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Updated: Sep 17, 2025

Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp
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Singularity formation in 3D Euler equations with smooth initial data and boundary.

Jiajie Chen1, Thomas Y Hou2

  • 1Courant Institute of Mathematical Sciences, New York University, New York, NY 10012.

Proceedings of the National Academy of Sciences of the United States of America
|June 27, 2025
PubMed
Summary
This summary is machine-generated.

Researchers reviewed a computer-assisted proof showing finite-time singularity blowup in the 2D Boussinesq and 3D Euler equations. This addresses a fundamental problem in mathematical fluid dynamics and nonlinear partial differential equations.

Keywords:
Euler equationscomputer-assisted prooffluid dynamicssingularity formation

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Area of Science:

  • Mathematical fluid dynamics
  • Nonlinear partial differential equations
  • Computational mathematics

Background:

  • The 3D incompressible Euler equations, introduced by Leonhard Euler in 1757, are fundamental to fluid dynamics and turbulence.
  • A key unsolved problem is whether smooth solutions can develop finite-time singularities.
  • These equations are closely related to the Navier-Stokes equations.

Purpose of the Study:

  • To review a recent computer-assisted proof concerning finite-time singularity formation.
  • To investigate singularity development in the 2D Boussinesq and 3D axisymmetric Euler equations.
  • To analyze the nonlinear stability of approximate self-similar blowup profiles.

Main Methods:

  • Computer-assisted proof techniques.
  • Dynamical rescaling formulation for analyzing blowup.
  • Numerical construction of approximate self-similar profiles.
  • Analysis of nonlinear stability.

Main Results:

  • Demonstration of finite-time, nearly self-similar blowup for specific Euler and Boussinesq equations.
  • Establishment of a framework for analyzing (nearly) self-similar blowup.
  • Confirmation of the nonlinear stability of numerically constructed approximate self-similar profiles.
  • The general singularity formation problem for the 3D Euler equations remains open.

Conclusions:

  • The study provides significant progress on the long-standing problem of singularity formation in fluid dynamics.
  • The computer-assisted proof offers a new framework and demonstrates nonlinear stability for approximate self-similar blowup.
  • While a complete solution for the 3D Euler equations is not presented, the findings offer insights into potential singularity mechanisms.