On the Finite-Time Splash and Splat Singularities for the 3-D Free-Surface Euler Equations

On the Finite-Time Splash and Splat Singularities for the 3-D Free-Surface Euler Equations

We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time “splash” (or “splat”) singularity first introduced in Castro et al. (Splash singularity for water waves, http://arxiv.org/abs/1106.2120v2, 2011), wherein the evolving 2-D hypersurface, the moving boundary of the fluid domain, self-inte...

On the Finite-time Splash & Splat Singularities for the 3-d Free-surface Euler Equations

We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time “splash” (or “splat”) singularity first introduced in [9], wherein the evolving 2-D hypersurface, the moving boundary of the fluid domain, self-intersects at a point (or on surface). Such singularities can occur when the crest of a bre...

A Note on Singularities of the 3-d Euler Equation

In this paper, we consider analytic initial conditions with nite energy, whose complex spatial continuation is a superposition of a smooth background ow and a singular eld. Through explicit calculation in the complex plane, we show that under some assumptions, the solution to the 3-D Euler equation ceases to be analytic in the real domain in nite time. This research was partly supported by the ...

Finite-time Euler singularities: A Lagrangian perspective

Well-Posedness of the Free-Boundary Compressible 3-D Euler Equations with Surface Tension and the Zero Surface Tension Limit

We prove that the 3-D compressible Euler equations with surface tension along the moving free-boundary are well-posed. Specifically, we consider isentropic dynamics and consider an equation of state, modeling a liquid, given by Courant and Friedrichs [8] as p(ρ) = αρ − β for consants γ > 1 and α, β > 0. The analysis is made difficult by two competing nonlinearities associated with the potential...