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arXiv · 1906.02612

Self-intersecting interfaces for stationary solutions of the two-fluid Euler equations

Diego Cordoba·Alberto Enciso·Nastasia Grubic
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Abstract

We prove that there are stationary solutions to the 2D incompressible free boundary Euler equations with two fluids, possibly with a small gravity constant, that feature a splash singularity. More precisely, in the solutions we construct the interface is a $C^{2,α}$ smooth curve that intersects itself at one point, and the vorticity density on the interface is of class $C^α$. The proof consists in perturbing Crapper's family of formal stationary solutions with one fluid, so the crux is to introduce a small but positive second-fluid density. To do so, we use a novel set of weighted estimates for self-intersecting interfaces that squeeze an incompressible fluid. These estimates will also be applied to interface evolution problems in a forthcoming paper.

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Diego Cordoba, Alberto Enciso, Nastasia Grubic. 2021-03-24. Self-intersecting interfaces for stationary solutions of the two-fluid Euler equations. https://arxiv.org/abs/1906.02612

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