On the global existence for the Muskat problem
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TLDR
This work proves an L2(R) maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface, and takes advantage of the fact that the bound ‖∂xf0‖L∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.Abstract:
. The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L2(R) maximum principle, in the form of a new “log” conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ‖f ‖1 ≤ 1/5. Previous results of this sort used a small constant 1 which was not explicit [7, 19, 9, 14]. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy ‖f0‖L∞ < ∞ and ‖∂xf0‖L∞ < 1. We take advantage of the fact that the bound ‖∂xf0‖L∞ < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.read more
Citations
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Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves
TL;DR: In this paper, it was shown that the Rayleigh-Taylor condition may hold initially but break down in finite time, and that the existence of water waves turning can be proven.
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Well-posedness of the Muskat problem with H2 initial data
TL;DR: In this paper, the authors study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations, and show that solutions to the solution instantaneously become infinitely smooth.
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On the Muskat problem: Global in time results in 2D and 3D
TL;DR: In this paper, the authors considered the three-dimensional Muskat problem in the stable regime and obtained a conservation law which provides an $L 2$ maximum principle for the fluid interface.
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Breakdown of Smoothness for the Muskat Problem
TL;DR: In this article, the authors show that there exists analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down, that is, no longer belongs to C4.
Journal ArticleDOI
Global regularity for 2D Muskat equations with finite slope
TL;DR: In this paper, the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law, is considered.
References
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Book
Dynamics of fluids in porous media
TL;DR: In this paper, the Milieux poreux Reference Record was created on 2004-09-07, modified on 2016-08-08 and the reference record was updated in 2016.
Book
Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals
Elias M. Stein,Timothy S Murphy +1 more
TL;DR: In this article, the authors introduce the Heisenberg group and describe the Maximal Operators and Maximal Averages and Oscillatory Integral Integrals of the First and Second Kind.
Journal ArticleDOI
The Penetration of a Fluid into a Porous Medium or Hele-Shaw Cell Containing a More Viscous Liquid
TL;DR: In this paper, it was shown that a flow is possible in which equally spaced fingers advance steadily at very slow speeds, such that behind the tips of the advancing fingers the widths of the two columns of fluid are equal.