Turning waves and breakdown for incompressible flows

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.