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The one-dimensional quasi-geostrophic equation is the one-dimensionalFourier-space analogue of the famous Navier-Stokes equations. In their work Liand Sinai have proposed a renormalization approach to the problem of existenceof finite-time blow up solutions of this equation. In this setting, existenceof finite time blow ups is a consequence of existence of a fixed point for acertain renormalization operator on an appropriate functional space. They haveprovided a proof of existence of complex-valued finite time blow up solutionsof the quasi-geostrophic equation.  In this paper we revisit the renormalization problem for thequasi-geostrophic blow ups, prove existence of a family of renormalizationfixed 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 energyand enstrophy become unbounded in finite time, different from those found inthe previous work of Li and Sinai.

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