On the Impossibility of Finite-Time Splash Singularities for Vortex Sheets

Abstract

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet evolution, i.e. for the two-phase incompressible Euler equations. We prove this by contradiction; we assume that a splash singularity does indeed occur in finite time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allow us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, showing that our assumption of a finite-time splash singularity was false.Date: June 30, 2015. Key words and phrases. vortex sheets, Euler equations, water waves, blow-up, interface singularity, splash singularity, splat singularity. 1 4 D. COUTAND AND S. SHKOLLERThe first local well-posedness result for the 3-D incompressible Euler equations without the irrotationality assumption was obtained by Lindblad [34] in the case that the domain is diffeomorphic to the unit ball using a Nash-Moser iteration. Coutand & Shkoller [18] proved local well-posedness for arbitrary initial geometries that have at least H 3 -class boundaries without derivative loss. Shatah & Zeng [40] established a priori estimates for this problem using an infinite-dimensional geometric formulation, and Zhang & Zhang proved well-poseness by extending the complex-analytic method of Wu [44] to allow for vorticity. Again, in the latter case the domain was with infinite depth.1.4.2. Long-time existence. It is of great interest to understand if solutions to the Euler equations can be extended for all time when the data is sufficiently smooth and small, or if a finite-time singularity can be predicted for other types of initial conditions.Because of irrotationality, the water waves problem does not suffer from vorticity concentration; therefore, singularity formation involves only the loss of regularity of the interface or interface collision. In the case that the irrotational fluid is infinite in the horizontal directions, certain dispersivetype properties can be made use of. For sufficiently smooth and small data, Alvarez-Samaniego & Lannes [5] proved existence of solutions to the water waves problem on large time-intervals (larger than predicted by energy estimates), and provided a rigorous justification for a variety of asymptotic regimes. By constructing a transformation to remove the quadratic nonlinearity, combined with decay estimates for the linearized problem (on the infinite half-space domain), Wu [45] established an almost global existence result (existence on time intervals which are exponential in the size of the data) for the 2-D water waves problem with sufficiently small data. In a different framework, Alazard, Burq & Zuily [2] have also proven this result. Using position-velocity potential holomorphic coordinates, Hunter, Ifrim, & Tataru [28] have also proved almo...

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

“…A natural question that then arose was to extend this type of self-contact along a smooth curve in the two-phase context (with surface tension to make the problem locally well-posed in Sobolev spaces). With different methods, Fefferman, Ionescu and Lie [10] and Coutand and Shkoller [9], established that the two-phase vortex sheet problem with surface tension does not have finite-time formation of a splash or splat singularity so long as the natural norm of the problem for the velocity field in one phase stays bounded. The results of [9] and [10] however do not exclude such a loss of injectivity.…”

Weak–Strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Surface Tension

“…Using finally the Sobolev embedding to bound the L ∞ -norm of η on the interface (which is either one-or two-dimensional; note that the constant in the Sobolev embedding may be bounded by Cr −1 c for our geometry), we infer from this estimate the desired bound (57d), using also (36) and (35). This concludes the proof.…”

“…Existence results for the two-phase Stokes and Navier-Stokes equation in the absence of surface tension have been established by Giga and Takahashi [57] and Nouri and Poupaud [71]. Note that in contrast to the case of a single fluid in vacuum, for the flow of two incompressible immiscible inviscid fluids splash singularities cannot occur as shown by Fefferman, Ionescu, and Lie [50] and Coutand and Shkoller [35]; one would expect a similar result to hold for viscous fluids. However, solutions may be subject to the Rayleigh-Taylor instability as proven by Prüss and Simonett [74].…”

“…where in the last step we have used also (25). Together Young's inequality and the coercivity properties of the relative entropy (35) and (36) we then immediately get the estimate…”

“…First, we extend the normal vector field n v to a (space-time) tubular neighborhood of the evolving interfaces I v (t) by projecting onto the interface. Second, we multiply this construction with a cutoff which decreases quadratically in the distance to the interface of the strong solution (see (35)).…”