Nonexistence of self-similar singularities for the 3D incompressible Euler equations

Abstract

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in ${\mathbb{R}}^n$. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.

This version refines the previous one by relaxing the condition of compact support for the vorticity

References in corpus (1)

• On the Collapse of Tubes Carried by 3D Incompressible Flows

Cited by in corpus (24)

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• Liouville-type theorems for the forced Euler equations and the Navier-Stokes equations
• On formation of a locally self-similar collapse in the incompressible Euler equations
• Incompressible Euler Equations: the blow-up problem and related results
• Renormalization and universality of blowup in hydrodynamic flows
• Fluid dynamics on logarithmic lattices
• Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation
• On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations
• Self-similar Singularity of a 1D Model for the 3D Axisymmetric Euler Equations
• On discretely self-similar solutions of the Euler equations
• On the uniqueness of solution to the steady Euler equations with perturbations
• Renormalization group formalism for incompressible Euler equations and the blowup problem
• On the locally self-similar singular solutions for the incompressible Euler equations
• Discretely self-similar singular solutions for the incompressible Euler equations
• Continuation of the zero set for discretely self-similar solutions to the Euler equations
• On the behaviors of solution near possible blow-up time in the incompressible Euler and related equations
• Nonexistence of self-similar singularities in the ideal magnetohydrodynamics
• Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity
• On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations
• On the apparition of singularities of vector fields transported by volume preserving diffeomorphisms
• Remarks on the asymptotically discretely self-similar solutions of the Navier-Stokes and the Euler equations
• Global regularity for the 3D Navier-Stokes and the 3D Euler equations
• Leray self-similarity equations in fluid dynamics
• Energy concentrations and Type I blow-up for the 3D Euler equations