Finite time singularities for the free boundary incompressible Euler equations

2011/12/09 by Ángel Castro, Angel Castro, Castro, Angel +8 · 11 citations
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Aquatic and Environmental Studies #Backward Euler method #Boundary (topology) #Boundary value problem #Compressibility #Euler equations #Euler's formula #Gravitational singularity #Mathematical analysis #Mathematics #Mechanics #Navier-Stokes equation solutions #Physics #Semi-implicit Euler method #Singularity #Smoothness #math-ph #math.AP #math.MP #physics.flu-dyn

paper · pdf · doi:10.4007/annals.2013.178.3.6

published in Annals of Mathematics 178(3), 1061-1134 (Princeton University) · 70 pages, 9 figures. Minor revisions

arxiv created 2012/09/29 · arxiv updated 2012/10/02 · openalex publication_date 2013/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem) for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity.

Citations

• Well-posedness for the motion of an incompressible liquid with free surface boundary
• Global solutions for the gravity water waves equation in dimension 3

Cited by

• Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary
• On a class of self-similar 2D surface water waves
• Self-Similar Solutions to the Hele-Shaw Problem with Surface Tension
• Exponential Vorticity Hessian Growth in Capillary Liquid Drop in Two Dimensions
• Long-time dynamics for the Kelvin-Helmholtz equations close to circular vortex sheets
• Recent advances on the global regularity for irrotational water waves. [europepmc]
• Long-Time Dynamics for the Kelvin-Helmholtz Equations Close to Circular Vortex Sheets. [europepmc]

Related

• Splash singularities for the free boundary Navier-Stokes equations
• Formulation of finite-time singularity for free-surface Euler equations
• On the splash singularity for the free-surface of a Navier-Stokes fluid
• Finite time singularity formation for moving interface Euler equations
• On the existence of stationary splash singularities for the Euler equations