1203.3348 (Alexei A. Mailybaev)

Renormalization group formalism for incompressible Euler equations and
the blowup problem    [PDF]

Alexei A. Mailybaev

The paper develops the renormalization group (RG) theory for compressible and incompressible inviscid flows, which describes universal scaling 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 given by the RG equations with increasing scaling parameter. Stationary states of the RG equations correspond to self-similar singular solutions. If such a stationary state is an attractor, the corresponding self-similar solution describes universal asymptotic form of a singularity for generic initial conditions. First, we consider the inviscid Burgers equation, where the complete RG analysis is carried out. We prove that the shock formation is described asymptotically by the universal self-similar solution. Then the RG formalism is extended to incompressible Euler equations. Renormalization schemes with single and multiple spatial scales are developed, describing possible asymptotic forms of self-similar singular solutions. These 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 the multiple-scale self-similar asymptotic solution predicted by the RG theory. This solution describes an infinite time singularity developing exponentially in time.

View original: http://arxiv.org/abs/1203.3348