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

2019/06/06 by Diego Cordoba, Diego Córdoba, Cordoba, Diego +4
Earth and Planetary Sciences · Mathematics · #Analysis of PDEs (math.AP) #Aquatic and Environmental Studies #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #math.AP

paper · pdf · doi:10.48550/arxiv.1906.02612

31 pages

openalex publication_date 2019/06/06 · arxiv created 2021/03/24 · arxiv updated 2021/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

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