On a transport equation with nonlocal drift

In \cite{CordobaCordobaFontelos05}, C'ordoba, C'ordoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions $${\mathrm{\partial }}_t\theta +u{\mathrm{\partial }}_x\theta =0,u=H\theta ,$$ where $H$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible H"older regularization effects of this equation and its consequences to the equation with diffusion $${\mathrm{\partial }}_t\theta +u{\mathrm{\partial }}_x\theta +{\mathrm{\Lambda }}^{\gamma }\theta =0,u=H\theta ,$$ where $\mathrm{\Lambda }=(-\mathrm{\Delta }{)}^{1\mathrm{/}2}$, and $1\mathrm{/}2\le \gamma <1$. Our results also apply to the model with velocity field $u={\mathrm{\Lambda }}^{s}H\theta$, where $s \in (-1,1)$. We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the H"older class in $C^{(s+1)\mathrm{/}2}$, for all positive time.