On the absence of splash singularities in the case of two-fluid interfaces

Abstract

We show that "splash" singularities cannot develop in the case of locally smooth solutions of the two-fluid interfaces in two dimensions. More precisely, we show that the scenario of formation of singularities discovered by Castro-Córdoba-Fefferman-Gancedo-Gómez-Serrano [6] in the case of the water waves system, in which the interface remains locally smooth but self-intersects in finite time, is completely prevented in the case of two-fluid interfaces with positive densities.1 There is a slight imprecision in these equations, namely the function z(, .t) is determined only up to a reparametrization. This is consistent, of course, with the physical interpretation of the equations. To eliminate this imprecision, there are at least two natural parametrizations one could use: Eulerian coordinates in the case when the interface is a graph above a Euclidean space (notice that the vertical direction is determined invariantly by the choice of the vector-field e2 in the gravity term in (1.2)), or Lagrangian coordinates relative to one of the fluids. In this paper we work invariantly, i.e. we do not make any specific choice of coordinates on the interface. 2 We remark that the surface tension coefficient σ has to be strictly positive in order to prevent Kelvin-Helmholtz instabilities and have local well-posedness, in the case of two-fluid interfaces ρ 1 > 0, ρ 2 > 0.

Global Analysis of a Model for Capillary Water Waves in Two Dimensions

“…Surface tension effects have been considered in the works of Beyer and Gunther [8], Ambrose and Masmoudi [7], Coutand and Shkoller [14], Shatah and Zeng [39,40], Christianson, Hur, and Staffilani [10], and Alazard, Burq, and Zuily [1]. Recently some blowup scenarios have also been investigated [9,15,20]. The question of long-time regularity of solutions with irrotational, small, and localized initial data was also addressed in a few works, starting with [44], where Wu showed almost-global regularity for the gravity problem in two dimensions (one-dimensional interfaces).…”

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

“…The latter setup in turn admits the well-known water-wave, or air-water, limit, in which the density of the lighter fluid, air, is assumed to be negligible and enters the dynamics only as means of maintaining constant pressure on the water's free surface. There are some similarities between the "splash" and "splat" singularities studied in [12,21,22] (or their backward-time versions) and the focus of this work on the interaction of a fluid's density isoline with rigid boundaries. For instance, the splash example provided in [21], due to the reflectional symmetry across a vertical line, can be viewed as interaction of the free surface with a vertical wall.…”

“…In this work, we want to shift our attention to issues that concern more specifically the time evolution of data in this class. Much of the difficulties here stem from the mathematical assumptions underlying the foundation of the governing equations, specifically the property that "fluid particles on a wall stay on the wall " ( [13], p. 22). When this assumption is accepted, the shrinking or opening of "islands" of density variations along the confining boundaries becomes a rather subtle issue, even in limiting cases such as layered fluids separated by sharp interfaces (see figure 1 for a schematic).…”

Asymptotic models for free boundary flow in porous media

“…In the case of the one-phase Muskat problem, Castro, Córdoba, Fefferman, Gancedo [6] proved the existence of curves that self intersect in finite time in what is called a splash singularity (see also [19,[21][22][23]. It remains as an interesting open problem whether the Muskat problem can have a cusp singularity, i.e.…”