Renormalization and universality of blowup in hydrodynamic flows

TL;DR

This paper investigates self-similar solutions in turbulence models and their connection to blowup phenomena in hydrodynamic equations, revealing a universal wave profile and potential applications for detecting singularities in incompressible flows.

Contribution

It demonstrates the universality of wave profiles near blowup in hydrodynamic models and links turbulence intermittency to shock formation through renormalization group concepts.

Findings

Self-similar solutions closely match shock formation in Burgers equation.

Universal wave profile independent of initial conditions.

Potential for applying theory to detect blowup in incompressible flows.

Abstract

We consider self-similar solutions describing intermittent bursts in shell models of turbulence, and study their relationship with blowup phenomena in continuous hydrodynamic models. First, we show that these solutions are very close to self-similar solution for the Fourier transformed inviscid Burgers equation corresponding to shock formation from smooth initial data. Then, the result is generalized to hyperbolic conservation laws in one space dimension describing compressible flows. It is shown that the renormalized wave profile tends to a universal function, which is independent both of initial conditions and of a specific form of the conservation law. This phenomenon can be viewed as a new manifestation of the renormalization group theory. Finally, we discuss possibilities for application of the developed theory for detecting and describing a blowup in incompressible flows.