On a transport equation with nonlocal drift

In [16], Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocaldrift variant of the 1D Burgers equation does not have global classical solutions ∂tθ + u ∂xθ = 0, u = Hθ, 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 ∂tθ + u ∂xθ + Λ θ = 0, u = Hθ, where Λ = (−∆), and 1/2 ≤ γ < 1. Our results also apply to the model with velocity field u = ΛHθ, where s ∈ (−1, 1). We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in C, for all positive time. August 5, 2014.