# On the global existence for the Muskat problem

## Author(s): Constantin, Peter; Cordoba, Diego; Gancedo, Francisco; Strain, Robert M

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 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 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 ϵ≪1 which was not explicit. 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. Publication Date: 2013 Electronic Publication Date: 2013 Citation: Constantin, Peter, Cordoba, Diego, Gancedo, Francisco, Strain, Robert M. (2013). On the global existence for the Muskat problem. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 15 (201 - 227. doi:10.4171/JEMS/360 DOI: doi:10.4171/JEMS/360 ISSN: 1435-9855 Pages: 201 - 227 Type of Material: Journal Article Journal/Proceeding Title: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY Version: Author's manuscript