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

Peter Constantin, Diego Córdoba, Francisco Gancedo, Luis Rodríguez-Piazza, Robert M. Strain

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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 available estimates 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 spaces with critical regularity, giving solutions with bounded slope and time integrable bounded curvature.

Original languageEnglish (US)
Pages (from-to)1455-1494
Number of pages40
JournalAmerican Journal of Mathematics
Volume138
Issue number6
DOIs
StatePublished - Dec 2016

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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    Constantin, P., Córdoba, D., Gancedo, F., Rodríguez-Piazza, L., & Strain, R. M. (2016). On the Muskat problem: Global in time results in 2D and 3D. American Journal of Mathematics, 138(6), 1455-1494. https://doi.org/10.1353/ajm.2016.0044