Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T11:45:48.634Z Has data issue: false hasContentIssue false
  • English
  • Français

Well-posedness of the Muskat problem in subcritical Lp-Sobolev spaces

Part of: Miscellaneous topics - Partial differential equations Harmonic analysis in several variables Equations of mathematical physics and other areas of application Parabolic equations and systems

Published online by Cambridge University Press:  18 January 2021

H. ABELS Open the ORCID record for H. ABELS [Opens in a new window]  and
B.-V. MATIOC Open the ORCID record for B.-V. MATIOC [Opens in a new window]
H. ABELS
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germany, emails: [email protected]; [email protected]
B.-V. MATIOC
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, Germany, emails: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an Lp-setting with p ∈ (1, ∞). The Sobolev space $W_p^s(\mathbb R)$ with s = 1+1/p is a critical space for this problem. We prove, for each s ∈ (1+1/p, 2) that the Rayleigh–Taylor condition identifies an open subset of $W_p^s(\mathbb R)$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

Keywords

Muskat problemRayleigh–Taylor conditionsubcritical spacesingular integral

MSC classification

Primary: 35R37: Moving boundary problems 35K55: Nonlinear parabolic equations
Secondary: 35Q35: PDEs in connection with fluid mechanics 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 2 , April 2022 , pp. 224 - 266
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abels, H. (2012) Pseudodifferential and Singular Integral Operators, De Gruyter Graduate Lectures, De Gruyter, Berlin.Google Scholar
Alazard, T. & Lazar, O. (2020) Paralinearization of the Muskat equation and application to the Cauchy problem. Arch. Ration. Mech. Anal. 237, 545–583.CrossRefGoogle Scholar
Amann, H. (1995) Linear and Quasilinear Parabolic Problems. Vol. I, Monographs in Mathematics, Vol. 89, Birkhäuser Boston, Inc., Boston, MA.Google Scholar
Ambrose, D. M. (2004) Well-posedness of two-phase Hele-Shaw flow without surface tension. Eur. J. Appl. Math. 15, 597–607.CrossRefGoogle Scholar
Ambrose, D. M. (2014) The zero surface tension limit of two-dimensional interfacial Darcy flow. J. Math. Fluid Mech. 16, 105–143.CrossRefGoogle Scholar
Angenent, S. B. (1990) Nonlinear analytic semiflows. Proc. R. Soc. Edinburgh Sect. A 115, 91–107.Google Scholar
Bazaliy, B. V. & Vasylyeva, N. (2014) The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension. Zh. Mat. Fiz. Anal. Geom. 10, 3–43, 152, 155.Google Scholar
Berselli, L. C., Córdoba, D. & Granero-Belinchón, R. (2014) Local solvability and turning for the inhomogeneous Muskat problem. Interfaces Free Bound. 16, 175–213.CrossRefGoogle Scholar
Cameron, S. (2019) Global well-posedness for the two-dimensional Muskat problem with slope less than 1. Anal. PDE 12, 997–1022.CrossRefGoogle Scholar
Castro, A., Córdoba, D., Fefferman, C. L. & Gancedo, F. (2013) Breakdown of smoothness for the Muskat problem. Arch. Ration. Mech. Anal. 208, 805–909.CrossRefGoogle Scholar
Castro, A., Córdoba, D., Fefferman, C. L., Gancedo, F. & López-Fernández, M. (2011) Turning waves and breakdown for incompressible flows. Proc. Natl. Acad. Sci. USA 108, 4754–4759.CrossRefGoogle Scholar
Castro, A., Córdoba, D., Fefferman, C. L., Gancedo, F. & López-Fernández, M. (2012) Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. (2) 175, 909–948.CrossRefGoogle Scholar
Cheng, C. H. A., Granero-Belinchón, R. & Shkoller, S. (2016) Well-posedness of the Muskat problem with H 2 initial data. Adv. Math. 286, 32–104.CrossRefGoogle Scholar
Constantin, P., Córdoba, D., Gancedo, F., Rodrģuez-Piazza, L. & Strain, R. M. (2016) On the Muskat problem: global in time results in 2D and 3D. Am. J. Math. 138, 1455–1494.CrossRefGoogle Scholar
Constantin, P., Córdoba, D., Gancedo, F. & Strain, R. M. (2013) On the global existence for the Muskat problem. J. Eur. Math. Soc. (JEMS) 15, 201–227.CrossRefGoogle Scholar
Constantin, P., Gancedo, F., Shvydkoy, R. & Vicol, V. (2017) Global regularity for 2D Muskat equations with finite slope. Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 1041–1074.CrossRefGoogle Scholar
Córdoba, A., Córdoba, D. & Gancedo, F. (2011) Interface evolution: the Hele-Shaw and Muskat problems. Ann. Math. (2) 173, 477–542.CrossRefGoogle Scholar
Córdoba, D. & Gancedo, F. (2007) Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Comm. Math. Phys. 273, 445–471.CrossRefGoogle Scholar
Córdoba, D. & Gancedo, F. (2010) Absence of squirt singularities for the multi-phase Muskat problem. Comm. Math. Phys. 299, 561–575.CrossRefGoogle Scholar
Córdoba Gazolaz, D., Granero-Belinchón, R. & Orive-Illera, R. (2014) The confined Muskat problem: differences with the deep water regime. Commun. Math. Sci. 12, 423–455.CrossRefGoogle Scholar
