Renormalization group formalism for incompressible Euler equations and the blowup problem

TL;DR

This paper extends the renormalization group formalism to analyze potential singularities in 3D incompressible Euler equations, providing a universal framework for understanding blowup phenomena and comparing theoretical predictions with numerical simulations.

Contribution

It develops multi-scale RG schemes for power law and exponential scaling, offering a new approach to interpret singularity formation in Euler equations.

Findings

Supports the conjecture of exponential-in-time singularity development.

Shows the RG fixed point describes universal self-similar singularities.

Aligns RG theory predictions with numerical simulation results.

Abstract

The paper discusses extensions of the renormalization group (RG) formalism for 3D incompressible Euler equations, which can be used for describing singularities developing in finite (blowup) or infinite time from smooth initial conditions of finite energy. In this theory, time evolution is substituted by the equivalent evolution for renormalized solutions governed by the RG equations. A fixed point attractor of the RG equations, if it exists, describes universal self-similar form of observable singularities. This universality provides a constructive criterion for interpreting results of numerical experiments. In this paper, renormalization schemes with multiple spatial scales are developed for the cases of power law and exponential scaling. The results are compared with the numerical simulations of a singularity in incompressible Euler equations obtained by Hou and Li (2006) and Grafke et al. (2008). The comparison supports the conjecture of a singularity developing exponentially in infinite time and described by a multiple-scale self-similar asymptotic solution predicted by the RG theory.