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

The paper studies the 1D quasi‑geostrophic model ∂t u − (Hu·u)x = uxx with F[Hu] = −i sgn(y)F[u] (equation (2) and Fourier identity), derives a self-similar Fourier ansatz v(y,t)=ψ(y√(T−t)) (choosing c=γ−2=0 for γ=2), and reduces the profile equation to the Riccati ODE ϑ̂′(s) = −ϑ̂(s)^2 − (s/2)ϑ̂(s) + ϑ(0)/2 (their (77)–(78)) before solving it via confluent hypergeometric functions. This leads to u(x,t) = (T−t)^{−1/2}F^{-1}[ψ](x (T−t)^{−1/2}) and blow-up rates E[u](t) ~ (T−t)^{−1/2}, Ω[u](t) ~ (T−t)^{−3/2} as t↑T (their (28)–(29)). While the Main Theorem as printed contains a prefactor (T−t)^{+1/2}, that contradicts the stated blow-up and is evidently a typographical sign error, corrected by the derivation in Section 2 (self-similar formula and energy/enstrophy scalings) . By contrast, the candidate solution reproduces the Riccati reduction but asserts a physical‑space prefactor (T−t)^{−1} and inverted blow‑up exponents (E ~ (T−t)^{−3/2}, Ω ~ (T−t)^{−1/2}), which conflict with both the paper’s consistent scaling and the Fourier ansatz (constant amplitude in v implies a (T−t)^{−1/2} prefactor in u, not (T−t)^{−1}). Hence, the paper’s argument (modulo a typo in the theorem statement and a likely minor notational slip in the enstrophy definition) is essentially correct, while the model’s scaling and blow‑up rates are not.