TY - JOUR
T1 - Turning waves and breakdown for incompressible flows
AU - Castro, Angel
AU - Córdoba, Diego
AU - Fefferman, Charles L.
AU - Gancedo, Francisco
AU - López-Fernández, María
PY - 2011/3/22
Y1 - 2011/3/22
N2 - 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.
AB - 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.
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U2 - 10.1073/pnas.1101518108
DO - 10.1073/pnas.1101518108
M3 - Article
AN - SCOPUS:79953190074
SN - 0027-8424
VL - 108
SP - 4754
EP - 4759
JO - Proceedings of the National Academy of Sciences of the United States of America
JF - Proceedings of the National Academy of Sciences of the United States of America
IS - 12
ER -