Renormalization in hydrodynamic models

The objective of the project is to study applications of the renormalization theory in several hydrodynamic models, specifically, in relation to the existence of self-similar finite-time blow up solutions.The issue of existence of finite-time blow ups in the incompressible Euler equations is an outstanding open problem in modern PDE´s.

Many of the features of the Euler vortex dynamics are captured by simpler models, such as the 2D quasi-geostrophic equation, and its further 1D reductions.

We propose an exploration of singularity formations in such models through renormalization, which can be viewed either dynamically, as a study of successively small-scale vortex evolution at times approaching the blow up time, or, analytically, as a construction of ``renormalization fixed points´´ in appropriate functional spaces which result in blow up solutions.Successful renormalization theories have been developed for several one- and two-dimensional dynamical systems.

In those theories, renormalization is a method to investigate a transition from regular to chaotic behaviour, self-similar geometry and existence of attractors. The renormalization approach has been recently successfully used in certain hydrodynamic models by D. Li, Ya. Sinai, and, independently, by A.

Luque and the applicant.

We are optimistic that similar techniques will be effective in vortex formation models described in this proposal such as the 2D quasi-geostrophic equations.