fluid layer in a porous medium, Ann. Inst. H. Poincaré Anal Non Linéaire, in press.
36. R. Granero-Belinchón and O. Lazar, Growth in the Muskat problem, Math. Model. Nat. Phenom., 15 (2020), 7.
37. R. Granero-Belinchón and S. Scrobogna, Asymptotic models for free boundary flow in porous media, Phys. D, 392 (2019), pp. 1-16.
38. R. Granero-Belinchón and S. Scrobogna, Well-posedness of a water wave model with viscous effects, Proc. Amer. Math. Soc., in press.
39. R. Granero-Belinchón and S. Scrobogna, Models for damped water waves, SIAM J. Appl. Math., 79 (2019), pp. 2530-2550.
40. R. Granero-Belinchón and S. Shkoller, Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability, Trans. Amer. Math. Soc., 372 (2019), pp. 2255-2286.
41. R. Granero-Belinchón and S. Shkoller, A model for Rayleigh-Taylor mixing and interface turnover, Multiscale Model. Simul., 15 (2017), pp. 274-308.
42. R. Granero-Belinchón, G. Navarro, and A. Ortega, On the effect of boundaries in twophase porous flow, Nonlinearity, 28 (2015), pp. 435-461.
43. D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics, American Mathematical Society, Providence, RI, 2013.
44. M. Muskat, Two fluid systems in porous media. The encroachment of water into an oil sand, Physics, 5 (1934), pp. 250-264.
45. H. Q. Nguyen and B. Pausader, A paradifferential approach for well-posedness of the Muskat problem, Arch. Ration. Mech. Anal., 237 (2020), pp. 35-100.
46. C. W. Oseen, Sur les formules de Green généralisées qui se présentent dans l'hydrodynamique et sur quelquesunes de leurs applications, Acta Math., 35 (1912), pp. 97-192
47. J. Simon, Compact sets in the space Lp(0, T;B), Ann. Mat. Pura Appl. (4), 146 (1987), pp. 65-96.