On the global existence for the Muskat problem

By Peter Constantin, D. C'ordoba, Francisco Gancedo, Robert M. Strain. 2010

150 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/184e4ced497f961d6d206816b3a6e8fe49902dcf

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Citations in the corpus

On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data (2016, methodology): “This is one of the keys in [13] [14] where they established global solutions evolved from an exact class of small initial data.”
A Paradifferential Approach for Well-Posedness of the Muskat Problem (2019, background): “lts on singularity formation and to [34, 30] for recent reviews on the Muskat problem. In the case of small data and inﬁnite depth, global strong solutions have been constructed in subcritical spaces [11, 14, 15, 16, 17, 18, 21, 24, 52] and in critical spaces [33]. We note in particular that [24] al”
Global well-posedness for the two-dimensional Muskat problem with slope less than 1 (2017, unknown): “The Muskat problem is known to be locally well-posed in H k for k ≥ 3 with solutions satisfying L ∞ and L 2 maximum principles, but neither imply any gain of derivatives; see [Córdoba and Gancedo 2009; Constantin et al. 2013].”
Endpoint Sobolev Theory for the Muskat Equation (2020, background): “In this direction, we are strongly influenced by many earlier works about global existence results in critical spaces under some smallness assumptions (see in particular [24, 23, 14, 32, 35]).”
Well-posedness of the Muskat problem with H2 initial data (2014, unknown): “The global weak solution of [13] was later extended to the confined case by Granero-Belinchón in [38].”
Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves (2011, background): “= ‖f0‖ 2 L(2) , which does not imply, for large initial data, a gain of derivatives in the system (see [8]).”