Finite-Time Singularity Formation for Incompressible Euler Moving Interfaces in the Plane

Abstract

This paper provides a general method for establishing finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane. This methodology is applied to two different problems.The first problem considered is the two-phase vortex sheet problem with surface tension, for which, under suitable assumptions of smallness of the initial height of the heaviest phase and velocity fields, is proved the finite-time singularity of the natural norm of the problem. This is in striking contrast with the case of finite-time splash and splat singularity formation for the one-phase Euler equations of [4] and [8], for which the natural norm (in the one-phase fluid) stays finite all the way until contact.The second problem considered involves the presence of a heavier rigid body moving in the inviscid fluid for which well-posedness was recently established in [16]. For a very general set of geometries (essentially the bottom of the symmetric domain being a graph) we first establish that the rigid body will hit the bottom of the fluid domain in finite time. This result allows for more general geometries than the ones first considered by [19] for this problem, as well as for small square integrable vorticity. Next, we establish the blow-up of a surface energy and a characterization of acceleration at contact: It opposes the motion, and is either strictly positive and finite if the contact zone is of non zero length, or infinite otherwise.

Recent advances on the global regularity for irrotational water waves

“…The two-dimensional result of [69] was extended to three dimensions and to some other related models by Coutand-Shkoller [70]. See also the related work [97] on the formation of singularities.…”

A priori estimates for the free-boundary Euler equations with surface tension in three dimensions

“…Combining (4.54), (4.55), and (4.56), we find 24 . We now estimate the remaining boundary terms in (4.53).…”

“…It has not been until fairly recently, with the works of Lindblad [61] for σ = 0, Coutand and Shkoller [25] for σ ≥ 0, and Shatah and Zeng [72,73], also for σ ≥ 0, and more recently by the first author and Ebin [34] for σ > 0, that existence and uniqueness for (1.4) have been addressed in full generality. Since the early 2000's, research on (1.4) has blossomed, as is illustrated by the sample list [1,3,4,5,6,7,10,8,2,9,12,11,15,16,17,18,19,20,21,22,23,24,26,27,29,30,31,32,33,40,41,42,44,45,46,47,49,50,51,52,54,55,56,…”

Global Well-posedness of Free Interface Problems for the incompressible Inviscid Resistive MHD

“…It is natural to consider the question whether there is a global well-posedness for free boundary problems or not. The recent works, Castro, Córdoba, Fefferman, Gancedo and Gómez-Serrano [5,6], Fefferman, Ionescu and Lie [18] and Coutand [12], imply the development of singularities in finite time of free boundary problems for some large initial data. For the irrotational incompressible Euler equations in the horizontally nonperiodic setting, certain dispersive effects can be used to establish the global well-posedness for the small initial data; we refer to Wu [56,57], Germain, Masmoudi and Shatah [20,21], Ionescu and Pusateri [29,30], Alazard and Delort [1] and Deng, Ionescu, Pausader and Pusateri [15].…”