Renormalization group formalism for incompressible Euler equations and the blowup problem

Details Renormalization group formalism for incompressible Euler equations and the blowup problem Renormalization group formalism for incompressible Euler equations and the blowup problem

2012-03-15

arxiv.org/abs/1203.3348v2

Physics

Fluid Dynamics

15 pages, 5 figures

Scientific paper

The paper develops the renormalization group (RG) theory for 3D incompressible Euler equations. It describes possible types of singularities developing in finite (blowup) or infinite time from smooth initial conditions of finite energy. In this theory, the time evolution is substituted by the equivalent evolution for renormalized solutions governed by the RG equations. Fixed points of the RG equations correspond to self-similar singular solutions, which describe universal asymptotic form of singularities. Renormalization schemes with single and multiple spatial scales are developed. 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 provides strong evidence in favor of a multiple-scale self-similar asymptotic solution predicted by the RG theory. This solution describes a singularity developing exponentially in infinite time.

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