Opened Access On the global existence for the Muskat problem

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Autor: Constantin, Peter
Córdoba Gazolaz, Diego
Gancedo García, Francisco
Strain, Robert M.
Departamento: Universidad de Sevilla. Departamento de Análisis Matemático
Fecha: 2013
Publicado en: Journal of the European Mathematical Society, 15 (1), 201-227.
Tipo de documento: Artículo
Resumen: 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 $L^2(\R)$ maximum principle, in the form of a new ``log'' conservation law (???) which is satisfied by the equation (???) for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for 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. Lastly, we prove a global existence result 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.
Cita: Constantin, P., Córdoba Gazolaz, D., Gancedo García, F. y Strain, R.M. (2013). On the global existence for the Muskat problem. Journal of the European Mathematical Society, 15 (1), 201-227.
Tamaño: 232.2Kb
Formato: PDF

URI: http://hdl.handle.net/11441/45145

DOI: 10.4171/JEMS/360

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