## The Muskat problem with $C^1$ data

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- by Ke Chen, Quoc-Hung Nguyen and Yiran Xu PDF
- Trans. Amer. Math. Soc.
**375**(2022), 3039-3060 Request permission

## Abstract:

In this paper we prove that the Cauchy problem of the Muskat equation is wellposed locally in time for any initial data in $\dot C^1(\mathbb {R}^d)\cap L^2(\mathbb {R}^d)$.## References

- Helmut Abels and Bogdan-Vasile Matioc.
*Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces.*European Journal of Applied Mathematics, 1–43, 2021. - Thomas Alazard and Omar Lazar,
*Paralinearization of the Muskat equation and application to the Cauchy problem*, Arch. Ration. Mech. Anal.**237**(2020), no. 2, 545–583. MR**4097324**, DOI 10.1007/s00205-020-01514-6 - Thomas Alazard and Quoc-Hung Nguyen.
*Endpoint Sobolev theory for the Muskat equation*. arXiv: 2010.06915. - Thomas Alazard and Quoc-Hung Nguyen,
*On the Cauchy problem for the Muskat equation. II: Critical initial data*, Ann. PDE**7**(2021), no. 1, Paper No. 7, 25. MR**4242131**, DOI 10.1007/s40818-021-00099-x - Thomas Alazard and Quoc-Hung Nguyen.
*On the Cauchy problem for the Muskat equation with non-Lipschitz initial data*. Communications in Partial Differential Equations, 46(11):2171–2212. DOI: 10.1080/03605302.2021.1928700. 2021. - Thomas Alazard and Quoc-Hung Nguyen,
*Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem*, Advances in Math (to appear), arXiv:2103.02474 - Thomas Alazard, Omar Lazar and Quoc-Hung Nguyen,
*On the dynamics of the roots of polynomials under differentiation*, Journal de mathematiques pures et appliquées (to appear), arXiv:2104.06921. - David M. Ambrose,
*Well-posedness of two-phase Hele-Shaw flow without surface tension*, European J. Appl. Math.**15**(2004), no. 5, 597–607. MR**2128613**, DOI 10.1017/S0956792504005662 - David M. Ambrose,
*Well-posedness of two-phase Darcy flow in 3D*, Quart. Appl. Math.**65**(2007), no. 1, 189–203. MR**2313156**, DOI 10.1090/S0033-569X-07-01055-3 - R. E. Caflisch, O. F. Orellana, and M. Siegel,
*A localized approximation method for vortical flows*, SIAM J. Appl. Math.**50**(1990), no. 6, 1517–1532. MR**1080505**, DOI 10.1137/0150089 - Stephen Cameron,
*Global well-posedness for the two-dimensional Muskat problem with slope less than 1*, Anal. PDE**12**(2019), no. 4, 997–1022. MR**3869383**, DOI 10.2140/apde.2019.12.997 - Stephen Cameron,
*Global wellposedness for the 3D Muskat problem with medium size slope*. arXiv:2002.00508. - Ángel Castro, Diego Córdoba, Charles Fefferman, and Francisco Gancedo,
*Breakdown of smoothness for the Muskat problem*, Arch. Ration. Mech. Anal.**208**(2013), no. 3, 805–909. MR**3048596**, DOI 10.1007/s00205-013-0616-x - Ángel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, and María López-Fernández,
*Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves*, Ann. of Math. (2)**175**(2012), no. 2, 909–948. MR**2993754**, DOI 10.4007/annals.2012.175.2.9 - Ke Chen and Quoc-Hung Nguyen,
*The Peskin problem with $BMO^1$ initial data*. arXiv:2107.13854. - C. H. Arthur Cheng, Rafael Granero-Belinchón, and Steve Shkoller,
*Well-posedness of the Muskat problem with $H^2$ initial data*, Adv. Math.**286**(2016), 32–104. MR**3415681**, DOI 10.1016/j.aim.2015.08.026 - Peter Constantin, Diego Córdoba, Francisco Gancedo, and Robert M. Strain,
*On the global existence for the Muskat problem*, J. Eur. Math. Soc. (JEMS)**15**(2013), no. 1, 201–227. MR**2998834**, DOI 10.4171/JEMS/360 - Peter Constantin, Diego Córdoba, Francisco Gancedo, Luis Rodríguez-Piazza, and Robert M. Strain,
*On the Muskat problem: global in time results in 2D and 3D*, Amer. J. Math.**138**(2016), no. 6, 1455–1494. MR**3595492**, DOI 10.1353/ajm.2016.0044 - Peter Constantin, Francisco Gancedo, Roman Shvydkoy, and Vlad Vicol,
*Global regularity for 2D Muskat equations with finite slope*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**34**(2017), no. 4, 1041–1074. MR**3661870**, DOI 10.1016/j.anihpc.2016.09.001 - Diego Córdoba and Francisco Gancedo,
