On a transport equation with nonlocal drift

In 2005, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions ∂ t θ + u ∂ x θ = 0 , u = H θ , where H is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion ∂ t θ + u ∂ x θ + Λ γ θ = 0 , u = H θ , where Λ = (−Δ)1/2, and 1/2 ≤ γ