  In this paper, we consider a 1D periodic transport equation with nonlocal
flux and fractional dissipation $$
u_{t}-(Hu)_{x}u_{x}+\kappa\Lambda^{\alpha}u=0,\quad (t,x)\in R^{+}\times S, $$
where $\kappa\geq0$, $0<\alpha\leq1$ and $S=[-\pi,\pi]$. We first establish the
local-in-time well-posedness for this transport equation in $H^{3}(S)$. In the
case of $\kappa=0$, we deduce that the solution, starting from the smooth and
odd initial data, will develop into singularity in finite time. If adding a
weak dissipation term $\kappa\Lambda^{\alpha}u$, we also prove that the finite
time blowup would occur.
