On the Cauchy Problem for the Muskat Equation. II: Critical Initial Data

Abedin, Farhan, Schwab, Russell W.: Regularity for a special case of two-phase hele-shaw flow via parabolic integro-differential equations. arXiv preprint arXiv:2008.01272, (2020)

Alazard, Thomas, Lazar, Omar: Paralinearization of the Muskat equation and application to the Cauchy problem. Arch. Ration. Mech. Anal. 237(2), 545–583 (2020)

Alazard, Thomas, Nguyen, Quoc-Hung: On the Cauchy problem for the Muskat equation with non-Lipschitz initial data. arXiv preprint arXiv:2009.04343, (2020)

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

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

Brué, Elia, Nguyen, Quoc-Hung: Advection diffusion equations with Sobolev velocity field. Comm. Math. Phys., (2021) https://doi.org/10.1007/s00220-021-03993-4

Brué, Elia, Nguyen, Quoc-Hung.: On the Sobolev space of functions with derivative of logarithmic order. Adv. Nonlinear Anal. 9(1), 836–849 (2020)

Brué, Elia, Nguyen, Quoc-Hung: Sharp regularity estimates for solutions to the continuity equation drifted by Sobolev vector fields. Anal. PDE, to appear (2021)

Caflisch, R.E., Orellana, O.F., Siegel, M.: A localized approximation method for vortical flows. SIAM J. Appl. Math. 50(6), 1517–1532 (1990)

Cameron, Stephen: Global well-posedness for the two-dimensional Muskat problem with slope less than 1. Anal. PDE 12(4), 997–1022 (2019)

Castro, Ángel., Córdoba, Diego, Fefferman, Charles, Gancedo, Francisco: Breakdown of smoothness for the Muskat problem. Arch. Ration. Mech. Anal. 208(3), 805–909 (2013)

Castro, Ángel., Córdoba, Diego, Fefferman, Charles, Gancedo, Francisco: Splash singularities for the one-phase Muskat problem in stable regimes. Arch. Ration. Mech. Anal. 222(1), 213–243 (2016)

Castro, Ángel., Córdoba, Diego, Fefferman, Charles, Gancedo, Francisco, López-Fernández, María: Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. of Math. (2) 175(2), 909–948 (2012)

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

Constantin, Peter, Córdoba, Diego, Gancedo, Francisco, Rodríguez-Piazza, Luis, Strain, Robert M.: On the Muskat problem: global in time results in 2D and 3D. Amer. J. Math. 138(6), 1455–1494 (2016)

Constantin, Peter, Córdoba, Diego, Gancedo, Francisco, Strain, Robert M.: On the global existence for the Muskat problem. J. Eur. Math. Soc. (JEMS) 15(1), 201–227 (2013)

Constantin, Peter, Gancedo, Francisco, Shvydkoy, Roman, Vicol, Vlad: Global regularity for 2D Muskat equations with finite slope. Ann. Inst. H. Poincaré Anal. Non Linéaire 34(4), 1041–1074 (2017)

Córdoba, Antonio, Córdoba, Diego, Gancedo, Francisco: Interface evolution: the Hele-Shaw and Muskat problems. Ann. of Math. (2) 173(1), 477–542 (2011)

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

Córdoba, Diego, Gancedo, Francisco: A maximum principle for the Muskat problem for fluids with different densities. Comm. Math. Phys. 286(2), 681–696 (2009)

Córdoba, Diego, Lazar, Omar: Global well-posedness for the 2d stable Muskat problem in $${H}^{\frac{3}{2}}$$. Ann. Sci. Éc. Norm. Supér., to appear (2021)

Darcy, Henry Philibert Gaspard: Les Fontaines publiques de la ville de Dijon. Exposition et application des principes à suivre et des formules à employer dans les questions de distribution d’eau, etc. V. Dalamont, (1856)

Deng, Fan, Lei, Zhen, Lin, Fanghua: On the two-dimensional Muskat problem with monotone large initial data. Comm. Pure Appl. Math. 70(6), 1115–1145 (2017)

Escher, Joachim, Simonett, Gieri: Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations 2(4), 619–642 (1997)

Gancedo, F., García-Juárez, E., Patel, N., Strain, R.M.: On the Muskat problem with viscosity jump: global in time results. Adv. Math. 345, 552–597 (2019)

Gancedo, Francisco, Garcia-Juarez, Eduardo, Patel, Neel, Strain, Robert: Global regularity for gravity unstable Muskat bubbles, (2020)

Gancedo, Francisco, Lazar, Omar: Global well-posedness for the 3d Muskat problem in the critical sobolev space. arXiv:2006.01787

Granero-Belinchón, Rafael, Scrobogna, Stefano: On an asymptotic model for free boundary Darcy flow in porous media. SIAM J. Math. Anal. 52(5), 4937–4970 (2020)

Matioc, Bogdan-Vasile.: The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results. Anal. PDE 12(2), 281–332 (2019)

Muskat, Morris: Two fluid systems in porous media. the encroachment of water into an oil sand. Physics 5(9), 250–264 (1934)

Nguyen, Huy Q., Pausader, Benoît: A paradifferential approach for well-posedness of the Muskat problem. Arch. Ration. Mech. Anal. 237(1), 35–100 (2020)

Nguyen, Quoc-Hung: Quantitative estimates for regular lagrangian flows with $$BV$$ vector fields. Comm. Pure Appl. Math. (2021)

Patel, Neel, Strain, Robert M.: Large time decay estimates for the Muskat equation. Comm. Partial Differential Equations 42(6), 977–999 (2017)

Prüss, Jan, Simonett, Gieri: Moving interfaces and quasilinear parabolic evolution equations, volume 105 of Monographs in Mathematics. Birkhäuser/Springer, [Cham], (2016)

Siegel, Michael, Caflisch, Russel E., Howison, Sam: Global existence, singular solutions, and ill-posedness for the Muskat problem. Comm. Pure Appl. Math. 57(10), 1374–1411 (2004)

Triebel, Hans: Theory of function spaces, volume 78 of Monographs in Mathematics. Birkhäuser Verlag, Basel, (1983)

Triebel, Hans: Characterizations of Besov-Hardy-Sobolev spaces: a unified approach. J. Approx. Theory 52(2), 162–203 (1988)

Yi, Fahuai: Global classical solution of Muskat free boundary problem. J. Math. Anal. Appl. 288(2), 442–461 (2003)