Renormalization and existence of the finite-time blow up solutions for a one-dimensional analogue of the Navier-Stokes equations

TL;DR

This paper investigates finite-time blow up solutions for a one-dimensional analogue of the Navier-Stokes equations, using renormalization techniques to establish the existence of new solutions with unbounded energy and enstrophy.

Contribution

It proves the existence of a family of renormalization fixed points and constructs real smooth solutions exhibiting finite-time blow up, extending prior complex-valued results.

Findings

Existence of renormalization fixed points for the quasi-geostrophic equation.

Construction of real-valued solutions with finite-time blow up.

Unbounded energy and enstrophy in the constructed solutions.

Abstract

The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier-Stokes equations. In their work Li and Sinai have proposed a renormalization approach to the problem of existence of finite-time blow up solutions of this equation. In this setting, existence of finite time blow ups is a consequence of existence of a fixed point for a certain renormalization operator on an appropriate functional space. They have provided a proof of existence of complex-valued finite time blow up solutions of the quasi-geostrophic equation. In this paper we revisit the renormalization problem for the quasi-geostrophic blow ups, prove existence of a family of renormalization fixed points, and deduce existence of real $C^\infty([0,T),C^\infty(\mathbb{R}) \cap L^2(\mathbb{R}))$ solutions to the quasi-geostrophic equation whose energy and enstrophy become unbounded in finite time, different from those found in the previous work of Li and Sinai.