On the Muskat problem: global in time results in 2D and 3D

2013/10/31 by Peter Constantin, Diego Cordoba, Francisco Gancedo +2 · 1 citation
Mathematics · #math.AP

paper · pdf · doi:10.1353/ajm.2016.0044

published as Amer. J. Math. (2016) · Added results on global existence in critical spaces. Included endpoint regularity theorem

arxiv created 2014/02/22 · arxiv updated 2019/05/02

Abstract

This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an L² maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the estimates from our paper \citePDPB to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature.

Cited by

• Existence, asymptotic behaviour and convergence of a generalised 3D Muskat problem in stable regime