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Turning waves and breakdown for incompressible flows

 

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Author: Castro Martínez, Ángel
Córdoba Gazolaz, Diego
Fefferman, Charles L.
Gancedo García, Francisco
López Fernández, María
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2011-03-22
Published in: Proceedings of the National Academy of Sciences, 108 (12), 4754-4759.
Document type: Article
Abstract: We consider the evolution of an interface generated between two immiscible, incompressible, and irrotational fluids. Specifically we study the Muskat and water wave problems. We show that starting with a family of initial data given by (α, f0(α)), the interface reaches a regime in finite time in which is no longer a graph. Therefore there exists a time ∗t where the solution of the free boundary problem parameterized as s (α, f(α, t)) blows up:: k∂αfkL∞(t∗) = ∞. In particular, for the Muskat problem, this result allows us to reach an unstable regime, for which the Rayleigh-Taylor condition changes sign and the solution breaks down.
Cite: Castro Martínez, Á., Córdoba Gazolaz, D., Fefferman, C.L., Gancedo García, F. y López Fernández, M. (2011). Turning waves and breakdown for incompressible flows. Proceedings of the National Academy of Sciences, 108 (12), 4754-4759.
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URI: http://hdl.handle.net/11441/45167

DOI: 10.1073/pnas.1101518108

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