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

Abstract

This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ 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 \cite{PDPB} 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.

Added results on global existence in critical spaces. Included endpoint regularity theorem

References in corpus (1)

• A maximum principle for the Muskat problem for fluids with different densities

Cited by in corpus (20)

• Global mild solutions of the Landau and non-cutoff Boltzmann equations
• The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results
• On The Muskat Problem With Viscosity Jump: Global In Time Results
• Global well-posedness for the 2D Muskat problem with slope less than 1
• A paradifferential approach for well-posedness of the Muskat problem
• Paralinearization of the Muskat equation and application to the Cauchy problem
• Piecewise constant subsolutions for the Muskat problem
• A note on stability shifting for the Muskat problem
• Large Time Decay Estimates for the Muskat Equation
• On the thin film Muskat and the thin film Stokes equations
• Lyapunov functions, Identities and the Cauchy problem for the Hele-Shaw equation
• Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces
• The Peskin Problem with Viscosity Contrast
• Weak solutions to the Muskat problem with surface tension via optimal transport
• The vanishing surface tension limit of the Muskat problem
• Local-existence for the Inhomogeneous Muskat problem
• Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data
• Entropy solutions to macroscopic IPM
• On a fourth order equation describing single-component film models
• The Second Iterate of the Muskat Equation in Supercritical Spaces