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 (α,f(α, t)) blows up: ∥∂αf∥L∞ (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.
|Original language||English (US)|
|Number of pages||6|
|Journal||Proceedings of the National Academy of Sciences of the United States of America|
|State||Published - Mar 22 2011|
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