Turning waves and breakdown for incompressible flows

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 $(\al,f_0(\al))$, the interface reaches a regime in finite time in which is no longer a graph. Therefore there exists a time $t^{\ast }$ where the solution of the free boundary problem parameterized as $(\al,f(\al,t))$ blows-up: $\|\da f\|_{L^\infty}(t^*)=\infty$. 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.

15 pages

References in corpus (6)

• Global well-posedness of the 3-D full water wave problem
• On the global existence for the Muskat problem
• Breakdown of smoothness for the Muskat problem
• A maximum principle for the Muskat problem for fluids with different densities
• Mathematics for 2d Interfaces
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Cited by in corpus (8)

• Breakdown of smoothness for the Muskat problem
• On the finite-time splash and splat singularities for the 3-D free-surface Euler equations
• The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results
• Splash singularity for water waves
• Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces
• On the Muskat flow
• Is dislocation flow turbulent in deformed crystals?
• The Second Iterate of the Muskat Equation in Supercritical Spaces