On a transport equation with nonlocal drift

Abstract

In \cite{CordobaCordobaFontelos05}, 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 \[ \partial_t θ+u \; \partial_x θ= 0, \qquad u = H θ, \] 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 \[ \partial_t θ+ u \; \partial_x θ+ Λ^γθ= 0, \qquad u = H θ, \] where $Λ= (-Δ)^{1/2}$, and $1/2 \leq γ<1$. Our results also apply to the model with velocity field $u = Λ^s H θ$, where $s \in (-1,1)$. 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.