%0 Journal Article
%A Castro, Angel
%A Córdoba, Diego
%A Fefferman, Charles L.
%A Gancedo, Francisco
%A López-Fernández, María
%T Turning waves and breakdown for incompressible flows
%D 2011
%R 10.1073/pnas.1101518108
%J Proceedings of the National Academy of Sciences
%P 4754-4759
%V 108
%N 12
%X 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.
%U https://www.pnas.org/content/pnas/108/12/4754.full.pdf