### Abstract

We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class W^{2,p}, 1<p≤∞. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.

Original language | English (US) |
---|---|

Pages (from-to) | 1041-1074 |

Number of pages | 34 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 34 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2017 |

### Fingerprint

### Keywords

- Darcy's law
- Maximum principle
- Muskat problem
- Porous medium

### ASJC Scopus subject areas

- Analysis
- Mathematical Physics

### Cite this

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*,

*34*(4), 1041-1074. https://doi.org/10.1016/j.anihpc.2016.09.001

**Global regularity for 2D Muskat equations with finite slope.** / Constantin, Peter; Gancedo, Francisco; Shvydkoy, Roman; Vicol, Vlad.

Research output: Contribution to journal › Article

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*, vol. 34, no. 4, pp. 1041-1074. https://doi.org/10.1016/j.anihpc.2016.09.001

}

TY - JOUR

T1 - Global regularity for 2D Muskat equations with finite slope

AU - Constantin, Peter

AU - Gancedo, Francisco

AU - Shvydkoy, Roman

AU - Vicol, Vlad

PY - 2017/7/1

Y1 - 2017/7/1

N2 - We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class W2,p, 1

AB - We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class W2,p, 1

KW - Darcy's law

KW - Maximum principle

KW - Muskat problem

KW - Porous medium

UR - http://www.scopus.com/inward/record.url?scp=85011115158&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85011115158&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2016.09.001

DO - 10.1016/j.anihpc.2016.09.001

M3 - Article

VL - 34

SP - 1041

EP - 1074

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 4

ER -