Abstract
This paper provides a new general method for establishing a 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 a very general set of geometries (essentially the contact zone being a graph) we first establish that the rigid body will hit the bottom of the fluid domain in finite time. Compared to the previous paper [20] for the rigid body case, the present paper allows for small square integrable vorticity and provides a characterization of acceleration at contact. A surface energy is shown to blow up and acceleration at contact is shown to oppose the motion: it is either strictly positive and finite if the contact zone is of non zero length, or infinite otherwise.
Highlights
Finite-time singularity formation in moving boundary problems has been an active field of research for at least the past 10 years
The first cases studied were contact problems for a symmetric rigid body moving in a fluid, which present the simplification at the level of the analysis of having a constant shape for the inclusion
The present paper presents a new methodology addressing finite-time singularity formation for any type of problems when the fluid equations are the incompressible Euler equations and the physical law of the included phase provides spatial control of the position of the interface
Summary
Finite-time singularity formation in moving boundary problems has been an active field of research for at least the past 10 years. This result is in striking contrast with splash and splat singularity formation for the one-phase water-waves problem introduced in [4], and treated with different methods in a more general context in [8], where the natural norm N (t) stays finite This was essential in the analysis of these papers in order to establish the finite-time contact, as this ensures that the magnitude of the relative velocity between two parts of an almost self-intersecting curve coming towards each other will be in magnitude greater than some strictly positive quantity. The differential inequality obtained appears in a way quite natural to the problem of a moving Euler phase, and is quite different from the pioneering works of Sideris [24] and Xin [26] for compressible Euler and Navier–Stokes equations We use this differential inequality to establish the first theorem on finite-time singularity formation for the Euler vortex sheet problem with surface tension and gravity effects: Theorem. Standard models of nonlinear elastodynamics (such as the quasilinear Saint-Venant Kirchhoff model) for the included phase Ω+ would be suitable
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