*Contour dynamics of incompressible 3-D fluids in a porous medium with different densities*, Comm. Math. Phys.**273**(2007), no. 2, 445–471. MR**2318314**, DOI 10.1007/s00220-007-0246-y - Diego Córdoba and Francisco Gancedo,
*A maximum principle for the Muskat problem for fluids with different densities*, Comm. Math. Phys.**286**(2009), no. 2, 681–696. MR**2472040**, DOI 10.1007/s00220-008-0587-1 - Diego Córdoba and Omar Lazar,
*Global well-posedness for the 2d stable Muskat problem in ${H}^{\frac {3}{2}}$*. To appear in Annales scientifiques de l’École normale supérieure, 2021. - Antonio Córdoba, Diego Córdoba, and Francisco Gancedo,
*Interface evolution: the Hele-Shaw and Muskat problems*, Ann. of Math. (2)**173**(2011), no. 1, 477–542. MR**2753607**, DOI 10.4007/annals.2011.173.1.10 - Fan Deng, Zhen Lei, and Fanghua Lin,
*On the two-dimensional Muskat problem with monotone large initial data*, Comm. Pure Appl. Math.**70**(2017), no. 6, 1115–1145. MR**3639321**, DOI 10.1002/cpa.21669 - Henry Philibert Gaspard Darcy,
*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. - Joachim Escher and Gieri Simonett,
*Classical solutions for Hele-Shaw models with surface tension*, Adv. Differential Equations**2**(1997), no. 4, 619–642. MR**1441859** - Francisco Gancedo and Omar Lazar.
*Global well-posedness for the 3d Muskat problem in the critical Sobolev space*. arXiv:2006.01787. - F. Gancedo, E. García-Juárez, N. Patel, and R. M. Strain,
*On the Muskat problem with viscosity jump: global in time results*, Adv. Math.**345**(2019), 552–597. MR**3899970**, DOI 10.1016/j.aim.2019.01.017 - Eduardo García-Juárez, Javier Gómez-Serrano, Huy Q. Nguyen, Benoît Pausader,
*Self-similar solutions for the Muskat equation.*arXiv:2109.02565. - Bogdan-Vasile Matioc,
*Viscous displacement in porous media: the Muskat problem in 2D*, Trans. Amer. Math. Soc.**370**(2018), no. 10, 7511–7556. MR**3841857**, DOI 10.1090/tran/7287 - Bogdan-Vasile Matioc,
*The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results*, Anal. PDE**12**(2019), no. 2, 281–332. MR**3861893**, DOI 10.2140/apde.2019.12.281 - Morris Muskat,
*Two fluid systems in porous media. The encroachment of water into an oil sand.*J. Appl. Phys.**5**(1934), no. 5, 250–264. - Huy Q. Nguyen and Benoît Pausader,
*A paradifferential approach for well-posedness of the Muskat problem*, Arch. Ration. Mech. Anal.**237**(2020), no. 1, 35–100. MR**4090462**, DOI 10.1007/s00205-020-01494-7 - Michael Siegel, Russel E. Caflisch, and Sam Howison,
*Global existence, singular solutions, and ill-posedness for the Muskat problem*, Comm. Pure Appl. Math.**57**(2004), no. 10, 1374–1411. MR**2070208**, DOI 10.1002/cpa.20040 - Hans Triebel,
*Characterizations of Besov-Hardy-Sobolev spaces: a unified approach*, J. Approx. Theory**52**(1988), no. 2, 162–203. MR**929302**, DOI 10.1016/0021-9045(88)90055-X - Fahuai Yi,
*Global classical solution of Muskat free boundary problem*, J. Math. Anal. Appl.**288**(2003), no. 2, 442–461. MR**2019452**, DOI 10.1016/j.jmaa.2003.09.003

## Additional Information

**Ke Chen**- Affiliation: Fudan University, 220 Handan Road, Shanghai 200433, China
- ORCID: 0000-0002-4560-2107
- Email: kchen18@fudan.edu.cn
**Quoc-Hung Nguyen**- Affiliation: ShanghaiTech University, 393 Middle Huaxia Road, Shanghai 201210, China
- Address at time of publication: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- Email: qhnguyen@amss.ac.cn
**Yiran Xu**- Affiliation: Fudan University, 220 Handan Road, Shanghai 200433, People’s Republic of China
- Email: yrxu20@fudan.edu.cn
- Received by editor(s): April 28, 2021
- Published electronically: February 9, 2022
- Additional Notes: The second author was supported by the ShanghaiTech University startup fund and the National Natural Science Foundation of China (12050410257). This work was finished during Ke Chen and Yiran Xu’s visit to ShanghaiTech.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 3039-3060 - MSC (2020): Primary 35Q35, 76S05
- DOI: https://doi.org/10.1090/tran/8559
- MathSciNet review: 4402655