Singularity formation in 3D Euler equations with smooth initial data and boundary
成果类型：
Article
署名作者：
Chen, Jiajie; Hou, Thomas Y.
署名单位：
New York University; California Institute of Technology
刊物名称：
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
ISSN/ISSBN：
0027-11459
DOI：
10.1073/pnas.2500940122
发表日期：
2025-06-27
关键词：
finite-time singularity
2d boussinesq
blow-up
implosion
摘要：
A long-standing fundamental open problem in mathematical fluid dynamics and nonlinear partial differential equations is to determine whether solutions of the 3D incompressible Euler equations can develop a finite-time singularity from smooth, finite-energy initial data. Leonhard Euler introduced these equations in 1757 [L. Euler, Memoires de lA'cademie des Sci. de Berlin 11, 274-315 (1757).], and they are closely linked to the Navier-Stokes equations and turbulence. While the general singularity formation problem remains unresolved, we review a recent computer-assisted proof of finite-time, nearly self-similar blowup for the 2D Boussinesq and 3D axisymmetric Euler equations in a smooth bounded domain with smooth initial data. The proof introduces a framework for (nearly) self-similar blowup, demonstrating the nonlinear stability of an approximate self-similar profile constructed numerically via the dynamical rescaling formulation. Significance The problem of singularity formation of the 3D incompressible Euler equations with smooth initial data has remained open since Leonhard Euler introduced the equations in 1757. Beyond its importance in partial differential equations (PDE) analysis, the problem has drawn interest from physicists and engineers due to the potential impact of singularities on fluid flow modeling. This paper provides a rigorous solution of the problem for flows in a domain with a smooth boundary. It introduces a framework for studying nearly self-similar blowup, advancing our understanding of singularity formation not only in fluid mechanics but also in broader classes of nonlinear PDEs.