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 (Formula Presented) 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 (Formula Presented) where Λ = (-Λ)^½, and ½ ≤ γ < 1. Our results also apply to the model with velocity field u = Λ^s Hθ, where s ∈ (−1, 1). We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in C^(s+1)/2, for all positive time.