Singularity formation in 3D Euler equations with smooth initial data and boundary.
basic_science · Level V
Where this comes from
- Record sourced from PubMed, PMID 40577113.
- Also identified by DOI 10.1073/pnas.2500940122 and PMC identifier 12260595.
- Licence recorded as CC BY.
- The licence permits redistribution, so the abstract is shown in full and the full text is available from the publisher.
Abstract
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, <i>Mémoires de l'Académie des Sci. de Berlin</i> <b>11</b>, 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.