Deng, F., Lei, Z. & Lin, F. (2017) On the two-dimensional Muskat problem with monotone large initial data. Comm. Pure Appl. Math. LXX, 1115–1145.CrossRefGoogle Scholar
Escher, J., Matioc, A.-V. & Matioc, B.-V. (2012) A generalized Rayleigh-Taylor condition for the Muskat problem. Nonlinearity 25, 73–92.CrossRefGoogle Scholar
Escher, J. & Matioc, B.-V. (2011) On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results. Z. Anal. Anwend. 30, 193–218.CrossRefGoogle Scholar
Escher, J., Matioc, B.-V. & Walker, C. (2018) The domain of parabolicity for the Muskat problem. Indiana Univ. Math. J. 67, 679–737.CrossRefGoogle Scholar
Escher, J. & Simonett, G. (1996) Analyticity of the interface in a free boundary problem. Math. Ann. 305, 439–459.CrossRefGoogle Scholar
Flynn, P. T. & Nguyen, H. Q. (2020) The vanishing surface tension limit of the Muskat problem. http://arxiv.org/abs/2001.10473arXiv:2001.10473.Google Scholar
Gancedo, F. (2017) A survey for the Muskat problem and a new estimate. SeMA J. 74, 21–35.CrossRefGoogle Scholar
Gancedo, F., García-Juárez, E., Patel, N. & Strain, R. M. (2019) On the Muskat problem with viscosity jump: global in time results. Adv. Math. 345, 552–597.CrossRefGoogle Scholar
Gancedo, F., Granero-Belinchón, R. & Scrobogna, S. (2020) Surface tension stabilization of the Rayleigh-Taylor instability for a fluid layer in a porous medium. Annales de l’Institut Henri Poincaré, Analyse Non Linéaire 37, 1299–1343.Google Scholar
Gómez-Serrano, J. & Granero-Belinchón, R. (2014) On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof. Nonlinearity 27, 1471–1498.CrossRefGoogle Scholar
Granero-Belinchón, R. (2014) Global existence for the confined Muskat problem. SIAM J. Math. Anal. 46, 1651–1680.CrossRefGoogle Scholar
Granero-Belinchón, R. & Lazar, O. (2020) Growth in the Muskat problem. Math. Model. Nat. Phenom. 15, 7.Google Scholar
Granero-Belinchón, R. & Shkoller, S. (2019) Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability. Trans. Am. Math. Soc. 372, 2255–2286.CrossRefGoogle Scholar
Lunardi, A. (1995) Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, Vol. 16, Birkhäuser Verlag, Basel.Google Scholar
Matioc, A.-V. & Matioc, B.-V. (2019) Well-posedness and stability results for a quasilinear periodic Muskat problem. J. Differ. Equ. 266, 5500–5531.CrossRefGoogle Scholar
Matioc, B.-V. (2018) Viscous displacement in porous media: the Muskat problem in 2D. Trans. Am. Math. Soc. 370, 7511–7556.CrossRefGoogle Scholar
Matioc, B.-V. (2019) The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results. Anal. PDE 12, 281–332.CrossRefGoogle Scholar
Matioc, B.-V. (2020) Well-posedness and stability results for some periodic Muskat problems. J. Math. Fluid. Mech. 20(3), 31, 45.CrossRefGoogle Scholar
Matioc, B.-V. & Walker, C. (2020) On the principle of linearized stability in interpolation spaces for quasilinear evolution equations. Monatshefte für Mathematik 191, 615–634.CrossRefGoogle Scholar
Meyer, Y. & Coifman, R. (1997) Wavelets, Cambridge Studies in Advanced Mathematics, Vol. 48, Cambridge University Press, Cambridge. Calderón-Zygmund and multilinear operators, Translated from the 1990 and 1991 French originals by David Salinger.Google Scholar
Murai, T. (1986) Boundedness of singular integral operators of Calderón type. V. Adv. Math. 59, 71–81.CrossRefGoogle Scholar
Murai, T. (1986) Boundedness of singular integral operators of Calderón type. VI. Nagoya Math. J. 102, 127–133.Google Scholar
Muskat, M. (1934) Two fluid systems in porous media. The encroachment of water into an oil sand. Physics 5, 250–264.CrossRefGoogle Scholar
Nguyen, H. Q. (2020) On well-posedness of the Muskat problem with surface tension. Adv. Math. 374, 107344.CrossRefGoogle Scholar
Nguyen, H. Q. & Pausader, B. (2020) A paradifferential approach for well-posedness of the Muskat problem. Arch. Ration. Mech. Anal. 237, 35–100.CrossRefGoogle Scholar
Patel, N. & Strain, R. M. (2017) Large time decay estimates for the Muskat equation. Comm. Partial Differ. Equ. 42, 977–999.CrossRefGoogle Scholar
Prüss, J., Shao, Y. & Simonett, G. (2015) On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension. Interfaces Free Bound. 17, 555–600.CrossRefGoogle Scholar
Prüss, J. & Simonett, G. (2016) Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, Vol. 105, Birkhäuser/Springer, Cham.Google Scholar
Prüss, J. & Simonett, G. (2016) On the Muskat flow. Evol. Equ. Control Theory 5, 631–645.CrossRefGoogle Scholar
Siegel, M., Caflisch, R. E. & Howison, S. (2004) Global existence, singular solutions, and ill-posedness for the Muskat problem. Comm. Pure Appl. Math. 57, 1374–1411.CrossRefGoogle Scholar
Tofts, S. (2017) On the existence of solutions to the Muskat problem with surface tension. J. Math. Fluid Mech. 19, 581–611.CrossRefGoogle Scholar
Yi, F. (1996) Local classical solution of Muskat free boundary problem. J. Partial Differ. Equ. 9, 84–96.Google Scholar