  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.
